K-Theory and stable algebra

H. Bass

doi:10.1007/BF02684689

K-Theory and stable algebra

Bass, H. 2007-08-06 00:00:00 by H. BASS CONTENTS Px~zs INTRODUCTION ................................................................................. 5 CHAPTER [, ---. Stable Structure of the linear groups ......................................... 8 w i. Notation and lemmas .............................................................. 8 w 2. The affine group .................................................................. Ix w 3. Structure of GL(A) ................................................................ 13 w 4. Structure in the stable range ....................................................... 14 w 5- Dimension o ...................................................................... 17 CIIAPTER II. -- Stable structure of projective modules ...................................... 19 w 6. Semi-local rings ................................................................... i9 w 7. Delocalization. ]'he maximal spectrum .............................................. 21 w 8. Serre's theorem .................................................................... 23 w 9. Cancellation ........................................................................ 25 w io. Stable isomorphism type ............................................................. 28 w I I. A stable range for GL(A), and a conjecture ......................................... 29 CHAP'rER III. --- The functors K ............................................................ 3~ w x2. K~ and KI(A,q) ................................................................ 3 I w 13. The exact sequence ................................................................ 33 w 14. Algebras .......................................................................... 37 w z 5. A filtration on K ~ ................................................................. 38 CIIAPTER IV. -- Applications ................................................................ 42 w i6. Multiplicafive inverses. Dedekind rings .............................................. 4~ w 17. Some remarks on algebras .......................................................... w 18. Finite generation of K ............................................................. w 19. A finitene.ss theorem for SL(n, A) ................................................... w 2o. Groups of simple homotopy types ................................................... w 2i. Subgroups of finite index in SL(n, A) ................................................ w 22. Some remarks on polynomial rings .................................................. 58 INTRODUCTION A vector space can be viewed, according to one's predilections, either as a module over a ring, or as a vector bundle over a space (with one point). If one seeks broad generalizations of the structure theorems of classical linear algebra, however, the satis- faction afforded by the topologists has, unhappily, no algebraic counterpart. 489 6 H. BASS Our point of departure here is the observation that the topological version of linear algebra is a particular case of the algebraic one. Specifically, the study of continuous real vector bundles over a compact X, for example, is equivalent to the study of finitely generated projective C(X)-modules, C(X) being the ring of continuous real functions on X. This connection was first pointed out by Serre [28] in algebraic geometry, and recently in the above form by Swan [34]. Serre even translated a theorem from bundle theory into pure algebra, and he invented the techniques to prove it [29] (see w 8). His example made it clear how to translate large portions of homotopy theory into the same setting, and thus discover, if not prove, an abundance of natural theorems of topo- logical origin. What follows is the result of a first systematic attempt to exploit this idea. This investigation was inaugurated jointly by S. Schanuel and the author, and an announcement of the results of that earlier work was made in [9]- In particular, the topological background for the results of w167 1o-xI is pointed out there. Generally speaking the method is as follows. The problem at hand is " locally trivial ", i.e. locally it can be solved by a simple parody of classical linear algebra. Furthermore, one proves an " approximation " lemma which asserts the existence of global data with prescribed behavior in a given local situation. Finally, in order to piece these together one uses a kind of " general position " argument. The ability to put things in general position imposes a dimensional restriction on the conclusions, and thus we determine a " stable range " for the problem. The structures investigated here are those of projective modules (Chapter II) and of the general linear group (Chapter I). This can be thought of as analogous to the study of vector bundles on a space, and on its suspension, respectively. If we " stabilize ", we are led to consider analogues of the functors K ~ and K 1, respectively, of Atiyah and Hirzebruch [2], and a good deal of the formalism, in particular the exact sequence, of that analogy is developed in Chapter III. The natural extension of the functors K i, i>2, to our algebraic context has so far evaded a definitive appearance. In another direction, this point of view should be ti'uitful in studying other classical linear groups. For example, thinking of bundles with reduced structure group, one can consider non degenerate quadratic forms on projective modules, and the associated orthogonal groups. K ~ would then be the " Witt ring ", and Witt's theorem, for example, is analogous to our Theorem 9-3- The local part of this theory has recently been done by Klingenberg [25], but several serious problems continue to obstruct its globalization. The most important applications of our machinery are to the linear groups over orders in semi-simple algebras finite dimensional over the rationals (w167 19-21 ). When applied to Z,~, ~z a finite group, they give quantitative information onJ. H. C. Whitehead's groups of simple homotopy types [36], results which elaborate on some earlier work of G. Higman [38]. They also shed some new light on the structure of SL(n, A), with A the ring of integers in a number field. Most striking is the fact that, for n suitably large, ifH is a non central normal subgroup, then SL(n, A)/H is afinite central extension 490 K-THEORY AND STABLE ALGEBRA of PSL(n, A/q) for some ideal q. This information is the starting point for the proof, in [4o], that every subgroup of finite index in SL(n, Z), n~3 , contains a congruence subgroup (see w 2I). The author owes most of his mathematics to Scrre's genius for asking the right question at the right time, and he records here his gratitude for having so profited. He is particularly thankful also to A. Heller, who has endured long audience on the present work, and who is responsible for numerous improvements in its exposition. 491 CI~APTER I STABLE STRUCTURE OF THE LINEAR GROUPS w I. Notation and lemmas. The objectives of this chapter are Theorems 3-x and 4.2 below, which purport to describe all the normal subgroups of the general linear group. We begin in this section by establishing notation and some general trivialities on matrices. Let A be a ring. GL(n, A) is the group of invertible n by n matrices over A. Let ei~ denote the matrix with i in the (i, j)th coordinate, and zeroes elsewhere; we recall that eif~=Sjkeo,. Let a, b~A and i~j. Then (I +ae~i)(i +be#)= I + (a-:-b)eii; here i -- x, denotes the n by n identity matrix. Thus, for i and j fixed, the matrices x 4_ aei, form a subgroup of GL(n, A) isomorphic to the additive group of A. A matrix of the form I -t- aeij, i +j, is called elementary, and we denote by E(n, A) the subgroup of GL(n, A) generated by all elementary matrices. Now let q (possibly -= A) be a two sided ideal in A. The q-congruence group is GL(n, A, q) ----ker(GL(n, A) -,~ GL(n, A/q)). Moreover we denote by E(n, A, q) the normal subgroup of E(n, A) generated by all elementary matrices in GL(n, A, q). We shall identify GL(n, A) with a subgroup of GL(n_m, A) by identifying ~.eGL(n,a) with ]-~o ~ m1 ~GL(n+m,A). This done, we set GL(A, q) :-[,J,GL(n, A, q) and E(A, q) ---- O,E(n, A, q). When q--A we write GL(A):=:GL(A,A) and E(A)=E(A,A). GL(A) is called the stable general linear group over A. Lemma (x.x) (" Homotopy Extension "). -- If A~B is a surjective ring homo- morphism, then E(n, A, q) -*E(n, B, qB) is surjective for all n and q. Proof. -- If ] +be# is B-elementary, and b is the image of aeA, then x +aet, maps onto I T beii. (There is an abuse of notation here which one should excuse, in that " I " and " e~i " have two senses.) This shows that E(n, A)-+E(n, B) is surjective. Now E(n, B, qB) is generated by elements -~ : ~-lv~ with ~eE(n, B) and -:. qB-elcmentary;i.e, z: i +qeij , qeqB. Wecanfind q'+_-q with image q, and ~'eE(n, A) with image ~ (by the first part of the proof). Clearly (i + q'eii ) ~ lifts v ~ If H 1 and H 2 are subgroups of a group we denote by [HI, H2] the subgroup generated by all [hl, he] -=hl-~h.,-lhlh2, with hieH i. 492 K-THEORY AND STABLE ALGEBRA Lemma (x.2). -- If i, j, and k are distinct, then [i --aeii , I --bejk ] = I -~- ab%. Pro@ -- (I § a@ (I "@ bej,~) (I -- a@ (I -- b%) = ( I + a% § beik + ab%) ( I -- a%-- beik + ab%) = ( I + ae~i § beik + ab%) -- (a@ -- (beik § ab%) -t- (ab%) = I -t- ab%. Corollary (x.3). -- If q and q" are ideals and n> 3, then E(n, A, qq') c[E(n, A, q), E(n, A, q')]. In particular, E(n, A, q) ---- [E(n, A), E(n, A, q)]. Pro@ -- The right side is a normal subgroup of E(n, A) which, by (I. 2), contains all qq'-elementary matrices. Corollary (x.4). -- Suppose n>_3, and let H be a subgroup of 13L(n, A) normalized by E(n, A). /f T is a family of elementary matrices in H, then H~E(n, A, q), where q is the two-sided ideal generated by the coordinates of I--'~ for all -~eT. Proof. -- If z = I -/- cehzeT then, since n > 3, we can commutate with other elementary matrices and obtain all elementary matrices of the form i + acb%. The E(n, A)-invariant subgroup generated by these, as z varies in T, is clearly E(n, A, q). Corollary (I.5). -- (i) E(n, A) = [E(n, A), E(n, A)] for n>3. (ii) If A is finitely generated as a Z-module, then E(n, A) is a finitely generated group for all n. (iii) If A is finitely generated as a Z-algebra, then E(n, A) is a finitely generated group for all n> 3. Proof. -- (i) is immediate from (I.3) , setting q~A=q'. (ii) Is obvious, since E(n, A) is generated by a finite number of subgroups, each isomorphic to the additive group of A. (iii) Let ao----I, al,..., a r generate A as a ring, and consider the elementary matrices I +a~eik , o<i<r, and all j+k. These generate a group which, by (1.2), contains all i + Mejk , where M is a monomial in the a i. Since A is additively generated by these M we catch all elementary matrices. In special cases we get something for n = 2. Lemme (x.6). -- Suppose i =u + v with u and v units in A. Then F.(2, A, q) A), A, q)]. If, further, u ---- w ~ with wecenter A, then E(2, A, q) = [E(2, A), E(2, A, q)]. Pro@ -- Given qeq, set b = v-lq, ~ = ;: and ~ = eE(e, A, q). Then [e'~]= ; (I~IU)b =/: Iqi e[GL(2'A)'I~'(2'A'q)]" M~176 by Lemma I. 7 below, and [w-le,~]----[e,~]. W--I(x ~ 0 W -1 493 IO H. BASS Examples. -- i. If A contains atield with more than two elements, I is a sum of two units. Likewise if 2 is a unit in A. 2. If A is an integral domain containing a primitive n th root, u, of i, then I+U+...-ku i is a unit if i+I is relatively prime to n. Since I +u+...+d=I-t-u(I-t-...-kd -~) we can write i as a sum of two units provided both i and i+I are prime ton. Such an i always exists unless n is a power of two. 3. The commutator quotient of GL(2, Z) is a group of type (2, 2), and that of SL(2, Z) is cyclic of order 12. More generally, the commutator quotient of SL(2, Z/qZ) is cyclic of order d, where d= gcd(q, 12). IfZp denotes the p-adie integers then the commutator quotient of $L(2, Zp) is cyclic of order 4 for p = 2, 3 for p = 3, and I for p4=2 or 3. In all of these examples E=SL. 4. For any ring A I is a sum of two units in M(n,A) for all n>I. For if n>I let B=A[~] =A[t]/(f(t)), where f(t) =t" -t-~-I. Then I =0~(I--~X n-I) =~(I -e)g(0~) SO ~ and I--0~ are units in B. Therefore they define A-automorphisms of B_~_-A " whose sum is I. The next lemma plays a fundamental role in what follows. Lemma (I "7) (" Whitehead Lemma "). -- Let aeGL(n, A) and bEGL(n, A, q). Then ab o I a o I ba o I mod E(2n, A, q) _= --1 o I ~o b[ 'o I and =1o a' mod E(2n, A) -[--b o: (The congruences hold for either left or right cosets.) Let Proof. - - Write b:-I-[ q; q is an n� matrix with coordinates in q. !ba oJ ~z [3= ao ~ "rj = I (ba)-lqi, "r2= I --a--lq :0 I '~ 0 I 0 I ~ I I I I o o and , = ~. j--b-lqa I ja I, Then clearly ": = vl~vaeE(2n, A, q). We begin by showing that ~z = ~. ]ba q 5"71 o:.1 _1= baTqa q'= a q[ i ,--a I ,i' ~0 I[ ~ __q+ql ]a o , i(l 0r 1"72 = ]--a q= 121 - ,--a b: o! 'a o b ; ~'71~- Ix2(3" ~ ~a- a b = !qa o:~ = ~. 494 K-THEORY AND STABLE ALGEBRA ib -1 ol Taking a-----b-t we have io b l ~E(2n, A, q) and hence lab o,=iab o,[b 1 oi__la__ o! rood E(2n, A, q). O I' -o I~,O b' !o b' Finally, modulo E(2n, A), we have "a o; 'a olii iill o'!I r ! ' o a ! Io b ,o b, o I " -I I, 0 I; ,-- b o. ' -=' '! I', " '= " For left cosets we need only observe that all subgroups involved are invariant under transposition. Corollary (x. 8). -- [GL(n, A), GL(n, A, q)] cE(2n, A, q). Remark. -- It would be useful if one could strengthen this corollary to say: [GL(n, A, q), GL(n, A, q')] cE(2n, A, qq'), say even under the assumption that q -- q' = A. Combining (1.3) and (1.8) we have, for n>3, E(n, A, q) = [E(n, A), E(n, A, q)] c[GL(n, A), GL(n, A, q)] cE(2n, A, q). Letting n-+oo we conclude: Corollary (I.9). -- E(A, q) -= [E(A), E(A, q)] -- [GL(A), GL(A, q)]. In particular, E(A) = [GL(A), GL(A)]. Remark. -- The last conclusion is due to J. H. C. Whitehead [36, w I]. Indeed, essentially, everything in this section is inspired by Whitehead's procedure in [36]. w 2. The affine group. Lemma (2. I). -- For n>2 the additive group generated by E(n, A) is the full matrix algebra, M(n, A). Proof. -- It suffices to catch a% for all a EA, all i, j. If i+j, a%= (i +ae#)--I. For the diagonal elements we have, for example, aell = ( I 4- ae12) ( i 4- e~1 ) -- I -- ael2-- e2t. Viewing An as a right A-module we can identify GL(n, A) with Aub~(A" ). Corollary (2.2). -- For n>_2 the E(n, A) invariantsubgroups of A" are the qA", where q ranges over all left ideals. (Note that these are not sub-A-modules.) Pro@ -- By the Lernma we can replace E(n, A) by M(n, A). Let = (al, ..., a,)eA" and let q=YAa~. It clearly suffices to show that i(n, A)e--qA". But the obvious use of coordinate projections, permutations, and left multiplications makes this evident. Now Aft(n, A) is defined to be tile semi-direct product GL(n, A)� If eeGL(n, A) and ~eA" the multiplication is given by = + 495 i2 H. BASS In particular = 1, __ We identify GL(n, A) with GL(n, A) � and A" with i � The latter is an abelian normal subgroup. Since o)-1= (i, we see that for subgroups HcGL(n, A) and ScA n, invariance of S under H is the same in both of its possible senses. We shall often identify (a, a) eAff(n,A) with : ~I eGL(n+ i, A) (viewing 0r as a column vector). Note that this identifies E(n, A) � A n with a subgroup of l~.(n + I, A). Proposition (2.3). -- Let H be a subgroup 0fAff(n, A) with projection L in GLn(, A). Then (i) [H, A"]=[L, A"]=Z, eL(I---:)A n. (ii) If H is normalized by An then [H, An]oH so in this case HoA" trivial=>H trivial. (iii) If n22 and H is normalized by E(n, A), then [U, An]=qA n for a unique left ideal q. Proof.- (i)If (,,~)eH, so ,eL, and a= (I, a)eA", then [(':, ~), (I, ~)] "-- (I, (I--'--l)~), (i) ~ (ii) is trivial, and (iii) is a consequence of (2.2). We shall use this proposition to show that, under suitable conditions, a subgroup HcGL(n,A) normalized by E(n,A) contains E(n,A,q) forsome q4=o. By (i.4) it will be sufficient to show that H contains a single non trivial elementary matrix. Of course we assume n> 3 in order to invoke (I.4). Suppose first that there is a eeH and a unit uecenterA, such that ~:t=u.I, but e--u. I has a zero row or column. We first apply the following: Remark. -- Since Z is a Euclidean ring l~.(n, Z) =SL(n, Z) for all n>2. Hence every " permutation " (i.e. a matrix with one non zero coordinate equal to + I in each row and column) of determinant I lies in E(n, Z). By specialization, every such permu- tation lies in E(n, A) for any A. Now back to ~ and u above. We can conjugate with a permutation in E(n, A) and assume e--u. i has either the last row or column zero; say the last row. Then ~=ue' with I4=e'eAff(n--I,A). Since commutating with e neglects u we can use (2 . 3) to produce lots of elementary matrices (in the last column). If the last column of o--u. I vanishes we transpose the above argument. Before proceeding further with our problem, let us compute the centralizer of {x= i +ae12]aeA }. Given ~, let ~ be the first column of~ and ~ the second row of ~-t. Then axa-t= I -~0r Hence, if a-:=vo we have xa~ =ae12. If a= i we conclude 496 K-THEORY AND STABLE ALGEBRA I3 that a = and ~ = (o, u- 1, o, ..., o). Then allowing general a we find uau- 1 = a so that uecenter A. Now changing e12 to eq we can already record the following: Corollary (2.4). -- If n>2, the centralizer of E(n, A) consists oj all matr#es u. I with u a unit in the center of A. Continuing our argument above, our objective is: Lemma (2.5). -- Let n>3 and H a subgroup of GL(n, A) normalized by E(n, A). If H contains a non central element ~ with some coordinate zero, then HZE(n, A, q) for some q+o. Proof. -- After conjugating a with a permutation in E(n, A) we can make a 1 or a, zero, where ~ ----- is the first column of a. If a commutes with all z = i + ae~2 then a has the same first column as u. I for some uEcenter A, by the last paragraph, and this case was handled already above. Hence we may assume a'~ can be chosen so that p=~w-l":-~:#I. Then p=v-l+~-t ", where y~a~,: -1, and ~ is the second row of e-a (see computation above). Case I. a,=o. Then ~y has zero last row, so p=z-l+0~y and ,~-t have the same last row. Hence we have 1 4: peArl(n-- I, A) and we can use (2.3) to get elementary matrices. Case 2. a, = o. Then the first rows of p and z-1 agree, so p is not central and has an off diagonal zero (since v -1 has first row (I,--a, o, ..., o)). Hence we can replace , by p and obtain Case i again. w 3- Structure of GL(A). Theorem (3. I) (Stable Structure Theorem.) Let A be any ring and GL(A) the stable general linear group over A (see w I for definition). a) For all two-sided ideals q, E(A, q) = [E(A), E(A, q)] ---- [GL(A), GL(A, q)]. b) A subgroup HcGL(A) is normalized by E(A)c:>for some (necessarily unique) q, E(A, q)cHcGL(A, q). H is then automatically normal in GL(A). c) If A--->B is a surjective ring homomorphism and if H is normal in GL(A), the image of H is normal in GL(B). (Note that GL(A)-+GL(B) need not be surjective.) Proof. -- a) is just (I.9). c) is a consequence of b) since the image of H will be normalized by the image of E(A), which, by (i. i), is E(B). Moreover, the uniqueness 497 ~4 H. BASS of q and normality of H in part b) both follows from [GL(A), H] =E(A, q)oH, which is a consequence of a). It remains to prove that if HcGL(A) is normalized by E(A), and if q is the ideal generated by the coordinates of i--z for all z~H, then E(A, q) cH. Let H,,=HnGL(n, A) and let qn be the ideal generated by all coordinates of i --v, ":ell,,. Viewing H,,cAff(n-f- I, A) it follows from (2.3) and (I-4) that H,, .-1 contains enough elementary matrices to capture E(n+ I, A, q,). Hence HD U,E(n + i, A, %) = U,,E(A, %) =E(A, q). w 4. Structure in the stable range. One would like to recover the stable structure theorem for GL(n, A), n<~c. Fortunately a sufficient condition for this can be formulated as a very simple axiom, one which we will verify in a rather general setting in Chapter II (Corollary 6.5 and Theorem t I. i). Definition. -- Let v. = (al, 9 9 at) be an element of the right A-module A r. We call ~ unimodular (in A r) if ZiAa i=A. This is clearly equivalent to the existence of a linear functional f: Ar--->A such that f(a) -- i. Let n> i ; we say n defines a stable range forGL(A) if, for all r> n, given a. = (al, ..., at) unimodular in A r, there exist bl, ..., b r_ 1 in A such that (al+blar, ..., ar_t+b r_ lar) is unimodular in A T Examples. -- If A is a semi-local ring, then n = i defines a stable range. If A is a Dedeking ring n=2 works. If A is the coordinate ring of a d dimensional affine algebraic variety, e.g. if A is a polynomial ring in d variables over a field, then n -- d-? I defines a stable range for GL(A). Lemma (4. x). -- If n defines a stable range for GL(A) it does likewise for GL(A/q) for all ideals q. Proof.--Suppose r>n. Writing A' =A/q, suppose a' = (a~, ..., a'~) is unimodular in (A')r; say i=Zt~a~. Lift t~ and a~ to ti and ai in A, so I=Ztiai+ q with qeq. Then (at,..., at, q)eA r-'* is unimodular; so, by hypothesis, there exist qeA such that (a I + qq, . .., a, + Crq ) is unimodular. Replacing a~ by a i+ qq, then, we can assume is unimodular. Again by hypothesis there exist b~eA rendering (al -~- b~ar, 9 9 at- 1 + b~_ tar) unimodular. The images, b~, of the b i in A' now satisfy our requirements. Theorem (4.o). -- Suppose n defines a stable range for GL(A). For r>n and for all ideals q: a) The orbits of E(r, A, q) on the unimodular elements of A r are the congruence classes modulo q. In particular E(r, A) is transitive. b) GL(r, A, q) =GL(n, A, q)E(r, A, q). c) E(r, A, q) is a normal subgroup of GL(r, A). For r>max(n, 2): d) E(r, A, q) = [E(r, A), E(r, A, q)] = [E(r, A), GL(r, A, q)] for all ideals q. 498 K-THEORY AND STABLE ALGEBRA I5 e) If H r A) is normalized by E(r, A), then, for a unique ideal q, E(r, A, q) cH and the image of H in GL(r, A/q) lies in the center. For r>max(2n, 3), and for all ideals q: f) E(r, A, q) -- [GL(r, A), GL(r, A, q)]. Proof. -- a) We first show that if ~ = (al, . .., at) -- (I + ql, q2, 9 9 qr) is unimodular, with qieq, then there is a veE(r,A,q) such that v0(=(i,o, ...,o). Writing I as a left linear combination of al, ..., ar and multiplying this equation on the left by q~ = at, the coefficient ofa~ in the new equation is a right multiple, say q, of q~, and hence in q. Thus a~ is in the left ideal generated by al, 9 9 ar_l, qa~, so (al, 9 9 a~_l, qar) is unimodular. Our hypotheses now say that we can find a~ =a i+ bda,. , i <i< r--I, t , = r--1 such that (al, ..., a~_l) is unimodular. Set %'1 I @Y~i=abiqeir6E(r, A, q), and write (a~,...,a;_~)=(I+q~,...,q;_i). Then q'~q,I<i<r--I, and "~10( = (I + q;, ..., q;_~, q~). Writing I as a left linear combination of a~, ..., a' r - l, and left multiplying this equation by qt--q,, we canwrite q;--qr=Zr--lcia~ with cie q. Then if ~,r--I E % I + ~=lcle~ E(r,A,q), t p p wehave z2z~0(= (I § ..., q~-t, q~). If G= I --% then ez,,-~l~ = (i, qs ..., q~-l, q;). Let %=I--(qs247247 ; then ":a(~v~z~0(=~=(I,o, ...,o), and v3eE(r, A, q). The presence of e, which need not belong to E(r, A, q), is harmless, since e-l~=~. Hence v=e-lv3(~v2-rleE(r, A, q) solves our problem. Now for the general case. Setting q = A, the case above was already general, and we have thus shown that E(r, A) is transitive. Now let q be arbitrary, and ~-~ mod q unimodular elements ofA r. We can find ~E(r, A) so that ~ = (I, o, . .., o). Since ~0(--~ mod q the argument above provides a -:~E(r, A, q) with -:~0(=~; hence = [~. b) Given aeGL(r, A, q), the last column of a is congruent, mod q, to (o, . .., o, I). Hence, by part a) above, there is a vleE(r , A, q) such that via= 2l with aleGL(r--i, A, q), 0(sqA ~-~. Set ~:~= eE(r, A, q); then By induction we can continue reducing until reaching GL(n, A, q). c) E(r, A, q) is generated by all z~=(r-~w with ~ q-elementary and ~eE(r, A). Given eEGL(r, A) we must show ~-lv~0(sE(r, A, q). But 0(-t':"0(= (~-~z~) ~, where =0(~-'. Since (by definition) E(r, A, q) is normalized by E(r, A), it suffices to show ~-t-r~eE(r, A, q). By part b) we can write ~=5.p, with u A) and peE(r, A). c , ceGL(r-- ~, A), Again, then, it suffices to show ,(-1T'~E(r, A, q). Write "F: and ": = t +-q%, qeq. (I--1%'(;0( ~6 H. BASS Case I. iorjis--r, say i=r. Then -:-- t I teqA '-1 so y~-y-1----- I[ which is clearly in E(r, A, q). Similarly for j=r. Case 2. i,j<r. Let a be the i th column ofc and b the 9 row ofc -1. Then ba=o, being the (i,j)th coordinate of c-lc= i (recall i4:j). One sees easily that +aqb o Let ":1-~ ~E(r, A, q), and y'ry- t = I + aqb) - l ; ;b/i ~ i al (~ = eE(r, A). We conclude by showing that (-~ly~y-t)~ A, q). I --a I o I+aqb O I ('L'1"~'7"~--1)~ O I qb(I +aqb) -I I ,o I --(l I+aqb o I a] o i llqb i i[o I --all a=~ b -~a=~ b o] qb I o I I+ a I1' since ba -~ o. we have, from (I.3) , d) Using r_>3 E(r, A, q) = [E(r, A), E(r, A, q)] c[E(r, A), GL(r, A, q)]. Let T= I +aeii be one of the generators of E(r, A). Suppose, for some ~, that [-~~ GL(r, A, q)] CE(r, A, q). Then [% GL(r, A, q)] = [.~o, GL(r, A, q)O]o-,= [~:o, GL(r, A, q)]~ A, q)~ A, q). Being free thus to choose e, we can transform -~ and assume ~ = i +aeri. Now by part b), using r>n, GL(r, A, q) =GL(r--I, A, q)E(r, A, q), so it suff� to check that c o GL(r_i,A,q),andwrlte~:= ~ o. [% GL(r-- I, A, q)] cE(r, A, q). Let T = i Then j o o oI:o ; o i [-, Y] 2t O I t I . I) I This is in E(r, A, q) since c- i mod q. f) Since GL(r, A) =GL(n, A)E(r, A) and GLir, A, q) =GL(n, A, q)E(r, A, q) there remains only, by virtue of part d), to show that [GL(n, A), GL(n, A, q)] r A, q). But this follows from (1.8), since r>2n. e) We begin by showing that if H contains a non central element, ~, then HDE(r, A, q') for some q'4:o. By (2.5) it suffices to produce, in H, a non central (a:) element with at least one coordinate zero. Let 0~= be the first column of ~. I' Since r>n, our hypothesis on n permits us to add multiples of a, to the other coordinates 500 K-THEORY AND STABLE ALGEBRA and render them unimodular. This can be accomplished by conjugating ~ with a matrix ~Ir--1 ~I j elZ.(r, A), so we can reduce to the case where (al, ..., at_t) is of the form l o Ii unimodular. We can then write a r=dla x 5... f_dr _tar_l, so that if ),-- i a r-1 where d=(dl, ...,dr t), we have ~-.1~__ 9 Now in the paragraph preceding Corollary 2.4 we showed that if ~ commutes with all -: ---- I + ael~ , then ~ has only one non zero coordinate, so we can finish with (2.5). Hence we may assume there is such a z for which p =(~za-lv-l:V i. Let ~ denote the second row of e-1 and y=a~v --1. Then (just as in the proof of (2.5)) we have We now claim that O is not central. Otherwise p=U.l with uecenterA, by(2.4). With ), as above, the last row of (~,-~,)y iszero, sothat X-lp=Z-~-:-t-~-(Z-~,)y and ?,-az -x have the same last row. Since I--V -t is concentrated in the upper left 2 � 2 corner, and I --X- t in the last row, we see, using r> 3, that the (r, r)thcoordinate of X-~'~- ~ is I. But the same coordinate of uX-t is u. This shows that p central ~-,o == I, contrary to our choice of-:. Now consider X-apZ----;~-tv-tz + (Z-~)(TZ). This is a non central element of H whose last row agrees with that of X-%-tX= I --~a3, where e is the first column of X-t and 3 is the second row of X= I. "-t o . Hence 3 has only one non zero coordinate d I (using r>2), so the last row of X-~z-lX has at most two non zero coordinates. We conclude then that X-lpX has some coordinate zero, and again we finish with (~.5). For the proof of e) now, choose q maximal so that E(r, A, q) cH. Our problem is to show that the image, H', of H in GL(r, A/q) lies in the center of GL(r, A/q). Since E(r, A) normalizes H, H' is normalized by the image of E(r, A), which, by (x. ~), is E(r, A/q). Hence if H' is not central Lemma 4. I permits us to apply the first part of our argument to H' and conclude that E(r, A/q, q'/q) cH' for some q'4 = q. Then the inverse image, L=HGL(r,A, q), of H' contains E(r, A, q'). By part d) now E(r, A, q') = [E(r, A), E(r, A, q')] r A), L] c tiE(r, A), H][E(r, a), GL(r, A, q)]" = [E(r, A), H]E(r, A, q) oH, contradicting the maximality of q. w 5- Dime.slon o. Corollary 6.5 tells us that n = I defines a stable range for GL(A) when A is semi- local, and it is in precisely this case that the restrictions r> 3 intervene effectively. The following refinements can be made: Proposition (5. z ). -- Suppose n = I defines a stable range for GL(A) (e. g. A can be any semi-local ring). 3 i8 H. B A S S a) If A is commutative, then, for all ideals q and all r> 2, E(r, A, q) = SL(r, A, q)(= SL(r, A) r~GL(r, A, q)). In particular E(r, A) = SL(r, A). b) Suppose I --w" + v with w and v units in the center of A. Then, for all r> 2, E(r, A, q) ---- [E(r, A), E(r, A, q)] -- [GL(r, A), GL(r, A, q)]. Pro@ -- a) Clearly F.(r, A, q) cSL(r, A). Now an element of GL(r, A, q) is, by (4.2) b), reducible modulo E(r, A, q) to GL(I, A, q), i.e. to a unit, and that unit is evidently the determinant of the original matrix. This proves a). b) For r>3 b) is just parts d) and f) of Theorem 4.2. However, the proof there uses r>3 only to invoke (1.3) and conclude that l~.(r, A, q) ~[E(r, A), F.(r, A, q)]. Our hypothesis permits us to use (i .6) instead for the same purpose when r: 2. zks a consequence of part a) above and (i. i) we have: Corollary (5.2). -- Let q be an ideal in the commutative ring A for which A/q is semi-local. Then, ]or all q'Dq, E(r, A, q') ~ SL(r, A/q, q'/q) is subjective for all r. In particular, SL(r, A) ~ SL(r, A/q) is subjective. Remarks. -- i. For local rings some further refiniments of our results can be found in Klingenberg [24]. 2. Let GL(r, A, q)' be the inverse image in GL(r, A) of the center of GL(r, A/q). One would like the following converse to part e) of Theorem 4.2: If E(r, A, q) cH cGL(r, A, q)' then H is normal. This would follow from: E(r, A, q) = [GL(r, A), GL(r, A, q)']. For a commutative local ring this follows from (4.2)f), since GL(r, A, 0)' is then generated by GL(r, A, q) and the center of GL(r, A). Klingenberg's proof of this for non commu- tative A appears to contain a gap, due to the erroneous equation, " gj(a-lb-~a) -- I " on p. 77 of [24]. 3. IfA is the ring of integers in a number field then n =-= I defines a stable range for GL(A). Our results in this setting are, to some extent, well known. For example, the last part of Corollary 5.2 (familiar to function theorists for A =Z) and condi- tion (4..2) b) for q=A were known long ago by Hurwitz. Moreover, Brenner [I3] recognized (4.2) b) for A=Z. However the commutator formulae appear to have escaped notice even for A=Z. They turn out to be essential in the proof (see w 2i) that every subgroup of finite index in SL(n, Z), n>3, contains a congruence subgroup. The discovery of the generality of these formulae lies ultimately in J. H. C. Whitehead's work on simple homotopy types [36]. 502 CHAPTER II STABLE STRUCTURE OF PROJECTIVE MODULES w 6. Semi-local rings. Let A be a ring. Throughout this chapter, " A-module " means right A-module, and " ideal ", unqualified, means two-sided ideal. Let P be an A-module, and ~eP. We write P .... Homn(P , A) and define o(:~) = oe(~) -= {f(:()!feP*}. o(:r is a left ideal in A, and it is clear that o(:r --A if, and only if, the homomorphism g : A-~P, g(a) = ~(a, has a left inverse. In this case we call :r a unimodular element of P. Lemma (6.x). -- Let ~ : Q---~ P be a homomorphism of A-modules with Q finitely generated and projective. Then ~ has a left inverse (i.e. is a monomorphism onto a direct summand) zf, and only zf, ~* : P*-~Q* /s an epimorph#m. Proof. -- ~ has a left inverse ~ ~* has a right inverse =-~** has a left inverse -~ has a left inverse, since Q= Q** is reflexive. Moreover, ~" has a right inverse ,~-~* is an epimorphism, since Q* is projective. Denote by rad A the Jacobson radical of A. Corollary (6.2). -- Let e,, : Q-~P as in (6.I), and assume Im(~---:)CP.radA. Then ~ has a left inverse ,.~ v does. Proof. -- Let fsIm(~--,)*cQ*; then f(Q) Crad a. Since Q if finitely generated and projective this implies that ferad A.Q*. Hence Im ~*CIm v*+ rad A.Q*. Since Q* is finitely generated, Nakayama's Lemma tells us that ~* surjective =--:* smjective. Using (6. I) now, this completes the proof. Definition. -- Call A semi-local if A/rad A is an Artin ring. It follows then that A/rad A is a finite product of full matrix algebras over division rings. For the balance of this section, A always denotes a semi-local ring. The lemmas which follow contain the " zero dimensional " case of the general results to follow. If S is a subset of an A-module, P, denote by (S) the submodule generated by S. We shall say f-rankA(S ; P) > r if (S) contains a direct summand of P isomorphic to A ~. We will suppress the subscript " A " when A is fixed by the context. In what follows A is a fixed semi-local ring, and P denotes an A-module. Corollary (6.3). -- f-rank(S; P) =f-rank((S) -!- P. rad A; P). 503 ~o H. BASS Proof. -- If f-rank((S) +P.rad A; P)>r there is a homomorphism a :Ar-+P having a left inverse, and with Im at(S)+P.rad A. Hence we can find v:Ar~P with Im-:r and Im((~--z) CP.radA. By (6.2) -~ has then also a left inverse, sol-rank(S; P)> r. The reverse inequality is obvious. The following simple result will play a fundamental role in what follows. Lemma (6.4). -- If beA and a is a left ideal such that Ab+a=A, then b+a contains a unit. Proof. -- Since units in A/rad A lift automatically to units in A we may assume A is semi-simple. Passing then to a simple factor we can reduce to the case A = EndD(V), where V is a finite dimensional right vector space over a division ring D. In this case a can be described as the set of endomorphisms which annihilate some subspace, W, of V (a=Ae, e2=e, and W=kere). The fact that Ab§ guarantees that ker bnW=o. Write V=W| bW| Now we can clearly construct an automorphism u such that u]W=blW and u(W')=U. (Note that W~bW=~W'~U.) Since a----u--b annihilates W, aea and we're done. Corollary (6.5). -- n=I defines a stable range for GL(A) (in the sense ofw 4). Pro@ -- By definition, we must show that if Aa 1 +... -}- Aa, = A, r> I, then we can add multiples of a, to al, ..., a~_ 1 so that the resulting r-- I elements still generate the unit left ideal. Let b =a 1 and a=Aa2+.. 9 +Aar; then (6.4) provides us with a unit u = al + b2a~ +. 9 9 + b,a~. Hence ue (A(a 1 + b~ar) + Aa~ +... + Aar_l) , as required. The following is a technical little argument with two important corollaries: Let and ~ be unimodular in ~A| Writing a=~b+~p we have A=o(~) =Ab+o(0@. By (6.4) u = b + a is a unit for some a~o(~e). Let f be an endomorphism of ~A| such that f(~)=o,f(P)c~A, and f(0@ =~a. The existence off follows from the definition of o(~1~ ) =Op(~p). If q%= i+f then ~o1(o~ ) =~u-{-~p. Now let g be the endomorphism killing P such that g(~) -- --~pu- 1, and set q~2 ---- I -~g. If ~ -- ~q% then (i) q~(0~) = ~u, (ii) ~ leaves invaHant all submodules containing ~A-t-~A, and (iii) since f2=o=g2, ~ is an automorphism. Corollary (6.6). -- A|174 Proof. -- Using the hypothesized isomorphism to identify the two modules, we can write ~A| = eA| With the ~0 constructed above we have, from (i), ~0(eA) ---- ~A, so P----~ (,~A| P)/~A = q0 (~A| (~A) ~ (~A| P')/~A~P'. Corollary (6.7). -- If M is a submodule of P, then f-rank(A'| A'| P). Proof. ~ It suffices to treat the case r-=--i. Moreover, the left side is clearly at least as large as the right. Let e~, ..., ~,~A| be a basis for a free direct summand of ~A| (~ is assumed unimodular here of course.) Now construct (p as above with ~----e~. By (ii) above q~(@e~A| also, I <i<s, so we can replace ~by q)(~) and, by (i), reduce to 504 K-THEORY AND STABLE ALGEBRA oi the case ~A = elA. But then we can subtract multiples of a 1 from as, ..., a8 and further render ~2, ...,%eM. Thus, f-rank(.3A| P)>s--i, as required. Corollary (6.8). -- If ~ is an element, and S a subset, of P, then f-rank(S, e; P)<I +f-rank(S; P). Proof. -- Suppose f-rank(S, e; P)>r; i.e. there is a a : Ar-+P with a left inverse, and Imac(S) 40~A. Let f:A| by f(a,p)=,.a+p. Then Im(f](A| so we can find g:A'-~A| such that ~=fr and ImgcA| The left invertibility of ~ implies that of g, and hence f-rank(A| A| >r. The present Corollary now follows from the preceeding one. Lemma (6.9).- Suppose oh, ...,areP and f-rank(~l, ..., ~,; P) = t<s<r. Then we can find ~1, ..., ~, of the form ~-= ~i + Z~.>~%.%., i <i <s, such that f-rank(~l, ..., ~; P)=t. Proof. -- We induce on t, the case t=o being trivial. If t>o we can write P=~A| with ~ a unimodular element of P in (~1, ...,~,). Writing ~--: ~b~ + e~ then, with c~ 9 P', we must have Z~b~A = A. By (6.4) (applied to the opposite ring of A, since we are now dealing with right ideals) we can find a unit u---bl+Ej>lb~alj. Set ~l--el+Zj>l%.alj.=~u+~. Then ~ is unimodular in P (since ueo(~a)), so P=~A@P, for some P~. Write ~i--~ci+~,i, 2ieP1, ~ <i<r. Note next that (~, ~.,, ..., a~) = (~, ~, ...,e~) = (~,y~, ...,y,). Hence, we conclude from (6.7) that f-rank(y~, . .., y,; Pa) =t--l. By induction, we can find ~---- ~ + X~>r 2 <i<s, such that f-rank(8~, ..., 88; Pt) = t-- i. Writing we see that (~, ~.,, ..., ~,) = (~, ~.o, ..., ~8), and this clearly solves our problem. w 7. Delocallzation. The maximal spectrum. As a basic reference for the material of this section, we refer to Bourbaki [12] or Grothendieck [22, Ch. o]. Let A be a commutative ring and X = max(A), the topological space whose points are the ma.'dmal ideals of A, and whose closed sets are the sets of all maximal ideals containing a given subset of A. If M is an A-module and xsX (x is a maximal idcal in A) we let M~=A~| denote the localization of M at x. Further, define supp M(= suppxM ) ={xEX I M,4: o}. 505 22 H. BASS The topology in X is better described, for our purposes, as follows: the closed sets in X are precisely those of the form supp M where M is some finitely generated A-module. Let A be a finite A-algebra (i.e. finitely generated as an A-module). If P is a A-module, Px is a module over the A,-algebra, Ax, and our finiteness condition on A gua- rantees that Ax is semi-local, for all xeX. Hence, if S is a subset of P, we can consider f-ranka~(S ; P~), as defined in w 6. (Here we are confusing S with its image in P~, but this should cause no difficulty.) We now define f-rankh(S; P) = inf, exf-rankA~(S; P~), and f-ranka(P ) =f-rankh(P; P). When A is fixed, we shall suppress the subscript. Since the definition is local, (6.7) immediately yields: Corollary (7. I ). -- If M is a submodule of the A-module P, f-rank(A*| A*| = r +f-rank(M ; P). More generally, we define the " singularities " of SoP by Fi(S; P)={xeXIf-ranka~(S; P~)<j}. Thus, F0(S; P)=0 for any S, and Fj(O; P)=X for all j>o, for example. It is essential to our method that these sets be closed, and for this we need a " coherence " condition on P. Lemma (7.2). -- Suppose P is a direct summand of a direct sum of finitely presented A-modules. Then, for all subsets S of P and for all j, Fi(S; P) is closed in X. (Recall that a finitely presented module is the cokernel ofa homomorphism Ar-+A~.) Proof. -- Let xeX--Fi(S; P). Then if Q= A j there is a homomorphism ~ : Q~P such that Im ac(S) and such that % : Qx-+Px has a left inverse. (One simply chooses elements in (S) whose images in Px are a basis for a direct summand.) It suffices now to show that {yeX[% has a left inverse} is open, for the latter will then be a neighborhood of x in X--Fj(S; P). Since Im a is finitely generated, our hypothesis on P permits us to reduce this last question to the case where P itself is finitely presented. Under these conditions the natural homomorphism (P*)y-+ (Py)* is an isomorphism for all yeX (see [5, Lemma 3.3]) ; the second * here refers, of course, to the duality of AN-modules. The same remark applies to the free module, Q, so that we can conclude from the commutative square, (a*)y (~y)* (Py)* > (Qy, 606 K-THEORY AND STABLE ALGEBRA an isomorphism Mu=~coker(%)* , where M=coker(a*). Now we have from (6.i) that {)!% has a left inverse} = {y i (%,)* is surjective}={y[Mu=o}--X--suppM. Since M is finitely generated (being a quotient of Q*), supp M is closed, and this completes the proof. It is useful to note above that, since M--o<>My=o for all y, ~ has a left inverse ~>~ru does for ally. Applying this to a:A~P with a(I)=a we obtain: Corollary (7-3)- --- With P as in (7.2) and ~eP, ~ is unimodutar in P if, and only iJ; F1(~ ; P)=O. Examkles. -- I) Lemma 7.2 applies, notably, when P is either projective or finitely presented. 2) The inclusion Z-~Z(p!, (rational p-adic integers) has a left inverse only at (p) emax(Z). 3) IfP is the Z-submodule of Q, generated by {p-tip prime}, and ifq is a fixed prime, then Fl(q-t; P)=max(Z)--{(q)} is not closed, whereas P is locally free (of rank one). Finally, we recall some topological notions. In any topological space, X, a closed set F is called irreducible if F4= O and if F--GuH with G and H closed=~F ---G or H. We then call codlin F(--codimxF ) the supremum of the lengths, n, of chains, F 0= FCF 1C. . . CF,, of distinct closed irreducible sets in X. For arbitrary closed F we define codlin F to be the infimum of the codimensions of the irreducible closed subsets of F, with the convention codlin f)- oo. The supremum of the codimensions of non empty closed subsets of X will be called, dim X. We call X noetherian if the closed sets satisfy the descending chain condition. In this case every non empty closed set F is a finite union of irreducible closed sets. Such a representation of F, when made irredundant, is unique up to order, and we call the intervening irreducible closed sets tile irreducible components of F. For example, if A is a commutative semi-local ring, max(A) is a finite discrete space, and hence a noetherian space of dimension zero. If A is only noetherian, then max(A) is a noetherian space of dimension <Krull dimension A=dim spec(A). w 8. Serre's theorem. For the next four sections we fix the following data: )IA is a commutative ring for which X =max(A) is a noetherian space. (8 I A is a finite A-algebra. The next theorem is a slight generalization of earlier versions. We include it for the sake of completeness, and also tbr the proof below, which is perhaps a little more manageable then its predecessors. 507 ~4 H. BASS Theorem (8.2). -- (Serre [29] ; see also [5]). Suppose P is a direct summand of a direct sum of finitely presented A-modules. Then, if f-rankA(P)>d = dim X, P~A| for some P'. We will derive the theorem from the two lemmas below, wherein P is always assumed to satisfy the coherence condition of the theorem. This assumption permits us to invoke (7.2) which ensures that the various " singular sets " intervening in the proofs are closed. All codimensions refer to codimension in X. Lemma L -- If f-rank P>r, there exist ~1,..., e,eP such that codim F~.(~I, ..., ~,; P)>(r+ I)--j, all j>o. Lemma II. -- If o~1,..., %eP, r> I, and k~Z, are such that codim Fj(~:, ..., %; P)>k--j, :<j_<r, then there exist ~=ai+~,ai, for suitable aieA , I<j<r--l, such that codim Fj(~I , ..., ~r-,; P) >k--j, I <j<r--:. Proof that I and II-~(8.2): Apply I to P, with r=d+I, to obtain ~x, ..-, ~,eP such that codimFj(~,, ..., e,; P) >(r+ ~)--j for all j>o. Now, with k=r+i, apply II, (r--I)-times. The result is a single element, ~eP, such that codim F,(~; P)>k--i =r---d+ I. Since d=dim X this implies FI(~; P) =0, so, by (7.3), ~ is unimodular, and this completes the proof. ProofofL -- We induce on r, the case r=o being trivial. Suppose 0~1, ..., % have been constructed as in the lemma, and we want e,+: (assuming f-rank P>r+ i). For o<j<r, let {D~.~} be the "largest" irreducible components of Fi_:(~, ..., ~,; P), i.e. those of smallest codimension, (rq- i)--(j+ i). Of course there may be none, but that's all the better. Since codim Fi(0h,..., 05; P) > (r-I,- ~) --j, Di~ r Fj(e,,..., ~; P) for all ~. It follows (since Di~ is irreducible) that we can choose x(j, v) eDit-- [Fi(~:, ..., ~; P) uU~.~Di~]. Since x(j, v)eFi+,, CF i we have f-ranka,~j.~)(al, ..., at; P*0".~;) =j<r, for o<j<r, and all v. It follows now from our hypothesis on P and (6.7) that we can choose ~j~ eP such that f-rankA, Ij.~)(ax, ..., ~,, ~,; P,:,.,~)) >j+ i. Since x(j, v) are (clearly) all distinct, we can write I = Zi,~ei~ in A, with ei~-- ~(i,~)(~,~) mod x(h, ~). Setting ~+ ~ -- Zi.~,3i~e~ we have a,+l-~i~modx(j, ~) so, by (6.3), f-ranka~(i.~)(~l, ..., ~,+l; P~(i.~))>j+ I, o<j<r. Hence x(j, ~)r ..., a,+a; P) (by definition) so also Di~4zFi+,(al, ..., a,+a; P). On the other hand, F~+~(a~, ..., e,+~; P) r ~a(a~, ..., ~,; P), and since the former contains no component of extreme codimension of the latter, we conclude, as desired, codim Fi+t(aa, ..., ~r+t; P) >(r+ I)--(j+ I) A y I = (r ~-2)--(j+ :), o_<_j<r. For the remaining values of j, F 0 = fO has infinite codimension, and if j> r, (r + ~) -- (j + I) <o, whereas all codimensions are >o. Proof of IL -- For o<.j<r let {Dj~} be the irreducible components of ..., P) of lowest possible codimension, k--(j+ I). Then D~r ..., ~,; P) 508 K-THEORY AND STABLE ALGEBRA 25 so we can find xlj, ..., %; P) u ; thus f-ranka,(i,,)(al, ..., a,; P~/i.~)) =j<r, for o~j<r. By (6.9) we can find elements %,~A,(y.~) such that, if ~,---~+a,aq~eP,(i,~l , we have f'rankA,(j.~l(~:J~, "", ~i,-~i~; P*(i,*I)=-J" If we modify our choice of aqy modulo the radical of Ax(i.,) we can even choose the %,EA, and hence ~i,~P. This is permissible by (6.3). Now choose e~ -- 3(/,~/(h,~)mod x(h, ~), as in the proof of I, set ai---E~.~aij~ej, , and put ~=a-i-ka-,ai, x<i<r--I. Since ~---- ~,~. mod x(j, ~) we have, by (6.3) again, f-rankA~(i.,l(~l , ..., ~,_1; P~(J,~I) ----J, o<j<r, all ~. The form of the ~'s makes it evident that (~1, ..-, ~,-t, a,) ---- (ul, ..., a,-t, a'r), SO it follows from (6.8) that F/(~x, ..., ~,_~; P) cF~.+~(0h, ..., ~,; P). Since we have arranged that the former contains no irreducible components of lowest possible codimension, k--(j+ ~), of the latter, we conclude, as desired, that codim F~.(~, ..., ~_~; P) >k--(j ~ ~) + i =k--j, o<j<r. Counterexamples. -- ~) To see the necessity of the coherence condition in Serre's Theorem, we can let P = Q| where Q is the Z-submodule of the rationals generated by {P-~IP prime}. Here f-rankz(P ) = ~ and dim max(Z) = ~, but P has no projective direct summands. 2) The following example of Serre shows that P can even be made finitely generated and locally free. Let X be a Cantor set on the real line andy a point of X which is a limit point from both the left and right in X. Set D0={x~X[x~y}, D a ={x~XIx<y }. With A= C(X), the ring of continuous real valued functions on X, it is well known that max(A)-:-X and is hence of dimension zero. Moreover A is locally a field so that A-modules are locally free. Let al be the ideal of functions vanishing on D~, i=o, x, and let P= (A/ao)| Since supp A/ai=D~_, we see that P is locally free of rank one except at {y}---- D0r~D~, where it has rank two. Hence, if P had a free direct summand, the complementary summand would have support {y}. If (f0,f~) eP were in this summand, f~A/a~= C(Dt_~), f~ would be a function on D~_~ vanishing everywhere except, perhaps, aty. Sincey is a linfit in D~_i, fi has to vanish also at y, and hence fi----o. Thus the complement would have to be zero, and we have a contradiction. w 9" Cancellation. A, X, and A are as in (8.~). " Cancellation " refers to Theorem 9.3 below. Theorem (9-I). -- Let Q and P be projective A-modules, a a left ideal, and a-= ~q+ a-vEQ| an element such that o(a)+a=A. (See w 6 for definition of o(a-).) Suppose, moreover, that f-rankP>d=dim X. Then there is a homornorphism f: Q--->P such that o(f(~)+~)+a=A. 4 '26 H. BASS Proof. -- We induce on d, and the case d--o will be subsumed in the general induction step. Our hypothesis makes Serre's theorem available, to the effect that P=f~'A| for some unimodular ~'eP; say ~v=~'b=- Then o(~q) +Ab~o(~') +a=A. Let F1, ..., F o be the irreducible components of X, and choose x~EF~--kJi.~Fi, I <i<s. Modulo the product of the xi, A is a semi-local ring, so we can apply (6.4) to find a~ + a' + ceO(aq) -~- o(~') + a such that the image of b + aq + a' + c is a unit in A modulo the xi, and hence already in Axi , I <i<s. Let g be an endomorphism of P such that g(I~')=o,g(P')c~'A, and g(~')=~'a'. The existence of g follows from the definition of o(~') (w 6). If ~=ip--g then ~ is an automorphism (since gZ=o) and a~v=.3'(b+a') ~-e'. Setting 3--~-'(~') and ~I=a-~(~')ePI=G-'(P ') we have P = ~A| and ~p = ~(b + a') + ~. Now choose fl : Q-+~A cP such that fl(~q) = ~aq; again, ft exists by the definition of O(eq). Putting b t =aq+ b +a', then, we have (*) f~(~q) + ~p = ~bt + 0~1, and bl ~c is a unit in A~i , i<i<s. If dim X=o then X={xl,...,x,}, and we're done. In general, we can find a teA belonging to none of the x; such that tA cAb 1+ a. (E. g. semi-localize at xl,..., x~, solve (bl+C)Z= i, and clear denominators.) Let A*=A/At, A*=A/At, C=image of a in A*, etc. Then X*=max(A*) is a closed subset of X containing no xi, hence no Fi, so dim X*<dim X. Hence, using (7- i) we have f-rankA.P~>f-rankAP 1--f-rankaP-I>dim X--I ~dim X*. * * * 9 * * Consider "./=%+~teO)s174 Since A=o(~q)+Ab~+O(el)-~-n we have, over A, o(.v*) + A*b~ + a* -- A'. Putting this together we are in a position to apply our induction hypothesis to y*eQ*| and the left ideal A*b~_ C, the result being a homomorphism h* : Q* ~-P~ such that o(h*~+:~;) -!-A*b~+o*=A*. Since Qis projective we can cover h. with an h : Q-~-P~ cP. Now, for the theorem we take f=fa+h : Q-+~A| It remains to show that b+o=A, where b=o(fxQ+~p). By (*)above we have f~Q -- ~p = (h~q -~-f~eq) + ~p = h~q + (~b I ]-~) --- ,~b~ + (h~Q-t- ~(1) e~A| It is thus evident that Abtcb , and that o(h~Q+~l)cb. But since P~ is projective it follows that the image of o(h~q+~l) in A* is o(h*~+ ~). By construction of h* this together with A*b~ ~ a* generates A*. Back in A, then, o(h~q + ~) + Abt -t- a + At = A. But AtcAbt-~-a so we have b 4-aDo(ho%~-~)+Ab~+ct--A, as required. Let P be a A-module. Recall that an element eeP is unimodular if there is an feP*=Hom~(P, A) for which f~= i. We shall similarly call feP* unimodular if there is an 7eP for which f~ =; ~ (i.e. iffis surjective). Further, we shall find it useful sometimes to identify P with Homa(A , P) ; i.e. we identify ~eP with g~ : A~P defined by g~,(a)-=~.a. Thus, if f: P-+A we can compose, ,~f: P-+P; ~f is defined by (~f) ~ - ~(f~) for ~ e P. An endomorphism z of P will be called a transvection if = -- ~i, + 7f, where ~eP, feP',f~.=o, and either e orfis unimodular. -: is then necessarily an auto- 510 K-THEORY AND STABLE AI.GEBRA _o 7 morphism, since (a f)" = o. If q is an ideal in A we call -: a q-transvection if Ira(a f) cPq. Denote by GL(P) the group of A-automorphisms of P, and by E(P) the subgroup generated by the transvections. More generally, if q is an ideal, let E(P, q) be the subgroup generated by the q-transvections. Suppose z= i +af is a q-transvection and ~GL(P). Then -:~ = I + (~-l~)(f~r) is clearly again a q-transvection; hence E(P, q) is a normal subgroup of GL(P). The next result should be compared with Theorem 4.2 a). Theorem (9.~). -- Let M be a A-module which has a projective direct summand of f-rank>d+ I (d-dim X). Then for any ideal q in A, the orbits of E(M, q) operating on the unimodular elements of M are precisely the congruence classes rood q. In particular, E(M) is transitive on the unimodular elements. Proof. -- By hypothesis, M=P'| with P' projective and f-rank P'>d+ ~. By Serre's theorem (twice), P'---fiA|174 with ~ and y unimodular. Let q be an ideal in A. We will first show that if a is a unimodular element of M and if ~=~modMe, then there is a -r~E(M,q) such that ":~=9. With P=3A| we have M=yA|174 Write a----yq+~,+a~. Since ~-~sPmodq, we have qeq. There is an h:M-+A with h~----i, by assumption, so we can write (*) ~ ---- ha = (hv)q + hap + ha~. Let Y=yr+:(~+as with r=q(hy)q. To see that E is also unimodular, we first note that o(~) --Ar+o(a~) +o(~s). Left multiplying (*) by q, then, we find qeo(E), so that o(~) ~Aq + O(ap) + o(a~) ---- o(~) ---- A. Now we apply Theorem 9. ~ above to Q~yA, P, yr+~p~Q.| and a=o(a~). The result is an f: Q--+P such that o((fu +o(~s) --A. Let ~---- (fy)q(hy)ePq, and let gl : M~A by g~(y) ---- i and gl(P| =o. Then z~---- IM+8~gl~E(M , q), and vl~=yq-!-@+aNeyA|174 where @=(fy)r+ap. We have arranged above that o(:@ +o(~y) =A, so @q--~ is unimodular. Hence we can find an h' : M-+A with h'(a~,+~N) -- I, and such that h'(y)=o. Write a~=~b+xleP- -~AC~)P 1. Since vla=~ mod Mq (recall ":1- I~mod q) we have (**) +a, + and in particular, I--b~q. Thus, if g2----((I--b)--q)h':M~A, Im g2Cq, so v2= I~+yg,,eE(M, q). T2Ti0~=y(x--b) +o~+a N. Let cr= IM+ggleE(M); then o-.r2~lq = y(I --b) + (~,+ 9(I --b)) + a~ ---- y(i --b) + ~ + al + aNEvAQgA|174 Let ga:M~A by ga~=I and ga(yA|174 and let ~a----(y(~--b)+~l+a~), which, by (**), is in (kerga) q. Then Ta---- ~M+Sag3cE( M, q) and TaO~Tia=~. The presence of a, which need not belong to E(M, q), is harmless because ~-1~ = ~_9g1~ = 9, so that -:=o-ITa(~T2zI~E(M, q) solves our problem. 511 ,~8 H. BASS Now for the general case, we note first (taking q-= A) that we have shown E(M) to be transitive on unimodular elements. Suppose given arbitrary unimodular elements and ~' in M with ~-~' rood Mq. Choose geE(M) with ~'=~ (~ as above). Then ~a-ea'=~ mod Mq so the argument above produces a v~E(M, q) with ~-ee=~'. Finally, ~-lwa--0:' does the trick. Theorem (9.3) (" Cancellation "). -- Let M be a A-module which has a projective direct summand of f-rank>d=dim X, and let Q be a finitely generated projective module. Then, if M' is another A-module, Q|174 ~ M--~M'. Proof. -- Since Q| for some n and Q' we can reduce, by induction on n, to the case Q= A. Then using the given isomorphism to identify A| with A| we can write ~A|174 with ~ and ~ unimodular. ~A| satisfies the hypo- thesis of (9.2), so there is a zeGL(~A| with w=~. Hence M ~ (}A| ---- v(0cA@ M')/v(aA) ~ (:cA| M')/aA ~-- M'. Remarks. ~ I) If M satisfies the coherence condition in Serre's Theorem, and if f-rank M>2d, then M has a free direct summand ~A ~+1 (by Serre's Theorem), and hence M fulfills the hypothesis of (9.3) above. 2) If d=o then A is semi-local, and (6.6) gives the conclusion of (9-3) with no restrictions on M. If d= I and A----A is commutative, then for M projective no further hypothesis is needed. For one can reduce this case further so that spec(A) is connected and M is finitely generated (using [6]). Then one applies Serre's Theorem to make M and M' each a direct sum of a free module and a projective module of rank one. The desired isomorphism then follows by taking suitable exterior powers (see [29, no. 8]). However, for d= i and A non commutative, (9.3) gives the best possible result even for M projective and finitely generated (see Swan [33]). For d>I (9.3) is best possible even with A=A commutative. For if A--R[x,y, z], x2-4-y2+z 2= I, is the algebraic coordinate ring of the real 2-sphere, and if P =A3/(x,y, z)A is the projective A-module corresponding to the tangent bundle on S 2, then P is not free, whereas A|174 2 (see Swan [34, Example I]). w IO. Stable isomorphism type. Keep A, X, and A as above (see (8. i)). The Proposition which follows is merely a reformulation of a special case of the preceding results. We include it for the sake of putting in evidence the faithfulness of the analogy between the present theory, and its topological source (see Introduction). Proposition (I0.I). -- Let P,(A) denote the isomorphism types of finitely generated projective A-modules P, such that P~-A~, for all x~X. Let f, : P,(A) -+ P,_t(A) be the map induced by | Then, if dim X = d: 512 K-TIIEORY AND STABLE ALGEBRA 29 I. (Serre's Theorem) f, is surjective for n>d. 2. (Cancellation) f, is injective for n>d. Of course our theorems are much more general. For example, one could formulate a similar result for all finitely presented modules, relative to f-rank. Thus, calling M and N " stably isomorphic " if Q|174 for some finitely generated projective module Q., we see that stable isomorphism=isomorphism for semi-local rings (6.6), and more generally also for M projective of f-rank>d, or finitely presented off-rank> 2d. In a special case of some interest we can make a mild improvement in Serre's Theorem for non projective modules. Proposition (IO.2). -- Let A be a Dedekind ring of characteristic zero and 7: a finite group. If M is a finitely generated torsion free Arc-module of f-rank~ I, then M = P| with P a projective Arc-module locally free of rank one. Proof. -- With L the field of quotients of A let A be a maximal order of Lrc containing Arc, and let a be the annihilator of the A-module A/A~. Characteristic zero guarantees a+o. If we semi-localize at the maximal ideals containing a we obtain a free summand of M (Serre's Theorem in dimension zero) generated, say, by eeM. Let P be the A-pure submodule of M generated by cAre; this is automatically an A=-submodule. Then o-+P~M~M/P~o is an exact sequence of torsion free Arc-modules which splits at all maximal ideals containing a. At all others it splits auto- matically because Arc there agrees with the hereditary ring A. Hence the sequence splits -- its is an element of Ext],(M/P, P) which vanishes at all localizations. P, being locally projective, is projective. Finally, since L| is Lrc-free of rank one, a theorem of Swan [32] (see also [5]) guarantees that P is locally free of rank one. Remark. -- Short of the last sentence, and its special conclusion, it is clear that we have invoked only very general properties of Arc. Corollary (IO.3). -- With A and 7: as above, if M is a finitely generated torsion free Arc-module of f-rank> 2, and Q a finitely generated projective An-module, then Q|174 => M~M'. This Corollary responds to a question of Swan [32] and Swan himself has shown it to be best possible [33]. He shows, moreover [33, Theorem 2], using a result of Eichler [i9] , that if A is a maximal order in a semi-simple algebra over a number field, then one can always cancel projectives unless R| has a quaternion factor. w xz. A stable range for GL(A), and a conjecture. We keep A, X, and A fixed as in (8.r). Theorem (ix.x). -- If d=dimX, then dq-I d~nes a stable range for GL(A), in the sense of w 4. Hence, the conclusions of Theorem 4.2 are valid for A with n = d + I. Proof. -- We must show that if r>d+ I and if Z~=IAai=A , then there exist 9 9 ", -- i,-l~xkai -V bx, b r 1r such that Z r-IAI 'bias) A. 513 3o H. BASS Note first that 0~ = (al, ..., at) is unimodular in A~= Ar-I| .3 = (o, ..., o, i); = 0~' q- ~a,. Since f-rank A r - 1 = r-- I >-d + i > dim X we can apply (9. i) and obtain f: ~A~A ~-I such that r is unimodular. Then f~= (bl, ..., br_l, o) solves our problem. Let q be an ideal in A, and write, in the notation of Chapter I, w i, Gr(q) = GL(r, A, q)/E(r, A, q). Let f~: G,(q)-->Gr + l(q) be induced by the inclusions. We obtain a direct sequence of sets with base points, and )im Gr(q)=GL(A, q)/E(h, q), which, by (3. I) a) is an abelian group. With this notation we can now translate parts of (4.2) in such a way as to exhibit the (partial) analogy with Proposition IO. i. Proposition (xI.2). -- Suppose dim X=d. Then for all ideals q, we have: a) ((4.2) c)) G,(q) is a group for r>d+I. b) ((4-2)b)) fr:G~(q)~G,+t(q) is surjective for r>-d+I. c) ((4.2) f)) G,(q) is an abelian group for r>-2(d§ and >--3. The missing link here is an assertion that f~ becomes injective. Our prevailing topological analogy suggests quite explicitly in this regard, the following, Conjecture. -- Under the conditions of (ii.2) f~:G,(q)-+G~+l(q) is injective for r>d+ I. In terms of matrix groups this says, for r>d+ I, E(r+ I, A, q) nGL(r, A, q) =E(r, A, q). When A is a division ring (so d= o) it is the affirmative solution of essentially this problem which constitutes Dieudonnd's theory of non commutative determinants [i 7]. Klingenberg [24] has generalized his solution to local rings (still d=o), although Klingenberg's proof is not valid when q--A. On the other hand it works in any ring provided q Crad A. This procedure of axiomatically constructing determinants (see Artin [I, Chap. V], for a good exposition) runs into severe computational difficulties if one tries to generalize it naively. The interest in the conjecture above stems from more than a simple love of symmetry. One can consult w 20 below and [7, w 1] for some rather striking consequences of its affirmative solution. 514 CHAPTER III THE FUNCTORS K w x2. K~ and K~(A, q). Let A be a ring and ~ =,~(A) the category of finitely generated projective right A-modules, and A-homomorphisms. Let "; = TA : obj ~K~ solve the universal problem for maps into an abelian group which satisy (A) (Additivity) If o->P'-+P~P"---~o is ancxact sequence (asA-modules) then TP = yP' + TP". Uniqueness of (TA, K~ is the usual formality, and existence follows by reducing the free abelian group generated by isomorphism types of obj ~ by the relations dictated by (A). Let T be an infinite cyclic group with generator t. We build now a new category, ~[T] =~r t-1], whose objects are A-automorphisms, ~, of modules P = dom c~eobj ~. If ~'eAutA(P' ) is another, a morphism, ~-~e', is an A-homomorphism, f: P--+P', such that fe=~J. If ~eAutA(P ) then e defines an A-representation ofT on P, t acting as c~. In this sense we can think of ~[T] as a category of A[T] =A[t, t-l]-modules, and as such we may speak of " exact sequences " of ~'s. Let q be an ideal in A (possibly q=A) and let ~q[T] be the full subcategory of ~[T] whose objects are those e for which " ~-: modq"; i.e. if P----dome, we require that c~(~AA/q = i on P| We define the group K~(A, q) by letting W~ : obj .~q['F]-+K~(A, q) solve the universal problem for maps into an abelian group which satisfy (A) (Additivity) If o-,='-.~-~0d'~o is exact then Wq==W,~'+W,=". (M) (Multiplicativity) If dom = =dom [~ then W,=~ =Wq~ +W,~. Existence and uniqueness are clear by a remark analogous to that above for K ~ Although we have no need for this fact, the reader will be able to determine easily that K ~ is unaltered if we relax (A) to apply only to split exact sequences. When q=A we call K:(A)=K~(A, A) the " Whitehead group " of A, and W=W a the " Whitehead determinant " 515 3 2 H. BASS Let ~ be a automorphism in :~q[T] and Peobj ~. Then (M) implies WqIe:-=o, so (A) further implies that Wq(a|162162 If 0r A, q) r A) =AutA(A" ) then we can regard aceobj ~q[T]. With our convention, GL(n, A)r A), ~ is identified with ~| The last para- graph shows that Wq respects this convention, and hence we have a map, also denoted Wq, W, : GL(A, q) = U, GL(n, A, q) -+KI(A, q), which, by (M), is a homomorphism. Suppose aeGL(A, q) and ~EGL(A). Then ~-t:r so Wq:c----Wq(~-%r This means that [GL(A), GL(A, q)] ---- E(A, q)Cker Wq, so we have an induced homomorphism f: GL(A, q)/E(A, q)--->Kt(A, q). Proposition (i~,. I ). -- The inclusions GL(n, A, q)Cobj ~q[t, t -t] induce an isomorphism f: GL(A, q)/E(A, q)~Kt(A, q). Pro@ -- Let xeobj ~q[T], P-=dom=. We can find a Q such that P| (some n). This isomorphism induces an isomorphism a| A, q). With Q fixed % varies in its conjugacy class in GL(n, A), so its image in G=OL(A, q)/E(A, q) doesn't change. If we replace Q by Q| a, is replaced by a conjugate of %| , and again its image in G is unaffected. Finally, if P| then Q|174174174 so we see that the image of % in G is independent of Q. We define thus a map g : obj ~q[T]--~G, and we propose to show that g is additive and multiplicative. Once shown, the uni- versality of W, produces a unique homomorphism Kt(A, q)-->G, which is manifestly an inverse for f. g is multiplicative. For if dora a = P -- dom ~ an isomorphism P~)Q~A" induces O~| ~IQ~n , and O~@IQ~OCn~ n. g is additive. Let o-->~'-->~-->0d'-+o be an exact sequence, with domains P', P, P", respectively. Choose Q' and Q" with isomorphisms P'|174 Since the exact sequence induces P-'~P'| we can choose an isomorphism P|174 compatible with the direct sum of the given sequence with o-->iq,--->Iq,| With these isomorphisms, we have ~'@IQ,~Q(n, ~"@IQ,,~gn' , a~, c(: I :' o I (%~)-1q and a| oq ..... a2,=[o ,, :r 0 I I " Since the second factor is manifestly in E(2n, A, q), and since 0r n OCnCX n 0 -- mod E(2n, A, q), ] o 1 o (x o I by the Whitehead Lemma (I.7), we see that ae, and %~' do indeed agree in G, as required. 516 K-THEORY AND STABLE ALGEBRA Suppose H is a normal subgroup of GL(A). Then, by Theorem 3.i, E(A, q)r q) for a unique q, and H/E(A, q) is, via Proposition i2. I above, a subgroup of K~(A, q). Conversely any subgroup of KI(A, q) defines in this way a normal subgroup of GL(A). Thus we see that a determination of all normal subgroups of GL(A) is equivalent to a determination of all subgroups of KX(A, q), for all q. Finally we note that K ~ and K 1 are functors. If q0 : A-+B is a ring homomorphism, then | : ~(A)-+~(B) induces ~~ : K~176 If q is an ideal in A and q' an ideal of B containing q~(q), then | induces qr K~(A, q)~Kt(B, q'). Note that the isomorphism of Proposition I2. I is an isomorphism of functors. w 13. The exact sequence. Let q~ :A-+B be a ring homomorphism. If P andf are a right A-module and A-homomorphism, we shall abbreviate PB=P| , and fB-f| Our objective is to construct a relative group, K~ ~0), which will fit into an exact sequence (Theorem I3. i, below). To this end we manufacture the category c~(~) whose objects are triples, ~ = (P, a, Q.), with P, Qeobj ~(A) (i.e. finitely generated projective right A-modules) and ~ a B-isomorphism, PB-+QB. If ~'= (P', 0~', Q') is another such triple, a morphism ~r'-+, is a pair, (f, g), of A-homomorphisms, f: P'-+P and g : Q.'-~Q, making P'B > Q'B /B : gB PB ~, QB commutative. We call ,(I',r a(t"'g';I .... " exact if P'LP2P" and Q'-~Q.~Q" are both exact sequences of A-modules. Note that the objects of ,~(q~) are a groupoid. Thus, if a=(P, e, P') and '= (P', e'P"), then we write a'e = (P, ~'~, P"). Now we define K~ ~) by letting R : obj c~(?)~K0(A, q~) solve the universal problem for maps into an abelian group which satisfy: (A) (Additivity) If o -+ a' -+ a -+ a" --+ o is exact then Rc = R~' + R~". (M) (Multiplicativity) If ~'~ is defined then R~'a = R~' + R~. 5 34 H. BASS We need to know K~ ?) in some detail, and it will be convient to introduce some provisional terminology for this purpose. A triple -: = (P, ~, P) will be called an " automorphism ". Since ~eAutB(PB), WB~eK~(B) is defined, and we shall write WB-:=Ws,8 in this case. ve=(P, ip, P) will be called an " identity ". Since ~1,~v--'P, (M) implies (I) Rcp=o. If there is an exact sequence then (A) further implies R~=o. Let t3~GL(B)=U,GL(n,B); say ~3~GL(n,B)=Aut~(A"B). Then = (A", [~, An) eobj ~(~). Viewing ~eGL(n+m, B) replaces . by u| , so R~ is unaltcred. Hence we have a well defined map GL(B) ~K~ ~), (s) which is a homomorphism, by (M). Now if ~ is elementary, then , appears in an exact sequence of type (2), with P--A "-I, Q----A. Hence E(B) is in the kernel of (3), so (3) iuduccs a homomorphism 8 : K~(B) = GL(B)/E(B)-+K~ ~). Let 9 = (P, ~, P) be an automorphism. If WBx=o then, by Proposition I2. I, we can find a Oso that T| y, An), with TeE(n, B), and hence Rv--o. Now if ~, a'cobj (~(~) write if there exist identities, v and z', and an automorphism ~ with W~r such that (a|174 A tedious, but straightforward, exercise shows that --- is an equi- valence relation. Our earlier remarks show that a~--a':~Ra=Ra'. Hence, if R' :obj ~(?)--->G=obj cd(?)/~-- is the natural projection, then R=hR' for a unique h : G-+K~ ?). An easy check shows that .-~ rcspccts | so that | induces on G the structurc of a monoid, with neutral elcment the class of the idcntitics, and rclative to which h is clearly a homomorphism. We propose to show that a) G is a group, and that R' is b) additive and c) muhipli- cative. Once shown, the universality of R produces a homomorphism h' : K~ ?)--->G which is clearly an inverse for h. This isomorphism will permit us to conclude: (4) Every element of K~ ~) has the form R~, ~eobj cg(~). R~=o-~-.o. a) G is a group. Given a=(P, 0c, Q), let a'~-(Q,--a -t,P). Then ~|174 L 0~ P| 0r -t O' 818 K-THEORY AND STABLE ALGEBRA It follows from the Whitehead Lemma (i.7) (using the fact that PB~QB) that =o. b) R' is additive. Let o -+ a' ---> z --.'- a" -+ o be exact. If ~'=-(P', cr Q') and a"= (P", 0r Q"), then ~=~= (P'| 0% Q'| where = = o = 0r [O Ip,, B Since W~s=o we have (h,~(~'| as required. c) R' is multiplicative. Suppose (~ = (P, e, P') and (~'= (P', ~', P"). We must show (~'~(;| From the commutative diagram PB ~ > P'B 1pB / (-- 1p)B PB- P'B we see that e~.--~=(P, --e, P'). Hence it will suffice fbr us to show a'~| = (P| [3 -- o Ip,~ ' P"| and ~, P"| (--q)|176 (P| Y= t, -- where ~= (POP', y-t~, P| Since PB~P'B it follows from the Whitehead Lemma (1.7) that W~,=o, and this completes the proof of c), hence also of (4). Finally, we define a homomorphism d: K~ q~)-+K~ by d(R(P, ~, Q)) =yAP--yAQ. It is clear that d is well defined. Theorem (,3.I). -- Let (p :A~B be a ring homomorphism. Then the sequence ~1 8 0 o KI(A) --->KI(B) ~K (A, q~) s176 ~K~ is exact. If q~ : A~A/q is the natural map, then KI(A, q) ~KI(A) -~K*(A/q) ~K~ q) -~K~ ->K~ is exact. (In the second sequence, K~ q) =K~ 9), and dl is induced by the inclusion GL(A, q) cGL(A).) Pro@ -- Exactness at K~ v~ P, :% Q)) -~ q~~ = YB PB-'(BQ B -- o 519 3 6 H. BASS (since ~ : PB~QB). Suppose q0~ =yBPB--~,BQB=o. Then there exists an 0c : PB|174 for some n, and hence Y~P--'(AQ= d(R( P| 0~, Q| Exactness at K~ q~). If ~eGL(n, B) then dS(WB~ ) = d(R(A", ~, An)) = y~A"--yAA" = o. Suppose d(R(P,e,Q))=gaP--TaQ=o. Then P|174 for some P', and we can even arrange P@P'~-A". Hence (P| 0~| Q| f~, A") for some ~GL(n, B) and R(P, 0~, Q)-----=~(W~(~)). Exactness at Kt(B). If ~eGL(n, A), then ~qr = ~(W~q~) = R(A", 0~B, A") (note ~e=eB). But the commutative square A"B ~B> A"B ~B I l~nB A,B ~*"~ A,B shows that R(A n, eB, A n ) =o. Now suppose ~eGL(n, B) and 3(WB~ ) =o. Then, by (4) above, a = (A ~, ~, A "),-,o. This means that a|162 for some automorphism s, with WB~---o. We can add an identity to both sides and further assume P=A" and ~ = (A "+m, y, A~+"). Then the isomorphism a|162 is given by isomorphisms f, geGL(n+m, A) making g,+m f~ 1,~> B,+, ' gB fB gn+ m __ "( ~ Br~ +m commutative. Hence, in GL(n+m,B),~=(?g)-l"~(?f). Since WBy=o we have = = Now suppose q0 : A-+A/q. Exactness at KI(A). If eeGL(A, q) then q~4(Wqe) =q~l(WAa ) =WA!o(q)~ ) =% since q~e----I. Suppose aeGL(A) and qo~(Wxa)----WA!q(q~e)=o. Then q)aeE(A/q), so it follows by Homotopy Extension (I.I) that there is an seE(A) with q~,=q)e. Hence ~-~EGL(A, q) and WAe----WA(~s -~) =da(Wq(~s-1)). If we knew how to define, and extend the exact sequence to, K 2 the next result would be an immediate corollary. Proposition (x3.2). -- Suppose A=A0| q (as abelian group) with A 0 a subring and q an ideal. Then there are split exact sequences o-+K'(i, q)~K'(A)~K'(h0) -~o, i=o, I. 520 K-THEORY AND STABLE ALGEBRA Proofi -- Let r o be the retraction with kernel q. Since r has a right inverse, so does r i=o, :. Hence the Proposition follows from (I3.:) provided we show Ki(A, q)-+KI(A) has a left inverse. If a~GL(A), a----~oa: where ~o=q~ceGL(Ao), and a:-----(q~o~)-lo~EGL(A, q). If also ~eGL(A) then (0C~) 1=q)(0~)-16r = (~9~)--1(q)0r = (q9{~)--10~1~ = [q)~, 6r = [~0, ~ll]gl~l 9 Now [D0, ~7']e[GL(A), GL(A, q)] =E(A, q). Hence W,(~):=Wq~+Wq~:, and we have constructed a homomorphism GL(A)->KI(A, q) whose restriction to GL(A, q) is Wq. This clearly induces the required retraction K:(A)--~K~(A, q). Examples. -- :) If A is a local ring (e.g. a field) then K~ and K~(A) is the commutator quotient group of A*---- GL(:, A). The latter is due to Dieudonn6 [i 7] for division rings, and to Klingenberg [24] in general. 2) If qCrad A (Jacobson radical) and r : A~A/q, then q~l is easily seen to be surjective, and (p0 to be injective (see Lemma I8. i below). Hence K~ q) -----o. The methods of Klingenberg [~4] adapt without essential change to compute KI(A, q) also in this case. In case A is q-adic complete, or if T has a right inverse, then cp ~ is even an isomorphism. 3) The following remark is often useful. Let A be semi-local and P, Q eobj ~(A). Then yAP=yAQ=>P-~--Q. For yAP=YAQ implies p| " for some n, so our conclusion follows from (6.6) by induction on n. 4) In w I6 we describe in detail the exact sequence associated with the embedding of a Dedeking ring in its field of quotients. w 14. Algebras. Tensor products endow our functors with various ring and module structures, and it is convenient to record these circumstances now. Let A be a commutative ring, A and A' A-algebras, and q an ideal in A. If P~(A) and P'e~(A') then P'QAPe~(A'QAA), and this induces a pairing, (: ') K~ ') x K~ ~ K~174 If ae~(A)q[t, t-l], then :e,Qoce~(A'|174 t-t] and this induces a pairing (2') K~ ') x K:(A, q)-+KI(A'QAA, a'| Taking A'----A=A in (i'), K~ becomes a ring. Then with A'=A in (i') and (2'), K~ and KI(A, q) become K~ Moreover, the pairings are K~ so they define (i) K~ ') |176 --> K~ (A '| and K~174 q)--+KI(A'| A'Nq). (2) the obvious naturality properties with respect to A-algebra These structures have homomorphisms. 521 3 8 H. BASS In order to treat K ~ and K 1 simultaneously we shall sometimes consider the following situation. Let q be an ideal in A. For an A-algebra, A, write K*(A, qA)= K~174 qA). K*(A, qA) is, as noted above, a K~ Moreover, (~) above gives us K'(A, q)|176 qa)= KX(A| qQA). Hence, if we view K*(A, q) as a graded ring, zero in degrees 52, then K*(A, qA) is a graded K*(A, q)-module. Finally, if A-+B is a homomorphism of commutative rings, then B@ A induces K*(A, q)--~K*(B| , B| In case B is finitely generated and projective as an A-module, then there is an obvious " restriction " functor ~(B|174 , t -t] --->~(A)q[t, t-t], and this induces a homomorphism K*(B| , B| q) which, following the one above, gives the homothetie of K'(A, q) defined by v A(B)eK~ w 15, A filtration on K ~ There seem to be several " geometrically reasonable " filtrations on K ~ (under suitable circumstances). We choose one here with the properties needed for our applica tions. Let A be a commutative noetherian ring, and X=max(A). All modules and A-algebras will be understood finitely generated as A-modules. Let A be an A-algebra. We shall consider complexes, C : ...-+Pi->Pi_l-+... of right A-modules which are finite and projective (i.e. Pi is projective, and = o for almost all i). Then Z(C)=zA(C)=Y~i(--I)'yAP~eK~ is defined. Our finiteness conditions guarantee that H(C) is a finitely generated A-module, so supp H(C) is a closed subset of X. Since localization, being exact, commutes with homology, we see that supp H(C) ={xeX I H(C)x+ o} ={xeX]Cx is not acyclic}. K~ consists of all ~eK~ with the following property: Definition. -- Given Y closed in X, there is a complex C, as above, such that z(C) =4, and codimy(Yn supp H (C))~i. Proposition (x 5. x). -- The K~ are subgroups of K~ which satisfy: :) K~ = K~ K~ K~ 2) If P is another A-algebra the pairing T : K~174176176174 (see w I4) 522 K-THEORY AND STABLE ALGEBRA induces K~174 K~ K~ O~,F);, j. In particular, K~ is a filtered ring, and K~ a filtered K~ 3) The .filtration is natural with respect to homomorphisms of A-algebras. 4) K~ for />dim X. A very useful consequence of this Proposition is Corollary (I5.2). --If d=dimX, then K~ Proof of 15. i. -- K~ is a subgroup because the support of a direct sum is the union of the supports, and the codimension of a (finite) union is the infimum of the codimensions. I) follows from the definition. 2) Let 4~K~176 and let Y be closed in X. Choose C' over A with zA(C') =4 and codimy(Yrasupp H(C'))>i. Choose C" over F with )q.(C") ----~ and codimyqsuppu(c,)(Yr~suppH(C')r~suppH(C"))>~. Then, if C=C'| over A| , it is clear that ZA|174 Moreover the inequality above implies codimy(Yc~supp H(C')r~supp H(C"))>i+j. Hence we can conclude by showing that supp H(C) csupp H(C')nsupp H(C"). But if, say, C~ is acyclic, then it is homotopic to zero (a finite acyclic complex of projective modules), so likewise for = cx. 3) Let f: A-+r be a homomorphism of A-algebras, and ~eK " 0 (A)i , . we want f~176 Given Y, choose C over A with 7,A(C)=4 and codimy(Ynsupp H(C))>i. Since f~ ) it suffices to note that supp H(C| Csupp H(C). But if C~ is acyclic it is homotopic to zero, so likewise for C~| ~ = (C| 4) Take Y=X in the definition. Proposition (z5.3). - K~ n ker(K~ -,K~ xEX Proof.- If 4eK~ and Y--{x} then 4 = 7.A(C) with codimy(Yr supp H(C))>I. This implies xr H(C), so C~ is acyclic. But then the image of ~ in K~ is z (cx) = o. Conversely, given 4 = Y.~.P--YAQeK~ to belong to the right side of our equation means y&P~=y~Q, for all x, and then by (6.6) (see Example 3) in w I4) , P~--~Q~. Now, given Y closed in X, let Yi, 9 9 Y, be the irreducible components of Y, and choose xieYi--U~..~Y i. Then, if we reduce modulo the product of the xi, P and Q become isomorphic. Lift such an isomorphism to f: Q~P; then f*i is an isomorphism, i <i<s. Let C be the complex with f the differential in degree I, and zero in all degrees 4=0 or I. Then z(C)=(--i)lyaQ+(--I)~ Since Y~r codimy(Ynsupp H(C))~I, as required. Proposition 15 . 3 gives a description of the first term of the filtration which behaves well without any finiteness assumptions on A, as we shall see below. Let A be now an arbitrary commutative ring, and let spec(A) denote its prime 523 4 ~ H. BASS ideal spectrum (Zariski topology). If Peobj ~(A) and xespec(A), then P, is a flee A,-module of rank pp(x), pv : spec(A)->Z. It is easy to see that pp is continuous (for the discrete topology on Z). Let C(A) denote the ring of all continuous functions from spec(A) to Z. Since Pv is (clearly) additive and multiplicative in P it induces a ring homomorphism p : K~ p even has a right inverse q~. To define % suppose f: spec(A)~Z is continuous, and let X,=f-l(n). Since spec(A) is quasi-compact, X~=D for almost all n. Now disjoint decompositions spec(A)=O,X~, all X, open, correspond, bijectively, to decomposi- tions l--E,e, of I as a sum of orthogonal idempotents, almost all zero (see e.g. [3% Chap. I]). Define ~(f)=Z,f(n)yA(Ae,) ; ~? is clearly the desired right inverse to p. We shall use q~ to identify C(A) with a subring of K~ Thus, K~ = C(A)| where J(A)=kero. Suppose ~=y,P--yAQeJ(A). Then p~=pe--pq=o. Thus for all xespec(A). Conversely, if P~Q~ for all xemax(A), then we see by localizing in two steps that Px~Q~ for all xespec(A), so yAP--yAQeJ(A). Let us summarize: Proposition (x5.4). --- Let A be an arbitrary commutative ring and let J(A)= fl ker(K~ -+K~ x C max (A) Then K~ = C(A)| where C(A) is isomorphic to the ring of all continuous functions from spec(A) to Z. J(A) is both the nil and Jacobson radical of K~ Proof. -- It remains to prove the last assertion, and it clearly suffices to show that every ~ = yP-yQeJ(A) is nilpotent. If A were noetherian and of finite Krull dimension this would follow from (x5.2) and (15.3). In general, however, ~ is induced from a finitely generated subring of A, and such a subring is of the latter type. Hence our conclusion is a consequence of the following lemma: Lemma. -- If f: B-+A is a homomorphism (commutative rings, then ( fo)- ~ (J ( A ) ) zJ(B), with equality if f is injective. Proof. -- If ~eJ(B) we want f~ i.e. if xespec(A) we want f0(~) to go to o in K~ But B--->A-+A x is the same as B-+Bv-+Ax, where y =f-l(x) ~spec(B) and ~ goes to o in K~ Conversely, if ~K~ we want to show" f~ Thus, given 524 K-THEORY AND STABLE ALGEBRA yespec(B), we must show that ~, the image of ~ in K~ is zero. Let S =f(B--y); since f is injective, this multiplicative set in A does not contain zero. Moreover B t> A Bu s-,~ S_IA is commutative. If S-1~ is the image off~ in K~ then S-I~=(S-~)~ Since By is local, ~=neZ=K~ and hence also S-1~=nEK~ But our hypothesis implies that this n vanishes in K ~ of any localization of S-1A, and hence n=o as desired. Nowfor any ~eK~ we can write ~=p~§ (~--p~), with ~ee(A), ~--p~eJ(A). We shall call 9~ the rank of ~, and often identify it with a function from spec(A) to Z. In particular, P(YP) =Pv is the " rank" of the projective module, P. Expressions like " rank ~> r ", reZ, make sense now, where we think of r as a constant function. If P is a projective A-module, rank P>r is equivalent to "f-rank P>r " in the sense ofw 7- Proposition (I5.5). -- Let A be a commutative ring and P a finitely generated projective A-module. Then the following conditions are equivalent: a) P is faithful. b) rank P>I. c) Every K~ annihilated by yAP is torsion. d) 1| generates the unit ideal in Q|176 Proof. -- a).r because P is locally free, and a finitely generated projective module is faithful if, and only if, it is locally non zero. b) => c). Since pp is everywhere positive and bounded, there is a function feC(A) with fpp=n>o. Hence fyP=n+j, with jeJ(A) nilpotent. Thus, modulo yP.K~ n is nilpotent, so n'~eyP.K~ for some m>o. c) => b). If xespec(A) and P~ = o, then K~ is a K~ annihilated by ~,P. c)~d) is evident. Proposition (x5.6). -- Let A be a commutative ring for which max(A) is a noetherian space of dimension <d. Then, if ~eK~ has rank>d, ~=yP for some P. If rank P>d and gP=yQ, then P~-Q. Proof.-- We can write ~=yQ--yA'. Then pq=p~§ Hence, by Serre's Theorem (8.2), Q~P| so ~=yP. If yP=yQ for some P and Q, then P|174 for some projective P'. If rank P>d we can invoke the Cancellation Theorem (9.3) and conclude that P=~Q. tl CHAPTER IV APPLICATIONS w I6. Multiplicative inverses. Dedekind rings. Throughout this section all modules are finitely generated right modules. Proposition (xfi. I). -- Let A be a commutative ring with max(A) a noetherian space of dimension d<oo, and let A be a finite A-algebra. If P is a projective A-module such that I@yA1 ) generates Q|176 as a (QQzK~ then there is a projective A-module Q, such that QQAP is A-free. Pro@ -- i| n is a Q|176 multiple of I| , and by choosing n large we can solve INyAAn=I| P) with ~eK~ Hence yAAn--~.yAP is a torsion element in K~ If then we replace n by a muldple of n, we can achieve yA A" = "~. YaP" Let xEmax(A); K~ and the image there of ~ is (p~,)(x) (see w 15 for the definition of p). The equation above together with (6.6) (see also Example 3) in w i3) implies that A~- (p~) (x). Px, and this evidently implies (p~) (x) >o. Thus, if we replace n by a further multiple, if necessary, we can achieve p~>d. It then follows from (I 5.6) that ~ =YAQ, for some projective A-module Q. Our equation above then becomes ya(A')=yA(Q| Since, by this time, n>d, we can invoke the Cancellation Theorem (9.3) and conclude that QNAP~A", as desired. Corollary (I6.2). -- Let A be any commutative ring and P a faithful projective A-module. Then QQA P is A-free fi~r some projective A-module Q. Proof. -- Since P is induced from a finitely generated subring of A, it suffices to solve our problem there, so we may assume A a noetherian ring of finite Krull dimension. We can now invoke (16.1) provided I| generates QQK~ but the latter is a consequence of (I5.4[). Let A be a commutative ring and m the class of faithful projective A-modules. Write P~Q if PQAA"=~Q| "~ for some n, m>o. Then ~ is an equivalence relation respecting | so that M(A) =m/~ is an abelian monoid with neutral element the class of the free modules. Corollary 16.2 says that M(A) is even a group. Proposition (x6.3). -- Let A be a commutative ring for which max(A) is a finite dimensional noetherian space. Then M(A) = GL(I, QQzK~ Q). 526 K-THEORY AND STABI.E ALGEBRA 43 Proof. -- Let uswrite B-~Q|176 and B*~-GL(t, B). If P~m, let ~P denote its class in M(A). By (15.4) i| let ~zP denote its class in B*/O*. Evidently f: M(A)~B*/Q* by f(~P) ==P is a well defined homomorphism. Suppose ~zP= i ; i.e. I| Since K~ ----- C(A)| (i 5. 4), we have B ---- (Q|174 (Q| and writing yP=~p+(-(P--D) we see that IQyP= IQppeQ'. Hence n(~P--pt~ ) =o for some n>o, so nyP=y(A"| Choosing n sufficiently large we conclude from (I5.6) that A"QP~A", so ~P= i. This shows thatfis a monomorphism. To see that f is surjective consider an element of B*. Modulo Q* we can assume it has the form i| with ~_K~ of rank~dim max(A). Hence ~----yP for some P by (I5.6), and the class of ~ mod Q* is nP--f(~P). The classical Steinitz-Chevalley theory [I5] of modules over a Dedekind ring furnishes a familiar setting in which to illustrate the general shape of our theory. We consider finitely generated torsion free (hence projective) modules P, over a Dedekind ring A. First P~aGF, a an ideal and F free -- Serre's Theorem. If POA:-~-P'OA then P-'~-P'; if we required rank P~2 this would be the Cancellation Theorem. The stronger conclusion here is possible only because of commutativity. If a and b are non zero ideals, a|174 As an equation in K~ we recover this from the nilpotency of J(A) (Proposition I5.4): (I--ya)(i--yb) =o in K~ We indicate below (Proposition i6.4) how to recover the actual isomorphism, as well as a variety of similar identities. With these facts we see easily that, as a ring, K~174 with J(A) an ideal of square zero, additively isomorphic to the ideal class group G of A. Alternatively, if I is the augmentation ideal of the integral group ring ZG, then K~ 2 as an augmented ring. Let ? : A-+L be the inclusion of A in its field of quotients. We propose now to interpret the exact sequence KI(A) ~K~(L)~K~ ~) -~K~ K~ q~o dim The composite K~176 is the augmentation, " rank ", with kernel J(A)~--G. Next we recall that K~ ?) is built out of triples (P, a, O) with P and O A-projective, and ~ : L|174 an L-isomorphism. Using the description of K~ ?) given in w 13 one can show easily that every element is represented by a triple (a, u, A) with a an ideal and ueL*, viewed as a homothetie of L. Moreover, the fractional ideal au is an invariant which defines an isomorphism of K~ ?) with the group of fractional ideals in L. With this identification, K~ ?)-+G=~-ker c? ~ assigns to each ideal its class. Moreover, K ~(L) d~,.~ L*, and KI(L)-,K~ ?) assigns to ucL* the principal ideal Au. Thus, thekernelis A*=GL(I,A)--Imq~ ~. Finally, kerq~a----SL(A)/E(A) is the commu- tator quotient group of SL(A). We shall see in w 19 that ifA is the ring of integers in a (finite) algebraic number field, then this group is finite. 527 44 H. BASS Now let A be any commutative ring. If P is an A-module, and n a non negative integer, let nP denote a direct sum of n copies of P, and P| a tensor product of n copies of P. Also, if Qis another A-module, write P+Q=P| and PQ=P| With these conventions, if f(T1, ..., T.) ---- Zaq...i,T lq. . .T~"eZ[TI, . .., "In] is a polynomial with non negative coefficients, and if P1, -.., P, are A-modules, we can write 9 ., pO q f(P1, 9 P~) =Y, ai,. ,~ 1 .-.e~'~. Proposition (16.4). -- Let A be a commutative ring such that max(A) is a noetherian space of dimension <n. Then, if P1, ..., P, are invertible A-modules and if S,(T1, ..., "In) is the i th elementary symmetric function, we have Z Si(P1, ..., Pn)= ~oSr ..., P~). i even i Proof. -- I -- yPieJ (A) for all i, so II~= 1 ( i -- YPi) = o by (i 5.2). Thus the equation is valid after applying y to it. Since the ranks of the two sides of the equation exceed dim max(A), (15.6) permits us to remove the y. w z 7. Some remarks on algebras. For a ring A let A-mod and mod-A denote the categories of left, respectively, right, A-modules. A generator for such a category is a module whose homomorphic images suffice to generate any other module. Let E be a right A-module and put F----HOmA(E , E) and E* ---- HOmA(E , A). We are in the situation (tEA, AE~.), and there are natural bimodule homomorphisms: E*| (~) EQAE*--~ I" Moreover, we shall consider the functors: (2) E*| r : F-mod~A-mod. E| A : A-mod--~F-mod The following basic result is due essentially to Morita [26], although the best exposition of these, and other matters in this section, is in Gabriel [21, Chapter V, w i]. Theorem (Morita). -- Let A be a ring, E a right A-module, F=HomA(E , E), and E*= HOmh(E , A). Then the following conditions are equivalent: a) E is a finitely generated projective generator for mod-A. b) The homomorphisms (i) are isomorphisms. (It suffices that they be epimorphisms.) c) The composites of the functors (~) are each naturally equivalent to the identity functors. In this case q<-->Eq defines a b~iection between the two-sided ideals of A and the I'-A-submodules of E. Conversely, if A and F are rings and A-rood is equivalent to P-mod, then any such equivalence is isomorphic to one as in (2) above, with E determined up to F-A-isomorphism. Thus, the situation above is entirely symmetric with respect to E and E* and to A and F. 528 K-THEORY AND STABLE ALGEBRA Now let A be a commutative ring and A an A-algebra. Then, in the above setting, A-rood is an " A-category " (i.e. all the Hom's are A-modules), likewise for the A-algebra P, and the functors in (2) are " A-functors" (i.e. induce A-homomorphisms on the Horn's). The converse part of the Morita Theorem remains true also in this sense, provided the given equivalence from A-mod to P-mod is assumed to be an A-functor. If A-mod and F-mod are A-equivalent we shall call A and P Morita equivalent, denoted A~F. Let E be a finitely generated, projective right A-module. It is easy to see (e.g. using Proposition 15.5) that E is then a generator for mod-A r E is faithful. Using the Morita Theorem we then see that, if F=HomA(E , E), F|174 , EQAA)~-~MA. In this situation we shall write A~B(F| ) and call the two algebras Brauer equivalent. Suppose A,-~MA' , say A'=HOmA(P , P) with P a finitely generated projective generator for mod-A. Can we show that A~BA' , i.e. that the two equivalence relations are the same ? Suppose there is a faithful, finitely generated, projective A-module Q, such that Q| A. Then we have A',--BHom * (Q, Q) | =~ Hom A (Q| P, Q| P) Hom A (A n, A")--~ Hom A (A", A") | A--,~A. Thus, modulo the existence of O, ~M and ~B agree. On the other hand, Proposition 16. i gives a criterion for the existence of Q which we will verify under suitable conditions below. Let A be an A-algebra, A ~ the opposite algebra, and A*=ANA A~ To avoid confusion we shall use E to denote A viewed as a right A-module. A is called an Azumaya algebra (=central separable algebra in [4], see also [21, Chapter V, w I]) if it satisfies the following equivalent conditions: (i) A is a projective generator for A%mod. (ii) a) E is a faithful, finitely generated, projective A-module. b) Ae~Homa(E, E) as A-algebras. In this case every ideal of A has the form qA for some ideal q of A [4, Corollary 3.2]. We propose now to consider the K*(A, q)-module K*(A, qA), with A an Azumaya algebra. For this purpose we use the diagram of functors mod-A mod-A l' (x)AeE @(A ~ AAe) (A '?" AE) mod-A ~*~~ mod-A * mod_(A| in which, by the Morita Theorem and condition (ii) a) above, the vertical arrows are equivalences. We obtain thus K*(A, q)-homomorphisms (*) K*(A, qA)LK*(A, q)--->K " " (A, qA), .:~;AA> 4 6 H. BASS To compute them, let M be a right A-module. Then (M| A~ ) | | ') (A| --~ (M| | (AeNh ~E ) ~-- M| It follows that gf is the homothetie defined by yA(E)~K~ Next we note that (M|176174174174176174174 (as A-modules), recalling that E =A viewed as a left A e-, right A-module. Hence f is the " restriction " map obtained by viewing A-modules as A-modules. In particular, yA(E)EIm(f). Now by Proposition :5.5, :|165 is a unit in Q|176 for any faithful, finitely generated, projective module P. Hence, using (ii) a), we see that O| and QGgf are isomorphisms, and we have proved: Theorem (I7.X). --- Let A be a commutative ring and A an Azumaya A-algebra. Then for all ideals q in A, Q QzK*(A, qA) is a free Q QzK*(A, q)-module generated by :| for any finitely generated faithful projective A-module P. Using Proposition I6. : and the discussion above we further obtain: Corollary (x 7. 2). -- Let A be a commutative ring for which max(A) is a finite dimensional noetherian space, and let A be an Azumaya A-algebra. Then: :) If P is a faithful, finitely generated, projective A-module, there is an A-module Q. of the same type, for which P| Q is a free A-module. 2) The class of A in the Brauer group of A (see [4]) depends only on the A-category A-rood. Theorem :7. : is not very useful for computing K~ since, in number theoretic contexts, all the interesting invariants are torsion. On the other hand the theorem is no longer true if we remove the " Q| ". To see this, let F=O(%/p) with p a prime - -- : mod 4, let I denote the ideal group of F, P the subgroup of principal ideals, and P+cP those principal ideals generated by totally positive elements. If A is the ring of integers in F, then %//~A~P, | For if %/fiA had a totally positive generator there would be a unit in A of norm --:, and by [23, p. 288] there is no such unit. Hence [P : P+]):, in fact, =2. Let Z be the standard quaternion algebra over F and A a maximal order in Z. Then Z is unramified except at ~, so A is an Azumaya A-algebra. On the other hand, Eichler [:9] has completely determined K~ (see Swan [33, Theorem 2]). Namely, K~174 +. Hence K~ is not isomorphic to K~174 (see w :6). This example was pointed out by Serre. A further amusing example in this connection is the following: Let Y. be the quaternion algebra over Q ramified atp and 0% p a prime -=-- : mod 4 and sufficiently large (e.g. p=:: will do). Z has a basis :, i,j, k with z'2=--:,j~=k2=--p, and ij = k =--ji. The result of Eichler-Swan quoted above shows that, if A is a maximal order in Z, then M(n, A) is a principal ideal ring for all n>__2. On the other hand, according to Eichler [I8, Satz 2], the class number of A itself is ):. Finally suppose A is a field and A is a central simple A-algebra. Then K~ so if B is any field extension of A, K~176174 is a monomorphism. What 630 K-THEORY AND STABLE ALGEBRA 47 about Kt? Dieudonn~ [17] shows that KI(A) is the commutator quotient group of A*=GL(I, A). Therefore, if B is a splitting field for A, we have homomorphisms A* ~ K1 (A) ---> K~ (B| ~ B *, and it is easy to see that the resulting map A*--~B* is the reduced norm (see Bourbaki, [1I, w I2]) and has image, therefore, in A*. Combining these remarks with the fact that every field is contained in a splitting field, we have: Proposition (x 7 . 3 ). -- If A is a central separable algebra over afield A, K I(A) ~ K l(B| x A) is a monomorphism for all field extensions B if, and only if, the commutator subgroup of A* is the kernel of the reduced norm. By a theorem of Wang [35], this is the case if A is a number field. w x8. Finite generation of K. We propose to show (Theorem 18.6 below) that if A is a finite Z-algebra, then K*(A, q) is a finitely generated abelian group for all ideals q. Lemma (I8. x). -- a) Let A be a ring. If N is an ideal in rad A, then K~ ~K~ is a monomorphism, and even an isomorphism if A is N-adic complete (e.g. if N is nilpotent). b) If A is a semi-local ring, K~ is a free abelian group of finite rank. Proof. -- a) Suppose yAP---~AQ~ker(K~176 After adding a free module to P and Q we can assume that P/PN~----Q/QN. Such an isomorphism lifts to a homomorphism f : P~Q. f is surjective mod N, hence surjective, by Nakayama. Hence kerf, being a direct summand of P, is finitely generated. Since kerf is zero mod N, it too is zero, by Nakayama. Suppose now that A is N-adic complete, and let P be a projective (A/N)-module, say P| (A/N)". If P =Homa(A" , A n) =M(n, A), then r/Nr =M(n, A/N). Let eeP/NF be an idempotent projection onto P. Since F is (NF)-adic complete, e lifts to an idempotent e'er (see e.g., [16, Lemma 77.4]). Now P'=Ime' is a direct summand of A n covering P, so ~,A/.~P is the image of yA P'. b) If A is semi-local, A/rad A is a finite product of simple Artin rings. Hence K~ A) is free abelian, of rank equal to the number of simple factors of A/rad A. By a) K~176 A) is injective, so our conclusion follows. Proposition (z8.2). -- Let A be a noetherian integral domain of Krull dimension one with field of quotients L. Let A be a finite A-algebra, N the nil radical of A, and T the torsion A-submodule 0fA/N. Then N is nilpotent, T is a semi-simple Artin ring, and A/N - T � r (product of rings), where P is an A-order in the semi-simple L-algebra L| A P. Proof. -- L| is a nil, hence nilpotent, ideal in the finite dimensional L-algebra L| Hence some power of N lies in the torsion submodule of A and is therefore a nil ideal of finite length. Therefore some further power of N is zero. For the rest we may assume N=o, i.e. A = A/N. Regard T as an A-algebra, possibly without identity. If J =rad T, J can be described as the intersection of all 531 4 8 H. BASS kerr, wheref is a T-homomorphism into a simple right T-module S, such that ST = S. (Note that S must be an A-module, andf compatible with this structure, in particular.) Suppose g : T-+T is a right T-endomorphism and f : T-+S as above. Then fg : T-+S so Jckerfg; i.e. g(J)Ckerf. Letting f vary we see that g(J)cj. Now letting g be left multiplication by an element of A we see that J is a left A-ideal (using the obvious fact that T is an ideal of A). Similarly, J is a right ideal. However, T has finite length as an A-algebra, so J is nilpotent. But A now has no nilpotent ideals . o, so J = o. Hence T is semi-simple, so it has an identity element, e. If a~A, then ae~T so ae = eae. Similarly ea=eae, so e is central. Thus A=T� where I ~=(I-e) A. Since I'is torsion free with zero nil radical, I'r174 and L| is a semi-simple L-algebra. Lemma (x8.3). -- Let A be a ring and N a nilpotent ideal finitely generated as a Z-module. Then, for all n> I, every subgroup of GL(n, A, N) is a finitely generated group. Proof. -- Induction on m, where N m = o, reduces us immediately to the case N ~ = o. Then GL(n, A, N) consists of all 1 -~- a where a is an n � n matrix with coordinates in N. If I 4- a' is another, then (I 4- a) (I 4- a') = I 4- (a 4- a'), so GL(n, A, N) ~N (the additive group). Theorem 18. 7 below will be proved by a reduction to the following classical results: Lemma (I8.4). -- Let A be an order in a finite semi-simple Q-algebra E=Q| o) (Jordan-Zassenhaus, see [37]). If M is a finitely generated E-module, there are only finitely many isomorphism types of finitely generated A-submodules of M. i) (See Siegel [3 I] or Borel-Harish-Chandra [9]) GL(n, A) is finitely generated for all n~ I. Moreover, if Z is simple, the subgroup of elements of reduced norm I in GL(n, A) is likewise finitely generated. Proposition (x8.5). -- Let A be a finite Z-algebra, and let q be an ideal in A. Then GL(n, A, q) is finitely generated for all n> I. Proof. -- If N is the nil radical of A, then GL(n, A, q)-+GL(n, A/N, q(A/N)) is surjective, and GL(n, A, q)r~GL(n, A, N) is finitely generated by (18.3). Hence we can reduce to the case N=o. Then A=T� as in (18.2) and GL(n, A, q) splits likewise into a product. It suffices then to treat T and r separately, and T, being finite, causes no problem. The result for P is a consequence of Siegel's theorem above (see Lemma I9.4 below). Theorem (x8.6). -- Let A be a finite Z-algebra and q an ideal in A. Then K~ and KI(A, q) are finitely generated abelian groups. Proof. -- Since dimmax(Z)=I it follows from (11.2) b) that KI(A,q) is a homomorphic image of 13I.(2, A, q), and (18.5) says the latter is finitely generated. Now for K ~ : Let N be the nil radical ofA and write A/N=T� asin(i8.2). Then, by (18.I), K~176176176 is an isomorphism, and K~ free abelian of finite rank. It remains to show K~ finitely generated. Let I" be a maximal order in O| containing r. Then I" is hereditary [3], so every projective I"-module is 532 K-THEORY AND STABLE ALGEBRA isomorphic to a direct sum of right ideals [I4, Chap. I, Theorem 5.3]. By Jordan- Zassenhaus (18.4, o)) there are only finitely many of these, up to isomorphism, so K~ ') is finitely generated. Now P has finite index, say m, in F'. Let S be the multiplicative set of integers prime to m. Then S-1F is semi-local, so K~ is free abelian of finite rank (18. r, b)). It will be sufficient for the theorem, therefore, to show that the homomorphism K~176176 induced by the inclusions, has finite kernel H. Let ,(rP--yrFncH; we may assume n>2. Then vr,(r'| and ys_.r(S-1P)=ys_,r((S-1F)"). Since F' and S-lP are algebras over rings with maximal spectra of dimensions I and o, respectively, and since n>2, (15.6) tells us that P| and S-1P~(S-1F) ". Let xemax(Z). If mex then P~ is a localization of S-1P, hence free. If mr then Px is a localization of P| r', hence free. Thus P~ is free of rank n for all xemax(Z), so we can apply Serre's Theorem (8.2) and write P~Q| "-t, with Q locally free of rank one, by (6.6). But then Qis isomorphic to an ideal in P, and again, byJordan- Zassenhaus (i8.4, o)), there are only finitely many such Q up to isomorphism. Since u F, this proves H is finite, as claimed. w 19. A finiteness theorem for SL (n, A). Let Y,=IIi~ ~ with Z i a central simple algebra over a finite algebraic number field C i. The reduced norms (see [ii, w I2]) give homomorphisms GL(n, Y~)-+C*=GL(I, C~) (compatible with the inclusions GL(n)r and their product defines a homomorphism GL(n, Z)-+C*, where C = II~C~ = center Z. We shall call this also the reduced norm, and denote its kernel by SL(n, 11). If A is an order in X and q an ideal[ in A we shall write SL(n, A) =GL(n, A)nSL(n, Y,), and SL(n, A, q) = GL(n, A, q)nSL(n, I1). Theorem (i9.i). -- Let 11 be a semi-simple algebra finite over Q, let A be an order in Z, and let SL(n, A) denote the elements of reduced norm one in GL(n, A) (in the sense defined above). Then there is an integer no=n0(11 ) such that, for all n>n o and for all ideals q in A, SL(n, A, q)/E(n, A, q) is finite. We shall begin by deriving a reformulation of this theorem which will be useful in its proof. The next two sections are devoted to some of its applications. Lemma (I9.2). -- If q is an ideal in A, there is another ideal q' for which qraq'=o and A/(q -f- q') is finite. Proof. -- Z -= qZ| being semi-simple, and q' = Z' n A clearly serves our purpose. Lemma (19.3). -- If q and q' are ideals with qnq'=o, then GL(n, A, q-k q')=GL(n, A, q) � GL(n, A, q') (direct product), and similarly for SL(n, A, q + q') and E(n, A, q + q'). In particular, K~(A, q +q')= K~(A, q)| q'). g33 7 5 ~ H. BASS Pro@ -- If I +q+q'eGL(n, A, q4-q'), where q and q' have coordinates in q and q', respectively, then I +q+q'= (i +q)(I +q')eGL(n, A, q) � A, q'), since qq'= o = q'q. The conclusion tbr SL follows from the factorwise definition of SL, and for E it follows by applying the reasoning above to its generators. Lemma (I9.4). -- GL(n, A, q) and SL(n, A, q) are finitely generated groups for all n and q. Proof. -- Lemmas 19.2 and 19. 3 permit us to assume A/q is finite, in which case GL(n, A, q) has finite index in GL(n, A), so it suffices to show the latter finitely gene- rated. If I" is a maximal order containing A, then mFcA for some m>o, so GL(n, I', mr) cGL(n, A), showing that GL(n, A) has finite index in GL(n, 1"). Since P is a product of maximal orders in the simple factors of Z, finite generation of GL(n, P) follows from Siegel's theorem (Lemma I8.4, i)). Exactly the same proof applies to SL. Now consider the direct system. ...SL(n, A, q)/E(n, A, q)-->SL(n+ I, A, q)/E(n+ I, A, q)-~.., with limit SL(A, q)/E(A, q). Thanks to Theorem I I . I we can apply Theorem 4.2 to A with n--- 2. Hence we know from (4.2, b)) that the maps above are surjective for n>2, and, from (4.2, f)) and (19.4), that the terms are finitely generated abelian groups for n_> 4. Consequently, since finitely generated abelian groups are noetherian, the system stabilizes; i.e. the maps are eventually all isomorphisms. (Indeed, the conjecture of w Ii alleges they ate isomorphisms already fbr n_> 3. If true, one could take n0= 3 in Theorem 19.1 , as the proof will show.) By the theorem of Wang [35] (see Proposition 17.3) SL(n, Y~) is the commutator subgroup of GL(n, 2;), and, by Dieudonn~ [17] (see Proposition 5. i, b)) the latter is just E(n, N) for n~2. Thus the reduced norm induces a monomorphism KI(Z) = GL(Z)/E(E) ---> C*. Moreover, the inclusion GL(A, q)cGL(E) induces an exact sequence o-)-SL(A, q)/E(A, q)--->GL(A, q)/E(A, q)-+GL(E)/E(Z) ' ,I I] KI(A, q)- > Kt(E) The next corollary summarizes some of these remarks: Corollary (x9.5). -- The following conditions on A and q are equivalent: (i) There exists an n0>2 such that SL(n0, A, q)/E(no, A, q) is finite (resp. trivial). (ii) For the same no, SL(n, A, q)/E(n, A, q) is finite (resp. trivial) for all n>n o. (iii) SL(A, q)/E(A, q) is finite (resp. trivial). (iv) Kt(A, q)-~KI(Y~) has finite (resp. trivial) kernel. Remark. ~ As noted above, the conjecture of w I I asserts that no = 3 already suffices in the corollary. 534 K-THEORY AND STABI,E AI.GEBRA Except for the dependence ot n o only on Z, Theorem 19. I is now seen to be contained in the following result, of which the last assertion has already been noted above. Theorem (x9.6). -- Let A be an order a semi-simple algebra Z, finite, over Q, with center C. Then, for any ideal q in A, KI(A, q) ->KI(Z) has finite kernel, and its image is isomorphic to the image of GL(2, A, q) in C* under the reduced norm. The following sequence of lemmas will permit various reductions in the proof of this theorem. Lemma (x9.7). -- Let G be a group and R a normal subgroup. Then, if n=G/R and G/[G, G] are finite, so also is G/[G, R]. Proof. -- R/[G, R]->G/[G, G] has finite image, so it suffices to see that it has finite kernel. But this is just the second map in the exact sequence H2(~z) ->H0(~., Hi(R))->HI(G) which comes from the Hochschild-Serre spectral sequence in homology with integral coefficients (see [14, XVI, w 6, (4 a)]) for the group extension I --> R --> G -> n --> 1. Since is finite, so is H2(=), and this proves the lemma. Corollary (i9.8). --/f A/q is finite, then so also is E(n, A)/E(n, A, q) for all n>_3. Pro@ -- Put G----E(n, A) and R=GL(n, A, q)nE(n, A). Then G/R is finite, and G= [G, G], by (i .5, (i)). By (4.2, d)) E(n, A, q) = [G, R], so the corollary now follows from (i 9 . 7). Corollary (x9.9). -- If, for some n0_>3, SL(n,,, A)/E(n0, A) is finite, then SL(n, A, q)/E(n, A, q) is finite for all q and all n2no. Proof. --By (19 . 2) and (19 . 3) the conclusion above for all q follows once we know it for q with A/q finite, so we now assume this. If n>no, then the finiteness of SL(n, A)/E(n, A) follows from our hypothesis and (I9.5) , and that of E(n, A)/E(n, A, q) from (19.8) above. Hence SL(n, A)/I~.(n, A, q) is finite, and this proves the corollary. Lemma (x 9. xo). -- Let A and A' be two orders in Z, and let q be an ideal in A for which A/q is finite. Then there is an ideal q' in A' with A'/q' finite such that, for all n> 4, E(n, A', q') CE(n, A, q). Proof.- mA'cA for some m:>o, so qx=A'mqmA' is a A' ideal contained in q, and clearly A'/qx is finite. Let H =GL(n, A, ql) =GL(n, A', ql). Then from (I .3) (using n>3) and (4.2, f))(using n>4) we have : E(n, A', q~)c[E(n, A', q~), E(n, A', q~)] r H] cE(n, A, ql)CE(n, A, q). Hence q'= (ql) ~" serves our purpose. Corollary (x 9. xx). -- If, for some order A' in Z, KI(A')--->KI(Z) has finite kernel, then there is an n o such that, for all n>no, for all orders A, and for all ideals q in A, SL(n, A, q)/E(n, A, q) is finite. 535 52 H. BASS Proof. -- Our hypothesis and (i9.5) imply SL(no, A')/E(n0, A') is finite for some n0_>3, so our conclusions for A' follow from (I9.9). Taking no~4, we can apply Lemma 19. Io to any other A, and conclude that E(n0, A', q')r A) for some q' with A'/q' finite. Since SL(n0, A')/E(n0, A, q') is finite, E(n0, A)r~SL(n0, A') has finite index in gL(n0, A) nSL(no, A'). But the latter contains SL(n0, Ar~A'), which has finite index in SL(n0, A). Therefore SL(no, A)/E(n0, A) is finite, and the corollary now follows from (i 9.9). Corollary 19. i i reduces Theorems 19. I and 19.6 to showing that KI(A) -+KI(Z) has a finite kernel for some A. Since KI(A) is a finitely generated abelian group (Theorem 18.6), we need only show the kernel is torsion, and for this the following crite- rion is useful. It is here that K ~ effectively intervenes in the proof. Proposition (xg.x2). -- Let AcB be commutative rings with B finitely generated and projective as an A-module. Then, if A is a finite A-algebra, the kernel of KI(A)-+Ka(B| is torsion. Proof. -- The nature of B provides us with a homomorphism KI(B| (see w I4) whose composite with the one above is the homothetie of the K~ K~(A), defined by TA(B)eK~ The Proposition now results from (I5.6), which tells us that anything killed by y,~(B) is torsion. We come now to the proof that ker(K~(A)--->K~(Z)) is torsion. Passing to ANzACL| , where A is the ring of integers in a splitting field L for Z, we can reduce, thanks to (I 9. I2) above, to the case where Z is split. By (i 9. iI), moreover, we may take for A a maximal order. But then A is a product of maximal orders in the simple factors of Z, and everything decomposes accordingly, so we reduce further to the case Z=EndL(V), V a vector space over the number field L, and then (see [3] or [I5] ) A----EndA(P), with P a projective module over the ring A of integers in L. Now the Morita theorem (w I7) gives us equivalences from the categories of A-modules to A-modules (| and from L-modules to Z-modules (| which commute with the passages from A to L and A to Z, respectively. Thus we have KI(A) -+K*(L) Kt(A) --> K* (Z) commutative, and it suffices, finally, to show that ker(Kl(A)-+KI(L)) is torsion. Using (I 9. I2) again, we see tlhat this is a consequence of the following proposition: Proposition (*9. x3). -- Let A be the ring of integers in a finite extension L of Q, and let ~eker(Kl(A)--~Kl(L)). Then there is a finite solvable extension F of L, such that ~eker(Kl(A)~KI(B)), where B is the ring of integers in F. Proof. -- By (iI.2, b)) ~=WAe with ~ an automorphism of A 2, and ~eker(Kl(A)---~Kl(L)) simply means det ~----I. Passing to a quadratic extension F 0 of L, with integers B0, we can give % = IB.| an eigenvalue. As an automorphism of F0 2, q. thus has a one dimensional invariant subspace, and since B o is a Dedekind ring, the 536 K-THEORY AND STABLE ALGEBRA latter contracts to a direct summand P0 of Bo 2. Po is invariant under %, and, having rank one, P0~--a, a an ideal in B 0. Since the class group of Bo is torsion (even finite) ah=(a) is principal for some h>o. Let F=F0(~/a ) have integers B. Then P=B| and P is invariant under ~= Is| IB| If we choose a basis for B ~" the first member of which generates P, then ~ is represented by a matrix of the form iu x i__iu oil u tx :i I Io v I Io v!o ij" The second factor is manifestly in E(2, B). Since det ~ = 1, we have v----u -t, so the first factor lies also in E(2, B) by the Whitehead lemma (I.7). Therefore WB(IB| ) =WB(~) = O, as required. Remarks. -- I) Theorems 19 . i and 19 . 6 are probably valid also for semi-simple algebras over a function field in one variable over a finite field. The proof above has two ingredients which are not known, to my knowledge, in that case. One is Wang's theorem. However, this can be circumvented easily since the discrepancy between E(n, Z) and SL(n, )3) is easily shown to be a torsion group for semi-simple algebras over any field. The second point, which I don't know how to supply or outmaneuver, is the finite generation of $L(n, A), say for n>3 (1). Similarly, this is the only point requiring attention if one works throughout, say, with orders in )2 over a ring of the form Z[n-t], for some nEZ. 2) Theorem x9.6 suggests an obvious analogue for K ~ Namely, one can ask that K~176 have finite kernel. Jan Strooker (Utrecht thesis) has pointed out that a necessary and sufficient condition for this is that every projective A-module P, for which Q| P is Z-free, be locally free. He gives examples for which this fails. w 2o. Groups of simple homotopy types. Theorem (2o. x ). Let ~ be a finite semi-simple Q-algebra with q simple factors, and suppose R| has r simple factors. Then, if A is an order in Z and q is an ideal in A, KI(A, q) is a finitely generated abelian group of rank<r--q, and =r--q if A/q is finite. Theorem (20.2). -- In the above setting the following conditions are equivalent: i) KI(A) is finite. 2) KI(A, q) is finite for all q. 3) An irreducible Z-module remains irreducible under scalar extension from Q to R. 4) The center of each simple factor of Z is either Q or an imaginary quadratic extension Of Q. Proof of (2o.i). -- By (~9.2) and (I9.3) we can assume A/q is finite. Let 1TM be a maximal order containing A. Then GL(n, A, q) cGL(n, A)r P) are both (1) This has recently been established by O'Meara (On the finite generation of linear groups over Hasse domains, to appear) for commutative A. g37 54 H. BASS subgroups of finite index, for all n. Thus KI(A, q)-+Kl(r) has finite cokernel. By (I9.6), moreover, the maps KI(A, q)-->Kl(r)-+Kl(Z) both have finite kernel. Hence rank KI(A, q) ----rank K~(r). Now P is a product of maximal orders in the simple factors of ~, and KI(I ') splits accordingly. Since the function r--q likewise adds over the simple factors we can reduce to the case where 52 is simple (i.e. q = I), say with center L. ll| = center R| has the same number r of simple factors as R| and we want to show that rank Kl(P)=r--i. We know from (I9.6) that rank Kl(P) is the rank of the image UCL* of GL(F) under the reduced norm. If A denotes the integers in L, then P being integral over A implies UcA*. On the other hand A*cF*----GL(I, P), so (A*)"cU, where [X:L]=n z. Hence rank U = rank A*. By the Dirichlet Unit Theorem, rank A*=r--i, and this completes the proof. Proof of (20.2). --The equivalence of I), 2), and 3) is an immediate consequence of the above theorem, and that of 3) and 4) is trivial. If rc is a finite group then Z~ is an order in the semi-simple algebra O,~, so we may apply the preceding results. Viewing -t-r~cGL(I, ZT~)cGL(Z~), it makes sense to write Kl(Zr~)/4-rq with a minor abuse of notation. J. H. C. Whitehead showed [36] that if X and Y are finite simplicial complexes of the same homotopy type and funda- mental group ~, then the simplicial homotopy equivalences from X to Y, modulo the simple homotopy equiw~lences, are classified by invariants which live in KI(Z~)/ Herein lies the principal interest of the next result, which elaborates on some earlier work of G. Higman [38]: Corollary (20.3). -- Let r: be a finite group, r the number of irreducible real representations of re, and q the number of irreducible rational representations of =. Then the commutator quotient group of GL(Zr:) is a finitely generated abetian group of rank r--q. There are well known group theoretic interpretations of r and q : q is the number of conjugacy classes of cyclic subgroups of ~z (Artin). Write a~b in 7: if a is conjugate to b Then r is the number of--~ classes (Berman-Witt). Both of these results can be found in Curtis-Reiner [16, Theorem, 42.8]. Examples. -- I) If r: is abelian, then r = q if, and only if, 7~ has exponent 4 or 6. For each simple factor of O~ is a cyclotomic field of n *t' roots of unity, where n lex p r:. These fields are either Q itself or totally imaginary. They have degree <2 precisely when n 14 or 6. 2) The rationals are a splitting field for the symmetric groups and O_v(%/~----i-) for the quaternions. Hence the Whitehead group is finite in these cases. For groups with this property the results of the next section can be used to give a crude bound on its order. It is not inconceivable that it even be trivial. 3) If rc is cyclic of order n, then 7: has 3(n) irreducible Q representations, ~(n) = the number of divisors of n. 7: has In/2] + I irreducible R representations, where [x] = the integral part of x. Hence the Whitehead group has rank In/2]-~ t--3(n) in this case. 538 K-THEORY AND STABLE ALGEBRA 4) In [7] it is shown that the Whitehead group is trivial when n is free abelian. Milnor has asked whether it is always a finitely generated abelian group if rc is. It seems reasonable, though difficult to show, that KI(A) is finitely generated for A any finitely generated commutative ring over Z with no nilpotent elements. The same statement for K ~ would generalize the Mordell-Weil Theorem. w 2x. Subgroups of finite index in SL(n, A). Does every subgroup of finite index in SL(n, Z) contain a congruence subgroup, SL(n, Z, qZ), tbr some q>o? The answer is easily seen to be " no " for n=2, as was already known to Klein. For n>3 , however, the solution is affirmative; a proof is outlined in [4o]. The method consists of an application of the present results to reduce the problem to a (rather formidable) cohomological calculation. The latter, in turn, depends heavily on some recent results of Lazard on analytic groups over p-adic fields. I shall summarize here, in a form adapted to this method, the information provided by the present material. Let A be an order in a simple algebra E, finite over Q. We introduce the following abbreviations in our notation: S = SL(n, A) E= E(n, A); for each ideal q, Sq=SL(n, A, q), Eq=E(n, A, q), and Fq=Ec~Sq. Theorem (2x. x). -- For n_> 2 center S = center E is isomorphic to the (cyclic) .group of n ~ roots of unity in the center of A. For n>_3, a non central subgroup of S normalized by E contains E 0 for some q 4= o, and E/Eq is finite. Hence a normal subgroup oat" E is either finite or of finite index, and the same is true of S as soon as S/E is finite. The latter holds for all sufficiently large n. Proof. -- By (2.4) an element of GL(n, A), n>2, centralized by E, has the form u. I, with uecenterA. Being in S means U"=l, and it then follows from (1. 7) that u. IeE. Since center Accenter Z, a field, the n th roots of unity form a cyclic group. The rest of the theorem is an immediate consequence of (4- 2, e)), (19.8) and ( 19. ~)- To avoid some technical difficulties we shall henceforth assume A is commutative, i.e. E is a number field. We shall be speaking of " profinite " (=compact, totally disconnected) groups, and their cohomology, for which we give Serre's notes [39] as a general reference. If H is any group we denote by t2I its completion in the topology defined by all subgroups of finite index. Since each of the latter contains a normal subgroup of finite index we can describe I2I by I2I= limwH~f nit~.H/H~ 9 This defines a functor from groups (and homomorphisms) to profinite groups (and conti- nuous homomorphisms) which evidently preserves epimorphisms. 5.39 56 H. BASS On the other hand, S and E above can be completed also in the " congruence topology " defined by taking the Sq, resp. Fq, q 4= o, as a basis for neighborhoods of the identity. By Corollary 5.2 the inclusion EcS induces isomorphisms 9 E F,-S/S,=$L(n, A/q), for each q 4= o. Since this group splits uniquely according to the primary decomposition of q, and since lim m SL(n, A/q") = SL(n, ./~,) for q prime, we conclude that S and E have the same congruence completion, II = II SL(n, .X.q). We shall write q prime, * 0 C -- ker(S-+ H) and C0 = ker(l~-+ II ). The question discussed above asks whether the congruence and profinite topologies in S coincide, i. e. whether C = o. Theorem (21.2), -- (i) There is a commutative diagram with exact rows, I ----~ C ----~ g --~ l-I ----> I t t ki I -+ Co-+g-+ II -+ I. Here l I = limq, oE/F, = lim,, oS/S, = 1-I $I.(n, ~). q prime, ~ 0 (ii) For n>_3 Co = limq. oFq/E,, and C ---- libra,, o(S,/E,), and the maps in both of these projective systems are all surjective. (iii) For n>_3, CoCcentcr E. For n>4 , Cffccnter S, and E-+S is a monomorphism. (iv) Consider the following conditions: a) Sq/Eo = {~ } for all q. b) A non central normal subgroup of S contains Sq for some q 4= o. c) A subgroup of finite index in S contains SQ for some q 4= o. d) C={I}. c) (SJE,) ---- {i } for all q. For we have a) b) and for n24 theyareaUequivatent. Similarly, they are all equivalent for n>_3 if we substitue E, Fq, and C O for S, Sq, and C, respectively. (v) Writing S=S(n), C----C(n), etc., to denote their dependence on n, the inclu- sions S(n--i)CS(n) and E(n--i)CE(n) induce homomorphisms C(n--i)-+C(n) and C0(n-- i) -+Co(n) which are surjective for n>_3. Proof. -- Part (i) is contained in the remarks preceding the theorem. It is immediate from (21. i) above that (*) E ---- 0E/E q for n>3, and this makes it evident that Co----li~_m,,oFJE q. S-+S/Eq induces g-->(S~q)-->I, and hence S-->limq+o(S/Eo). It follows from (2I'.I) again that this is injective. It < . 9 1. li+_m_m,. K-THEORY AND STABLE ALGEBRA is surjective since the image is (clearly) dense, and S is compact. Since S/Sq is finite for q 4: o it is now evident that C --ker(fim q . 0(S/Eq) -+ 0S/Sq) admits the description in (ii). The last part of (ii) follows once we show that, if o + q C q', then F,/E,--+Fq,/Eq, and S,/E,--+Sq,/Eq, are surjective. This means simply that FoE q, = Fq, and SqEq, = Sq,. The first of these equations is a consequence of the second, and the second is contained in Corollary 5.2. For n> 3 we know from (4.2, d)) that [E, Fq] =Eq, so it follows from (ii) that C0ccenterE. If n> 4 then from (4.2,f)) we have [S, Sq] =Eq, so it follows similarly from (ii) that CCcenter S. To show that g-+g is a monomorphism it suffices, by (*) above, to show that S/Eq is separated in its profinite topology. But for n>4 , S/E, is a central extension of a finite group S/Sq by a finitely generated abelian group SJEq, using (4.2, f)) and (i9.4), and such a group is clearly separated. This proves (iii). Now for (iv). Assume n>3: a)~b) follows from (2I.I) and (ii). b) ~c) since a subgroup of finite index contains a normal subgroup of finite index, and the center of S is finite. c).,:~d) since c) asserts the coincidence of the profinite and congruence topologies. d)<>e) follows from (ii). If n>4, then Sq/Eq is a finitely generated abelian group, as already noted in the last paragraph, so d):~a). The proof for E is parallel, but the last point is simplified since Fq/EQ is even finite already for n>3. For part (v) it suffices, by compactness, to show that C(n--x)--~C(n) has a dense image for n>3, and similarly for Co. Denseness means that C(n--I) projects onto every finite quotient of C(n). But it follows from (ii) that every finite quotient of C(n) has the form S0(n)/H with Eo(n ) oH. By (4.2, b)), S0(n ) -: S0(n-- r)E0(n ) = S0(n-- I)H, so S,(n--I)/HnSo(n--I ) coming from C(n--~) maps onto S0(n)/H , as required. The proof for Co is identical, after replacing S and S 0 by E and Fq, respectively. In [4 o] it is shown that, when A=Z, H2(II(2), O/Z)=o (cohomology in the sense of [39]), and on this basis that C(n)= o for n=>3. (By virtue of (2I .2, (iv)) this is equivalent to E(q)-= S(q) for all q> o. For q<__ 5 this had been shown by Brenner [I 3] by direct calculation.) In the general case one knows only the following result, which Serre has proved using recent results of Lazard and of Steinberg (Colloque de Bruxelles, I962 ). Theorem (2x.3)(Serre, unpublished) H2(II, Q,/Z) is finite. Plugging this into the argument of [4o], and using the information in (2I.~) and (21.2) above, one obtains: Corollary (2x.4). -- C o /s a finite group for n>3, and C is finite for n large enough, so that S/E is finite. li+.m_mq. 5 8 H. BASS w 22. Some remarks on polynomial rings. Let A be commutative and noetherian, and let B=A[tl, ..., tn] with tl,..., t n (n~I) indeterminates. Grothendieck has shown that, if A is regular, the homomorphism K~176 is an isomorphism (see [29] or [7]). It follows that if P is a projective B-module, then -(BP=yB(B| ) for the projective A-module Q= P/(tl, ..., tn)P. Now this equation in K~ can be replaced by the isomorphism P~BQAQ , provided rank P>dim max(B) (Proposition 15.6 ). At this point the unpleasant fact emerges that dim max(B) = dim spec(B) = n + dim spec(A). Thus, for example, if A is local (so dim max(A) =o), dim max(A[t]) can be arbitrarily large. In any event, we can record the following conclusion, using the fact that: A is regular and dim spec(A) = d r global dim A = d. Theorem (22. i ). -- Let A be a commutative noetherian ring of global dimension d, and let B = A[t~, ..., t,], the t~ being indeterminates. Then a projective B-module of rank > d + n has the form B| for some projective A-module Q. If A is a field we see that projective B-modules of rank>n are free, but we can't conclude this if A is only local. However, we can make a very small compensation in this case (Corollary 22.3 below). For here, in the equation ~.BP---fB(BQAQ), Qwill be free, so we can conclude that P| r, for some r, s; we want P to be free. If we write (P|174174 and apply induction, we are reduced to showing, under suitable hypotheses, that P| r =>P~B r-1. It is easy to see (cf. proof of the Cancel- lation Theorem, 9-3) that this conclusion is equivalent to the assertion that AutB(B ~) is transitive on the unimodular elements of B r. Proposition (22.2). -- Let A be commutative and noetherian and suppose d= dim spec(A)> dim spec(A/rad A). Then, if B--=A[tl, ..., tn], t~ indeterminates, E(r, B) is transitive on the unimodular elements of B y for r>d+n. Remark. -- If we replace d + n above by d-r n -t- i then this Proposition is contained in Theorems 11. I and 4.1, a), since dim max(B) =d+n. Pro@ -- Suppose e= (a~, ..., ar)EB ~ is unimodular; we seek /~E(r, B) such that r = (i, o, ..., o). The remark above, together with our hypothesis, shows that this can be done if we replace A by A/rad A. It follows, using Lemma i. i, that we can find r B) so that zl0~= (al, ..., a;)- (I, o, ..., o)mod radA.g r. Since a' r ~ I mod rad A. B, a maximal ideal of B containing a'~ cannot contract to a maximal ideal of A. It follows that dim max(B/a~B)<n+d--i. Hence we can again apply the remark above, this time to the (unimodular) image of (a~, ..., a;) in (B/a[B) "-t, and transform this image into (I, o, . .., o) with ~E(r-- I, B/a;B). By Lemma i. i , I O I again, ~2 lifts to r B), and we set ~2= ,, sE(r, B). Then ,2~1c~ has O ~2 the form (a[, I + bza'~, b3a'l, ..., b,a[), and it is now clear how to finish with elementary transformations. 542 K-THEORY AND STABLE ALGEBRA From the discussion preceding this proposition we derive the following corollary: Corollary (22.3). -- If, in Proposition 22.2, A is a regular ring for which K~ ~Z, then a projective B-module of rank> d + n is free. As a special case, we have the following corollary: Corollary (22.4). -- If A is a semi-local principal ideal domain (so d~I) then a projective A[q, ..., t,]-module (t i indeterminates) of rank >n is free. This last corollary has recently been strengthened, for n=e, by S. 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H. Bass, S. Schanuel (1962)
The homotopy theory of projective modules
Bulletin of the American Mathematical Society, 68
(1960)
GOLDMAN, Maximal orders
A. Grothendieck (1961)
Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : III. Étude cohomologique des faisceaux cohérents, Première partie
Publications Mathématiques de l'IHÉS, 11
H. Hasse (1949)
Zahlentheorie
M. Eichler (1938)
Über die Idealklassenzahl hyperkomplexer Systeme
Mathematische Zeitschrift, 43
J. Dieudonne (1943)
Les déterminants sur un corps non commutatif
Bulletin de la Société Mathématique de France, 71

Publisher: Springer Journals
Copyright: Copyright © 1964 by Publications mathématiques de l’I.H.É.S
Subject: Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
ISSN: 0073-8301
eISSN: 1618-1913
DOI: 10.1007/BF02684689
Publisher site: See Article on Publisher Site

Abstract

by H. BASS CONTENTS Px~zs INTRODUCTION ................................................................................. 5 CHAPTER [, ---. Stable Structure of the linear groups ......................................... 8 w i. Notation and lemmas .............................................................. 8 w 2. The affine group .................................................................. Ix w 3. Structure of GL(A) ................................................................ 13 w 4. Structure in the stable range ....................................................... 14 w 5- Dimension o ...................................................................... 17 CIIAPTER II. -- Stable structure of projective modules ...................................... 19 w 6. Semi-local rings ................................................................... i9 w 7. Delocalization. ]'he maximal spectrum .............................................. 21 w 8. Serre's theorem .................................................................... 23 w 9. Cancellation ........................................................................ 25 w io. Stable isomorphism type ............................................................. 28 w I I. A stable range for GL(A), and a conjecture ......................................... 29 CHAP'rER III. --- The functors K ............................................................ 3~ w x2. K~ and KI(A,q) ................................................................ 3 I w 13. The exact sequence ................................................................ 33 w 14. Algebras .......................................................................... 37 w z 5. A filtration on K ~ ................................................................. 38 CIIAPTER IV. -- Applications ................................................................ 42 w i6. Multiplicafive inverses. Dedekind rings .............................................. 4~ w 17. Some remarks on algebras .......................................................... w 18. Finite generation of K ............................................................. w 19. A finitene.ss theorem for SL(n, A) ................................................... w 2o. Groups of simple homotopy types ................................................... w 2i. Subgroups of finite index in SL(n, A) ................................................ w 22. Some remarks on polynomial rings .................................................. 58 INTRODUCTION A vector space can be viewed, according to one's predilections, either as a module over a ring, or as a vector bundle over a space (with one point). If one seeks broad generalizations of the structure theorems of classical linear algebra, however, the satis- faction afforded by the topologists has, unhappily, no algebraic counterpart. 489 6 H. BASS Our point of departure here is the observation that the topological version of linear algebra is a particular case of the algebraic one. Specifically, the study of continuous real vector bundles over a compact X, for example, is equivalent to the study of finitely generated projective C(X)-modules, C(X) being the ring of continuous real functions on X. This connection was first pointed out by Serre [28] in algebraic geometry, and recently in the above form by Swan [34]. Serre even translated a theorem from bundle theory into pure algebra, and he invented the techniques to prove it [29] (see w 8). His example made it clear how to translate large portions of homotopy theory into the same setting, and thus discover, if not prove, an abundance of natural theorems of topo- logical origin. What follows is the result of a first systematic attempt to exploit this idea. This investigation was inaugurated jointly by S. Schanuel and the author, and an announcement of the results of that earlier work was made in [9]- In particular, the topological background for the results of w167 1o-xI is pointed out there. Generally speaking the method is as follows. The problem at hand is " locally trivial ", i.e. locally it can be solved by a simple parody of classical linear algebra. Furthermore, one proves an " approximation " lemma which asserts the existence of global data with prescribed behavior in a given local situation. Finally, in order to piece these together one uses a kind of " general position " argument. The ability to put things in general position imposes a dimensional restriction on the conclusions, and thus we determine a " stable range " for the problem. The structures investigated here are those of projective modules (Chapter II) and of the general linear group (Chapter I). This can be thought of as analogous to the study of vector bundles on a space, and on its suspension, respectively. If we " stabilize ", we are led to consider analogues of the functors K ~ and K 1, respectively, of Atiyah and Hirzebruch [2], and a good deal of the formalism, in particular the exact sequence, of that analogy is developed in Chapter III. The natural extension of the functors K i, i>2, to our algebraic context has so far evaded a definitive appearance. In another direction, this point of view should be ti'uitful in studying other classical linear groups. For example, thinking of bundles with reduced structure group, one can consider non degenerate quadratic forms on projective modules, and the associated orthogonal groups. K ~ would then be the " Witt ring ", and Witt's theorem, for example, is analogous to our Theorem 9-3- The local part of this theory has recently been done by Klingenberg [25], but several serious problems continue to obstruct its globalization. The most important applications of our machinery are to the linear groups over orders in semi-simple algebras finite dimensional over the rationals (w167 19-21 ). When applied to Z,~, ~z a finite group, they give quantitative information onJ. H. C. Whitehead's groups of simple homotopy types [36], results which elaborate on some earlier work of G. Higman [38]. They also shed some new light on the structure of SL(n, A), with A the ring of integers in a number field. Most striking is the fact that, for n suitably large, ifH is a non central normal subgroup, then SL(n, A)/H is afinite central extension 490 K-THEORY AND STABLE ALGEBRA of PSL(n, A/q) for some ideal q. This information is the starting point for the proof, in [4o], that every subgroup of finite index in SL(n, Z), n~3 , contains a congruence subgroup (see w 2I). The author owes most of his mathematics to Scrre's genius for asking the right question at the right time, and he records here his gratitude for having so profited. He is particularly thankful also to A. Heller, who has endured long audience on the present work, and who is responsible for numerous improvements in its exposition. 491 CI~APTER I STABLE STRUCTURE OF THE LINEAR GROUPS w I. Notation and lemmas. The objectives of this chapter are Theorems 3-x and 4.2 below, which purport to describe all the normal subgroups of the general linear group. We begin in this section by establishing notation and some general trivialities on matrices. Let A be a ring. GL(n, A) is the group of invertible n by n matrices over A. Let ei~ denote the matrix with i in the (i, j)th coordinate, and zeroes elsewhere; we recall that eif~=Sjkeo,. Let a, b~A and i~j. Then (I +ae~i)(i +be#)= I + (a-:-b)eii; here i -- x, denotes the n by n identity matrix. Thus, for i and j fixed, the matrices x 4_ aei, form a subgroup of GL(n, A) isomorphic to the additive group of A. A matrix of the form I -t- aeij, i +j, is called elementary, and we denote by E(n, A) the subgroup of GL(n, A) generated by all elementary matrices. Now let q (possibly -= A) be a two sided ideal in A. The q-congruence group is GL(n, A, q) ----ker(GL(n, A) -,~ GL(n, A/q)). Moreover we denote by E(n, A, q) the normal subgroup of E(n, A) generated by all elementary matrices in GL(n, A, q). We shall identify GL(n, A) with a subgroup of GL(n_m, A) by identifying ~.eGL(n,a) with ]-~o ~ m1 ~GL(n+m,A). This done, we set GL(A, q) :-[,J,GL(n, A, q) and E(A, q) ---- O,E(n, A, q). When q--A we write GL(A):=:GL(A,A) and E(A)=E(A,A). GL(A) is called the stable general linear group over A. Lemma (x.x) (" Homotopy Extension "). -- If A~B is a surjective ring homo- morphism, then E(n, A, q) -*E(n, B, qB) is surjective for all n and q. Proof. -- If ] +be# is B-elementary, and b is the image of aeA, then x +aet, maps onto I T beii. (There is an abuse of notation here which one should excuse, in that " I " and " e~i " have two senses.) This shows that E(n, A)-+E(n, B) is surjective. Now E(n, B, qB) is generated by elements -~ : ~-lv~ with ~eE(n, B) and -:. qB-elcmentary;i.e, z: i +qeij , qeqB. Wecanfind q'+_-q with image q, and ~'eE(n, A) with image ~ (by the first part of the proof). Clearly (i + q'eii ) ~ lifts v ~ If H 1 and H 2 are subgroups of a group we denote by [HI, H2] the subgroup generated by all [hl, he] -=hl-~h.,-lhlh2, with hieH i. 492 K-THEORY AND STABLE ALGEBRA Lemma (x.2). -- If i, j, and k are distinct, then [i --aeii , I --bejk ] = I -~- ab%. Pro@ -- (I § a@ (I "@ bej,~) (I -- a@ (I -- b%) = ( I + a% § beik + ab%) ( I -- a%-- beik + ab%) = ( I + ae~i § beik + ab%) -- (a@ -- (beik § ab%) -t- (ab%) = I -t- ab%. Corollary (x.3). -- If q and q" are ideals and n> 3, then E(n, A, qq') c[E(n, A, q), E(n, A, q')]. In particular, E(n, A, q) ---- [E(n, A), E(n, A, q)]. Pro@ -- The right side is a normal subgroup of E(n, A) which, by (I. 2), contains all qq'-elementary matrices. Corollary (x.4). -- Suppose n>_3, and let H be a subgroup of 13L(n, A) normalized by E(n, A). /f T is a family of elementary matrices in H, then H~E(n, A, q), where q is the two-sided ideal generated by the coordinates of I--'~ for all -~eT. Proof. -- If z = I -/- cehzeT then, since n > 3, we can commutate with other elementary matrices and obtain all elementary matrices of the form i + acb%. The E(n, A)-invariant subgroup generated by these, as z varies in T, is clearly E(n, A, q). Corollary (I.5). -- (i) E(n, A) = [E(n, A), E(n, A)] for n>3. (ii) If A is finitely generated as a Z-module, then E(n, A) is a finitely generated group for all n. (iii) If A is finitely generated as a Z-algebra, then E(n, A) is a finitely generated group for all n> 3. Proof. -- (i) is immediate from (I.3) , setting q~A=q'. (ii) Is obvious, since E(n, A) is generated by a finite number of subgroups, each isomorphic to the additive group of A. (iii) Let ao----I, al,..., a r generate A as a ring, and consider the elementary matrices I +a~eik , o<i<r, and all j+k. These generate a group which, by (1.2), contains all i + Mejk , where M is a monomial in the a i. Since A is additively generated by these M we catch all elementary matrices. In special cases we get something for n = 2. Lemme (x.6). -- Suppose i =u + v with u and v units in A. Then F.(2, A, q) A), A, q)]. If, further, u ---- w ~ with wecenter A, then E(2, A, q) = [E(2, A), E(2, A, q)]. Pro@ -- Given qeq, set b = v-lq, ~ = ;: and ~ = eE(e, A, q). Then [e'~]= ; (I~IU)b =/: Iqi e[GL(2'A)'I~'(2'A'q)]" M~176 by Lemma I. 7 below, and [w-le,~]----[e,~]. W--I(x ~ 0 W -1 493 IO H. BASS Examples. -- i. If A contains atield with more than two elements, I is a sum of two units. Likewise if 2 is a unit in A. 2. If A is an integral domain containing a primitive n th root, u, of i, then I+U+...-ku i is a unit if i+I is relatively prime to n. Since I +u+...+d=I-t-u(I-t-...-kd -~) we can write i as a sum of two units provided both i and i+I are prime ton. Such an i always exists unless n is a power of two. 3. The commutator quotient of GL(2, Z) is a group of type (2, 2), and that of SL(2, Z) is cyclic of order 12. More generally, the commutator quotient of SL(2, Z/qZ) is cyclic of order d, where d= gcd(q, 12). IfZp denotes the p-adie integers then the commutator quotient of $L(2, Zp) is cyclic of order 4 for p = 2, 3 for p = 3, and I for p4=2 or 3. In all of these examples E=SL. 4. For any ring A I is a sum of two units in M(n,A) for all n>I. For if n>I let B=A[~] =A[t]/(f(t)), where f(t) =t" -t-~-I. Then I =0~(I--~X n-I) =~(I -e)g(0~) SO ~ and I--0~ are units in B. Therefore they define A-automorphisms of B_~_-A " whose sum is I. The next lemma plays a fundamental role in what follows. Lemma (I "7) (" Whitehead Lemma "). -- Let aeGL(n, A) and bEGL(n, A, q). Then ab o I a o I ba o I mod E(2n, A, q) _= --1 o I ~o b[ 'o I and =1o a' mod E(2n, A) -[--b o: (The congruences hold for either left or right cosets.) Let Proof. - - Write b:-I-[ q; q is an n� matrix with coordinates in q. !ba oJ ~z [3= ao ~ "rj = I (ba)-lqi, "r2= I --a--lq :0 I '~ 0 I 0 I ~ I I I I o o and , = ~. j--b-lqa I ja I, Then clearly ": = vl~vaeE(2n, A, q). We begin by showing that ~z = ~. ]ba q 5"71 o:.1 _1= baTqa q'= a q[ i ,--a I ,i' ~0 I[ ~ __q+ql ]a o , i(l 0r 1"72 = ]--a q= 121 - ,--a b: o! 'a o b ; ~'71~- Ix2(3" ~ ~a- a b = !qa o:~ = ~. 494 K-THEORY AND STABLE ALGEBRA ib -1 ol Taking a-----b-t we have io b l ~E(2n, A, q) and hence lab o,=iab o,[b 1 oi__la__ o! rood E(2n, A, q). O I' -o I~,O b' !o b' Finally, modulo E(2n, A), we have "a o; 'a olii iill o'!I r ! ' o a ! Io b ,o b, o I " -I I, 0 I; ,-- b o. ' -=' '! I', " '= " For left cosets we need only observe that all subgroups involved are invariant under transposition. Corollary (x. 8). -- [GL(n, A), GL(n, A, q)] cE(2n, A, q). Remark. -- It would be useful if one could strengthen this corollary to say: [GL(n, A, q), GL(n, A, q')] cE(2n, A, qq'), say even under the assumption that q -- q' = A. Combining (1.3) and (1.8) we have, for n>3, E(n, A, q) = [E(n, A), E(n, A, q)] c[GL(n, A), GL(n, A, q)] cE(2n, A, q). Letting n-+oo we conclude: Corollary (I.9). -- E(A, q) -= [E(A), E(A, q)] -- [GL(A), GL(A, q)]. In particular, E(A) = [GL(A), GL(A)]. Remark. -- The last conclusion is due to J. H. C. Whitehead [36, w I]. Indeed, essentially, everything in this section is inspired by Whitehead's procedure in [36]. w 2. The affine group. Lemma (2. I). -- For n>2 the additive group generated by E(n, A) is the full matrix algebra, M(n, A). Proof. -- It suffices to catch a% for all a EA, all i, j. If i+j, a%= (i +ae#)--I. For the diagonal elements we have, for example, aell = ( I 4- ae12) ( i 4- e~1 ) -- I -- ael2-- e2t. Viewing An as a right A-module we can identify GL(n, A) with Aub~(A" ). Corollary (2.2). -- For n>_2 the E(n, A) invariantsubgroups of A" are the qA", where q ranges over all left ideals. (Note that these are not sub-A-modules.) Pro@ -- By the Lernma we can replace E(n, A) by M(n, A). Let = (al, ..., a,)eA" and let q=YAa~. It clearly suffices to show that i(n, A)e--qA". But the obvious use of coordinate projections, permutations, and left multiplications makes this evident. Now Aft(n, A) is defined to be tile semi-direct product GL(n, A)� If eeGL(n, A) and ~eA" the multiplication is given by = + 495 i2 H. BASS In particular = 1, __ We identify GL(n, A) with GL(n, A) � and A" with i � The latter is an abelian normal subgroup. Since o)-1= (i, we see that for subgroups HcGL(n, A) and ScA n, invariance of S under H is the same in both of its possible senses. We shall often identify (a, a) eAff(n,A) with : ~I eGL(n+ i, A) (viewing 0r as a column vector). Note that this identifies E(n, A) � A n with a subgroup of l~.(n + I, A). Proposition (2.3). -- Let H be a subgroup 0fAff(n, A) with projection L in GLn(, A). Then (i) [H, A"]=[L, A"]=Z, eL(I---:)A n. (ii) If H is normalized by An then [H, An]oH so in this case HoA" trivial=>H trivial. (iii) If n22 and H is normalized by E(n, A), then [U, An]=qA n for a unique left ideal q. Proof.- (i)If (,,~)eH, so ,eL, and a= (I, a)eA", then [(':, ~), (I, ~)] "-- (I, (I--'--l)~), (i) ~ (ii) is trivial, and (iii) is a consequence of (2.2). We shall use this proposition to show that, under suitable conditions, a subgroup HcGL(n,A) normalized by E(n,A) contains E(n,A,q) forsome q4=o. By (i.4) it will be sufficient to show that H contains a single non trivial elementary matrix. Of course we assume n> 3 in order to invoke (I.4). Suppose first that there is a eeH and a unit uecenterA, such that ~:t=u.I, but e--u. I has a zero row or column. We first apply the following: Remark. -- Since Z is a Euclidean ring l~.(n, Z) =SL(n, Z) for all n>2. Hence every " permutation " (i.e. a matrix with one non zero coordinate equal to + I in each row and column) of determinant I lies in E(n, Z). By specialization, every such permu- tation lies in E(n, A) for any A. Now back to ~ and u above. We can conjugate with a permutation in E(n, A) and assume e--u. i has either the last row or column zero; say the last row. Then ~=ue' with I4=e'eAff(n--I,A). Since commutating with e neglects u we can use (2 . 3) to produce lots of elementary matrices (in the last column). If the last column of o--u. I vanishes we transpose the above argument. Before proceeding further with our problem, let us compute the centralizer of {x= i +ae12]aeA }. Given ~, let ~ be the first column of~ and ~ the second row of ~-t. Then axa-t= I -~0r Hence, if a-:=vo we have xa~ =ae12. If a= i we conclude 496 K-THEORY AND STABLE ALGEBRA I3 that a = and ~ = (o, u- 1, o, ..., o). Then allowing general a we find uau- 1 = a so that uecenter A. Now changing e12 to eq we can already record the following: Corollary (2.4). -- If n>2, the centralizer of E(n, A) consists oj all matr#es u. I with u a unit in the center of A. Continuing our argument above, our objective is: Lemma (2.5). -- Let n>3 and H a subgroup of GL(n, A) normalized by E(n, A). If H contains a non central element ~ with some coordinate zero, then HZE(n, A, q) for some q+o. Proof. -- After conjugating a with a permutation in E(n, A) we can make a 1 or a, zero, where ~ ----- is the first column of a. If a commutes with all z = i + ae~2 then a has the same first column as u. I for some uEcenter A, by the last paragraph, and this case was handled already above. Hence we may assume a'~ can be chosen so that p=~w-l":-~:#I. Then p=v-l+~-t ", where y~a~,: -1, and ~ is the second row of e-a (see computation above). Case I. a,=o. Then ~y has zero last row, so p=z-l+0~y and ,~-t have the same last row. Hence we have 1 4: peArl(n-- I, A) and we can use (2.3) to get elementary matrices. Case 2. a, = o. Then the first rows of p and z-1 agree, so p is not central and has an off diagonal zero (since v -1 has first row (I,--a, o, ..., o)). Hence we can replace , by p and obtain Case i again. w 3- Structure of GL(A). Theorem (3. I) (Stable Structure Theorem.) Let A be any ring and GL(A) the stable general linear group over A (see w I for definition). a) For all two-sided ideals q, E(A, q) = [E(A), E(A, q)] ---- [GL(A), GL(A, q)]. b) A subgroup HcGL(A) is normalized by E(A)c:>for some (necessarily unique) q, E(A, q)cHcGL(A, q). H is then automatically normal in GL(A). c) If A--->B is a surjective ring homomorphism and if H is normal in GL(A), the image of H is normal in GL(B). (Note that GL(A)-+GL(B) need not be surjective.) Proof. -- a) is just (I.9). c) is a consequence of b) since the image of H will be normalized by the image of E(A), which, by (i. i), is E(B). Moreover, the uniqueness 497 ~4 H. BASS of q and normality of H in part b) both follows from [GL(A), H] =E(A, q)oH, which is a consequence of a). It remains to prove that if HcGL(A) is normalized by E(A), and if q is the ideal generated by the coordinates of i--z for all z~H, then E(A, q) cH. Let H,,=HnGL(n, A) and let qn be the ideal generated by all coordinates of i --v, ":ell,,. Viewing H,,cAff(n-f- I, A) it follows from (2.3) and (I-4) that H,, .-1 contains enough elementary matrices to capture E(n+ I, A, q,). Hence HD U,E(n + i, A, %) = U,,E(A, %) =E(A, q). w 4. Structure in the stable range. One would like to recover the stable structure theorem for GL(n, A), n<~c. Fortunately a sufficient condition for this can be formulated as a very simple axiom, one which we will verify in a rather general setting in Chapter II (Corollary 6.5 and Theorem t I. i). Definition. -- Let v. = (al, 9 9 at) be an element of the right A-module A r. We call ~ unimodular (in A r) if ZiAa i=A. This is clearly equivalent to the existence of a linear functional f: Ar--->A such that f(a) -- i. Let n> i ; we say n defines a stable range forGL(A) if, for all r> n, given a. = (al, ..., at) unimodular in A r, there exist bl, ..., b r_ 1 in A such that (al+blar, ..., ar_t+b r_ lar) is unimodular in A T Examples. -- If A is a semi-local ring, then n = i defines a stable range. If A is a Dedeking ring n=2 works. If A is the coordinate ring of a d dimensional affine algebraic variety, e.g. if A is a polynomial ring in d variables over a field, then n -- d-? I defines a stable range for GL(A). Lemma (4. x). -- If n defines a stable range for GL(A) it does likewise for GL(A/q) for all ideals q. Proof.--Suppose r>n. Writing A' =A/q, suppose a' = (a~, ..., a'~) is unimodular in (A')r; say i=Zt~a~. Lift t~ and a~ to ti and ai in A, so I=Ztiai+ q with qeq. Then (at,..., at, q)eA r-'* is unimodular; so, by hypothesis, there exist qeA such that (a I + qq, . .., a, + Crq ) is unimodular. Replacing a~ by a i+ qq, then, we can assume is unimodular. Again by hypothesis there exist b~eA rendering (al -~- b~ar, 9 9 at- 1 + b~_ tar) unimodular. The images, b~, of the b i in A' now satisfy our requirements. Theorem (4.o). -- Suppose n defines a stable range for GL(A). For r>n and for all ideals q: a) The orbits of E(r, A, q) on the unimodular elements of A r are the congruence classes modulo q. In particular E(r, A) is transitive. b) GL(r, A, q) =GL(n, A, q)E(r, A, q). c) E(r, A, q) is a normal subgroup of GL(r, A). For r>max(n, 2): d) E(r, A, q) = [E(r, A), E(r, A, q)] = [E(r, A), GL(r, A, q)] for all ideals q. 498 K-THEORY AND STABLE ALGEBRA I5 e) If H r A) is normalized by E(r, A), then, for a unique ideal q, E(r, A, q) cH and the image of H in GL(r, A/q) lies in the center. For r>max(2n, 3), and for all ideals q: f) E(r, A, q) -- [GL(r, A), GL(r, A, q)]. Proof. -- a) We first show that if ~ = (al, . .., at) -- (I + ql, q2, 9 9 qr) is unimodular, with qieq, then there is a veE(r,A,q) such that v0(=(i,o, ...,o). Writing I as a left linear combination of al, ..., ar and multiplying this equation on the left by q~ = at, the coefficient ofa~ in the new equation is a right multiple, say q, of q~, and hence in q. Thus a~ is in the left ideal generated by al, 9 9 ar_l, qa~, so (al, 9 9 a~_l, qar) is unimodular. Our hypotheses now say that we can find a~ =a i+ bda,. , i <i< r--I, t , = r--1 such that (al, ..., a~_l) is unimodular. Set %'1 I @Y~i=abiqeir6E(r, A, q), and write (a~,...,a;_~)=(I+q~,...,q;_i). Then q'~q,I<i<r--I, and "~10( = (I + q;, ..., q;_~, q~). Writing I as a left linear combination of a~, ..., a' r - l, and left multiplying this equation by qt--q,, we canwrite q;--qr=Zr--lcia~ with cie q. Then if ~,r--I E % I + ~=lcle~ E(r,A,q), t p p wehave z2z~0(= (I § ..., q~-t, q~). If G= I --% then ez,,-~l~ = (i, qs ..., q~-l, q;). Let %=I--(qs247247 ; then ":a(~v~z~0(=~=(I,o, ...,o), and v3eE(r, A, q). The presence of e, which need not belong to E(r, A, q), is harmless, since e-l~=~. Hence v=e-lv3(~v2-rleE(r, A, q) solves our problem. Now for the general case. Setting q = A, the case above was already general, and we have thus shown that E(r, A) is transitive. Now let q be arbitrary, and ~-~ mod q unimodular elements ofA r. We can find ~E(r, A) so that ~ = (I, o, . .., o). Since ~0(--~ mod q the argument above provides a -:~E(r, A, q) with -:~0(=~; hence = [~. b) Given aeGL(r, A, q), the last column of a is congruent, mod q, to (o, . .., o, I). Hence, by part a) above, there is a vleE(r , A, q) such that via= 2l with aleGL(r--i, A, q), 0(sqA ~-~. Set ~:~= eE(r, A, q); then By induction we can continue reducing until reaching GL(n, A, q). c) E(r, A, q) is generated by all z~=(r-~w with ~ q-elementary and ~eE(r, A). Given eEGL(r, A) we must show ~-lv~0(sE(r, A, q). But 0(-t':"0(= (~-~z~) ~, where =0(~-'. Since (by definition) E(r, A, q) is normalized by E(r, A), it suffices to show ~-t-r~eE(r, A, q). By part b) we can write ~=5.p, with u A) and peE(r, A). c , ceGL(r-- ~, A), Again, then, it suffices to show ,(-1T'~E(r, A, q). Write "F: and ": = t +-q%, qeq. (I--1%'(;0( ~6 H. BASS Case I. iorjis--r, say i=r. Then -:-- t I teqA '-1 so y~-y-1----- I[ which is clearly in E(r, A, q). Similarly for j=r. Case 2. i,j<r. Let a be the i th column ofc and b the 9 row ofc -1. Then ba=o, being the (i,j)th coordinate of c-lc= i (recall i4:j). One sees easily that +aqb o Let ":1-~ ~E(r, A, q), and y'ry- t = I + aqb) - l ; ;b/i ~ i al (~ = eE(r, A). We conclude by showing that (-~ly~y-t)~ A, q). I --a I o I+aqb O I ('L'1"~'7"~--1)~ O I qb(I +aqb) -I I ,o I --(l I+aqb o I a] o i llqb i i[o I --all a=~ b -~a=~ b o] qb I o I I+ a I1' since ba -~ o. we have, from (I.3) , d) Using r_>3 E(r, A, q) = [E(r, A), E(r, A, q)] c[E(r, A), GL(r, A, q)]. Let T= I +aeii be one of the generators of E(r, A). Suppose, for some ~, that [-~~ GL(r, A, q)] CE(r, A, q). Then [% GL(r, A, q)] = [.~o, GL(r, A, q)O]o-,= [~:o, GL(r, A, q)]~ A, q)~ A, q). Being free thus to choose e, we can transform -~ and assume ~ = i +aeri. Now by part b), using r>n, GL(r, A, q) =GL(r--I, A, q)E(r, A, q), so it suff� to check that c o GL(r_i,A,q),andwrlte~:= ~ o. [% GL(r-- I, A, q)] cE(r, A, q). Let T = i Then j o o oI:o ; o i [-, Y] 2t O I t I . I) I This is in E(r, A, q) since c- i mod q. f) Since GL(r, A) =GL(n, A)E(r, A) and GLir, A, q) =GL(n, A, q)E(r, A, q) there remains only, by virtue of part d), to show that [GL(n, A), GL(n, A, q)] r A, q). But this follows from (1.8), since r>2n. e) We begin by showing that if H contains a non central element, ~, then HDE(r, A, q') for some q'4:o. By (2.5) it suffices to produce, in H, a non central (a:) element with at least one coordinate zero. Let 0~= be the first column of ~. I' Since r>n, our hypothesis on n permits us to add multiples of a, to the other coordinates 500 K-THEORY AND STABLE ALGEBRA and render them unimodular. This can be accomplished by conjugating ~ with a matrix ~Ir--1 ~I j elZ.(r, A), so we can reduce to the case where (al, ..., at_t) is of the form l o Ii unimodular. We can then write a r=dla x 5... f_dr _tar_l, so that if ),-- i a r-1 where d=(dl, ...,dr t), we have ~-.1~__ 9 Now in the paragraph preceding Corollary 2.4 we showed that if ~ commutes with all -: ---- I + ael~ , then ~ has only one non zero coordinate, so we can finish with (2.5). Hence we may assume there is such a z for which p =(~za-lv-l:V i. Let ~ denote the second row of e-1 and y=a~v --1. Then (just as in the proof of (2.5)) we have We now claim that O is not central. Otherwise p=U.l with uecenterA, by(2.4). With ), as above, the last row of (~,-~,)y iszero, sothat X-lp=Z-~-:-t-~-(Z-~,)y and ?,-az -x have the same last row. Since I--V -t is concentrated in the upper left 2 � 2 corner, and I --X- t in the last row, we see, using r> 3, that the (r, r)thcoordinate of X-~'~- ~ is I. But the same coordinate of uX-t is u. This shows that p central ~-,o == I, contrary to our choice of-:. Now consider X-apZ----;~-tv-tz + (Z-~)(TZ). This is a non central element of H whose last row agrees with that of X-%-tX= I --~a3, where e is the first column of X-t and 3 is the second row of X= I. "-t o . Hence 3 has only one non zero coordinate d I (using r>2), so the last row of X-~z-lX has at most two non zero coordinates. We conclude then that X-lpX has some coordinate zero, and again we finish with (~.5). For the proof of e) now, choose q maximal so that E(r, A, q) cH. Our problem is to show that the image, H', of H in GL(r, A/q) lies in the center of GL(r, A/q). Since E(r, A) normalizes H, H' is normalized by the image of E(r, A), which, by (x. ~), is E(r, A/q). Hence if H' is not central Lemma 4. I permits us to apply the first part of our argument to H' and conclude that E(r, A/q, q'/q) cH' for some q'4 = q. Then the inverse image, L=HGL(r,A, q), of H' contains E(r, A, q'). By part d) now E(r, A, q') = [E(r, A), E(r, A, q')] r A), L] c tiE(r, A), H][E(r, a), GL(r, A, q)]" = [E(r, A), H]E(r, A, q) oH, contradicting the maximality of q. w 5- Dime.slon o. Corollary 6.5 tells us that n = I defines a stable range for GL(A) when A is semi- local, and it is in precisely this case that the restrictions r> 3 intervene effectively. The following refinements can be made: Proposition (5. z ). -- Suppose n = I defines a stable range for GL(A) (e. g. A can be any semi-local ring). 3 i8 H. B A S S a) If A is commutative, then, for all ideals q and all r> 2, E(r, A, q) = SL(r, A, q)(= SL(r, A) r~GL(r, A, q)). In particular E(r, A) = SL(r, A). b) Suppose I --w" + v with w and v units in the center of A. Then, for all r> 2, E(r, A, q) ---- [E(r, A), E(r, A, q)] -- [GL(r, A), GL(r, A, q)]. Pro@ -- a) Clearly F.(r, A, q) cSL(r, A). Now an element of GL(r, A, q) is, by (4.2) b), reducible modulo E(r, A, q) to GL(I, A, q), i.e. to a unit, and that unit is evidently the determinant of the original matrix. This proves a). b) For r>3 b) is just parts d) and f) of Theorem 4.2. However, the proof there uses r>3 only to invoke (1.3) and conclude that l~.(r, A, q) ~[E(r, A), F.(r, A, q)]. Our hypothesis permits us to use (i .6) instead for the same purpose when r: 2. zks a consequence of part a) above and (i. i) we have: Corollary (5.2). -- Let q be an ideal in the commutative ring A for which A/q is semi-local. Then, ]or all q'Dq, E(r, A, q') ~ SL(r, A/q, q'/q) is subjective for all r. In particular, SL(r, A) ~ SL(r, A/q) is subjective. Remarks. -- i. For local rings some further refiniments of our results can be found in Klingenberg [24]. 2. Let GL(r, A, q)' be the inverse image in GL(r, A) of the center of GL(r, A/q). One would like the following converse to part e) of Theorem 4.2: If E(r, A, q) cH cGL(r, A, q)' then H is normal. This would follow from: E(r, A, q) = [GL(r, A), GL(r, A, q)']. For a commutative local ring this follows from (4.2)f), since GL(r, A, 0)' is then generated by GL(r, A, q) and the center of GL(r, A). Klingenberg's proof of this for non commu- tative A appears to contain a gap, due to the erroneous equation, " gj(a-lb-~a) -- I " on p. 77 of [24]. 3. IfA is the ring of integers in a number field then n =-= I defines a stable range for GL(A). Our results in this setting are, to some extent, well known. For example, the last part of Corollary 5.2 (familiar to function theorists for A =Z) and condi- tion (4..2) b) for q=A were known long ago by Hurwitz. Moreover, Brenner [I3] recognized (4.2) b) for A=Z. However the commutator formulae appear to have escaped notice even for A=Z. They turn out to be essential in the proof (see w 2i) that every subgroup of finite index in SL(n, Z), n>3, contains a congruence subgroup. The discovery of the generality of these formulae lies ultimately in J. H. C. Whitehead's work on simple homotopy types [36]. 502 CHAPTER II STABLE STRUCTURE OF PROJECTIVE MODULES w 6. Semi-local rings. Let A be a ring. Throughout this chapter, " A-module " means right A-module, and " ideal ", unqualified, means two-sided ideal. Let P be an A-module, and ~eP. We write P .... Homn(P , A) and define o(:~) = oe(~) -= {f(:()!feP*}. o(:r is a left ideal in A, and it is clear that o(:r --A if, and only if, the homomorphism g : A-~P, g(a) = ~(a, has a left inverse. In this case we call :r a unimodular element of P. Lemma (6.x). -- Let ~ : Q---~ P be a homomorphism of A-modules with Q finitely generated and projective. Then ~ has a left inverse (i.e. is a monomorphism onto a direct summand) zf, and only zf, ~* : P*-~Q* /s an epimorph#m. Proof. -- ~ has a left inverse ~ ~* has a right inverse =-~** has a left inverse -~ has a left inverse, since Q= Q** is reflexive. Moreover, ~" has a right inverse ,~-~* is an epimorphism, since Q* is projective. Denote by rad A the Jacobson radical of A. Corollary (6.2). -- Let e,, : Q-~P as in (6.I), and assume Im(~---:)CP.radA. Then ~ has a left inverse ,.~ v does. Proof. -- Let fsIm(~--,)*cQ*; then f(Q) Crad a. Since Q if finitely generated and projective this implies that ferad A.Q*. Hence Im ~*CIm v*+ rad A.Q*. Since Q* is finitely generated, Nakayama's Lemma tells us that ~* surjective =--:* smjective. Using (6. I) now, this completes the proof. Definition. -- Call A semi-local if A/rad A is an Artin ring. It follows then that A/rad A is a finite product of full matrix algebras over division rings. For the balance of this section, A always denotes a semi-local ring. The lemmas which follow contain the " zero dimensional " case of the general results to follow. If S is a subset of an A-module, P, denote by (S) the submodule generated by S. We shall say f-rankA(S ; P) > r if (S) contains a direct summand of P isomorphic to A ~. We will suppress the subscript " A " when A is fixed by the context. In what follows A is a fixed semi-local ring, and P denotes an A-module. Corollary (6.3). -- f-rank(S; P) =f-rank((S) -!- P. rad A; P). 503 ~o H. BASS Proof. -- If f-rank((S) +P.rad A; P)>r there is a homomorphism a :Ar-+P having a left inverse, and with Im at(S)+P.rad A. Hence we can find v:Ar~P with Im-:r and Im((~--z) CP.radA. By (6.2) -~ has then also a left inverse, sol-rank(S; P)> r. The reverse inequality is obvious. The following simple result will play a fundamental role in what follows. Lemma (6.4). -- If beA and a is a left ideal such that Ab+a=A, then b+a contains a unit. Proof. -- Since units in A/rad A lift automatically to units in A we may assume A is semi-simple. Passing then to a simple factor we can reduce to the case A = EndD(V), where V is a finite dimensional right vector space over a division ring D. In this case a can be described as the set of endomorphisms which annihilate some subspace, W, of V (a=Ae, e2=e, and W=kere). The fact that Ab§ guarantees that ker bnW=o. Write V=W| bW| Now we can clearly construct an automorphism u such that u]W=blW and u(W')=U. (Note that W~bW=~W'~U.) Since a----u--b annihilates W, aea and we're done. Corollary (6.5). -- n=I defines a stable range for GL(A) (in the sense ofw 4). Pro@ -- By definition, we must show that if Aa 1 +... -}- Aa, = A, r> I, then we can add multiples of a, to al, ..., a~_ 1 so that the resulting r-- I elements still generate the unit left ideal. Let b =a 1 and a=Aa2+.. 9 +Aar; then (6.4) provides us with a unit u = al + b2a~ +. 9 9 + b,a~. Hence ue (A(a 1 + b~ar) + Aa~ +... + Aar_l) , as required. The following is a technical little argument with two important corollaries: Let and ~ be unimodular in ~A| Writing a=~b+~p we have A=o(~) =Ab+o(0@. By (6.4) u = b + a is a unit for some a~o(~e). Let f be an endomorphism of ~A| such that f(~)=o,f(P)c~A, and f(0@ =~a. The existence off follows from the definition of o(~1~ ) =Op(~p). If q%= i+f then ~o1(o~ ) =~u-{-~p. Now let g be the endomorphism killing P such that g(~) -- --~pu- 1, and set q~2 ---- I -~g. If ~ -- ~q% then (i) q~(0~) = ~u, (ii) ~ leaves invaHant all submodules containing ~A-t-~A, and (iii) since f2=o=g2, ~ is an automorphism. Corollary (6.6). -- A|174 Proof. -- Using the hypothesized isomorphism to identify the two modules, we can write ~A| = eA| With the ~0 constructed above we have, from (i), ~0(eA) ---- ~A, so P----~ (,~A| P)/~A = q0 (~A| (~A) ~ (~A| P')/~A~P'. Corollary (6.7). -- If M is a submodule of P, then f-rank(A'| A'| P). Proof. ~ It suffices to treat the case r-=--i. Moreover, the left side is clearly at least as large as the right. Let e~, ..., ~,~A| be a basis for a free direct summand of ~A| (~ is assumed unimodular here of course.) Now construct (p as above with ~----e~. By (ii) above q~(@e~A| also, I <i<s, so we can replace ~by q)(~) and, by (i), reduce to 504 K-THEORY AND STABLE ALGEBRA oi the case ~A = elA. But then we can subtract multiples of a 1 from as, ..., a8 and further render ~2, ...,%eM. Thus, f-rank(.3A| P)>s--i, as required. Corollary (6.8). -- If ~ is an element, and S a subset, of P, then f-rank(S, e; P)<I +f-rank(S; P). Proof. -- Suppose f-rank(S, e; P)>r; i.e. there is a a : Ar-+P with a left inverse, and Imac(S) 40~A. Let f:A| by f(a,p)=,.a+p. Then Im(f](A| so we can find g:A'-~A| such that ~=fr and ImgcA| The left invertibility of ~ implies that of g, and hence f-rank(A| A| >r. The present Corollary now follows from the preceeding one. Lemma (6.9).- Suppose oh, ...,areP and f-rank(~l, ..., ~,; P) = t<s<r. Then we can find ~1, ..., ~, of the form ~-= ~i + Z~.>~%.%., i <i <s, such that f-rank(~l, ..., ~; P)=t. Proof. -- We induce on t, the case t=o being trivial. If t>o we can write P=~A| with ~ a unimodular element of P in (~1, ...,~,). Writing ~--: ~b~ + e~ then, with c~ 9 P', we must have Z~b~A = A. By (6.4) (applied to the opposite ring of A, since we are now dealing with right ideals) we can find a unit u---bl+Ej>lb~alj. Set ~l--el+Zj>l%.alj.=~u+~. Then ~ is unimodular in P (since ueo(~a)), so P=~A@P, for some P~. Write ~i--~ci+~,i, 2ieP1, ~ <i<r. Note next that (~, ~.,, ..., a~) = (~, ~, ...,e~) = (~,y~, ...,y,). Hence, we conclude from (6.7) that f-rank(y~, . .., y,; Pa) =t--l. By induction, we can find ~---- ~ + X~>r 2 <i<s, such that f-rank(8~, ..., 88; Pt) = t-- i. Writing we see that (~, ~.,, ..., ~,) = (~, ~.o, ..., ~8), and this clearly solves our problem. w 7. Delocallzation. The maximal spectrum. As a basic reference for the material of this section, we refer to Bourbaki [12] or Grothendieck [22, Ch. o]. Let A be a commutative ring and X = max(A), the topological space whose points are the ma.'dmal ideals of A, and whose closed sets are the sets of all maximal ideals containing a given subset of A. If M is an A-module and xsX (x is a maximal idcal in A) we let M~=A~| denote the localization of M at x. Further, define supp M(= suppxM ) ={xEX I M,4: o}. 505 22 H. BASS The topology in X is better described, for our purposes, as follows: the closed sets in X are precisely those of the form supp M where M is some finitely generated A-module. Let A be a finite A-algebra (i.e. finitely generated as an A-module). If P is a A-module, Px is a module over the A,-algebra, Ax, and our finiteness condition on A gua- rantees that Ax is semi-local, for all xeX. Hence, if S is a subset of P, we can consider f-ranka~(S ; P~), as defined in w 6. (Here we are confusing S with its image in P~, but this should cause no difficulty.) We now define f-rankh(S; P) = inf, exf-rankA~(S; P~), and f-ranka(P ) =f-rankh(P; P). When A is fixed, we shall suppress the subscript. Since the definition is local, (6.7) immediately yields: Corollary (7. I ). -- If M is a submodule of the A-module P, f-rank(A*| A*| = r +f-rank(M ; P). More generally, we define the " singularities " of SoP by Fi(S; P)={xeXIf-ranka~(S; P~)<j}. Thus, F0(S; P)=0 for any S, and Fj(O; P)=X for all j>o, for example. It is essential to our method that these sets be closed, and for this we need a " coherence " condition on P. Lemma (7.2). -- Suppose P is a direct summand of a direct sum of finitely presented A-modules. Then, for all subsets S of P and for all j, Fi(S; P) is closed in X. (Recall that a finitely presented module is the cokernel ofa homomorphism Ar-+A~.) Proof. -- Let xeX--Fi(S; P). Then if Q= A j there is a homomorphism ~ : Q~P such that Im ac(S) and such that % : Qx-+Px has a left inverse. (One simply chooses elements in (S) whose images in Px are a basis for a direct summand.) It suffices now to show that {yeX[% has a left inverse} is open, for the latter will then be a neighborhood of x in X--Fj(S; P). Since Im a is finitely generated, our hypothesis on P permits us to reduce this last question to the case where P itself is finitely presented. Under these conditions the natural homomorphism (P*)y-+ (Py)* is an isomorphism for all yeX (see [5, Lemma 3.3]) ; the second * here refers, of course, to the duality of AN-modules. The same remark applies to the free module, Q, so that we can conclude from the commutative square, (a*)y (~y)* (Py)* > (Qy, 606 K-THEORY AND STABLE ALGEBRA an isomorphism Mu=~coker(%)* , where M=coker(a*). Now we have from (6.i) that {)!% has a left inverse} = {y i (%,)* is surjective}={y[Mu=o}--X--suppM. Since M is finitely generated (being a quotient of Q*), supp M is closed, and this completes the proof. It is useful to note above that, since M--o<>My=o for all y, ~ has a left inverse ~>~ru does for ally. Applying this to a:A~P with a(I)=a we obtain: Corollary (7-3)- --- With P as in (7.2) and ~eP, ~ is unimodutar in P if, and only iJ; F1(~ ; P)=O. Examkles. -- I) Lemma 7.2 applies, notably, when P is either projective or finitely presented. 2) The inclusion Z-~Z(p!, (rational p-adic integers) has a left inverse only at (p) emax(Z). 3) IfP is the Z-submodule of Q, generated by {p-tip prime}, and ifq is a fixed prime, then Fl(q-t; P)=max(Z)--{(q)} is not closed, whereas P is locally free (of rank one). Finally, we recall some topological notions. In any topological space, X, a closed set F is called irreducible if F4= O and if F--GuH with G and H closed=~F ---G or H. We then call codlin F(--codimxF ) the supremum of the lengths, n, of chains, F 0= FCF 1C. . . CF,, of distinct closed irreducible sets in X. For arbitrary closed F we define codlin F to be the infimum of the codimensions of the irreducible closed subsets of F, with the convention codlin f)- oo. The supremum of the codimensions of non empty closed subsets of X will be called, dim X. We call X noetherian if the closed sets satisfy the descending chain condition. In this case every non empty closed set F is a finite union of irreducible closed sets. Such a representation of F, when made irredundant, is unique up to order, and we call the intervening irreducible closed sets tile irreducible components of F. For example, if A is a commutative semi-local ring, max(A) is a finite discrete space, and hence a noetherian space of dimension zero. If A is only noetherian, then max(A) is a noetherian space of dimension <Krull dimension A=dim spec(A). w 8. Serre's theorem. For the next four sections we fix the following data: )IA is a commutative ring for which X =max(A) is a noetherian space. (8 I A is a finite A-algebra. The next theorem is a slight generalization of earlier versions. We include it for the sake of completeness, and also tbr the proof below, which is perhaps a little more manageable then its predecessors. 507 ~4 H. BASS Theorem (8.2). -- (Serre [29] ; see also [5]). Suppose P is a direct summand of a direct sum of finitely presented A-modules. Then, if f-rankA(P)>d = dim X, P~A| for some P'. We will derive the theorem from the two lemmas below, wherein P is always assumed to satisfy the coherence condition of the theorem. This assumption permits us to invoke (7.2) which ensures that the various " singular sets " intervening in the proofs are closed. All codimensions refer to codimension in X. Lemma L -- If f-rank P>r, there exist ~1,..., e,eP such that codim F~.(~I, ..., ~,; P)>(r+ I)--j, all j>o. Lemma II. -- If o~1,..., %eP, r> I, and k~Z, are such that codim Fj(~:, ..., %; P)>k--j, :<j_<r, then there exist ~=ai+~,ai, for suitable aieA , I<j<r--l, such that codim Fj(~I , ..., ~r-,; P) >k--j, I <j<r--:. Proof that I and II-~(8.2): Apply I to P, with r=d+I, to obtain ~x, ..-, ~,eP such that codimFj(~,, ..., e,; P) >(r+ ~)--j for all j>o. Now, with k=r+i, apply II, (r--I)-times. The result is a single element, ~eP, such that codim F,(~; P)>k--i =r---d+ I. Since d=dim X this implies FI(~; P) =0, so, by (7.3), ~ is unimodular, and this completes the proof. ProofofL -- We induce on r, the case r=o being trivial. Suppose 0~1, ..., % have been constructed as in the lemma, and we want e,+: (assuming f-rank P>r+ i). For o<j<r, let {D~.~} be the "largest" irreducible components of Fi_:(~, ..., ~,; P), i.e. those of smallest codimension, (rq- i)--(j+ i). Of course there may be none, but that's all the better. Since codim Fi(0h,..., 05; P) > (r-I,- ~) --j, Di~ r Fj(e,,..., ~; P) for all ~. It follows (since Di~ is irreducible) that we can choose x(j, v) eDit-- [Fi(~:, ..., ~; P) uU~.~Di~]. Since x(j, v)eFi+,, CF i we have f-ranka,~j.~)(al, ..., at; P*0".~;) =j<r, for o<j<r, and all v. It follows now from our hypothesis on P and (6.7) that we can choose ~j~ eP such that f-rankA, Ij.~)(ax, ..., ~,, ~,; P,:,.,~)) >j+ i. Since x(j, v) are (clearly) all distinct, we can write I = Zi,~ei~ in A, with ei~-- ~(i,~)(~,~) mod x(h, ~). Setting ~+ ~ -- Zi.~,3i~e~ we have a,+l-~i~modx(j, ~) so, by (6.3), f-ranka~(i.~)(~l, ..., ~,+l; P~(i.~))>j+ I, o<j<r. Hence x(j, ~)r ..., a,+a; P) (by definition) so also Di~4zFi+,(al, ..., a,+a; P). On the other hand, F~+~(a~, ..., e,+~; P) r ~a(a~, ..., ~,; P), and since the former contains no component of extreme codimension of the latter, we conclude, as desired, codim Fi+t(aa, ..., ~r+t; P) >(r+ I)--(j+ I) A y I = (r ~-2)--(j+ :), o_<_j<r. For the remaining values of j, F 0 = fO has infinite codimension, and if j> r, (r + ~) -- (j + I) <o, whereas all codimensions are >o. Proof of IL -- For o<.j<r let {Dj~} be the irreducible components of ..., P) of lowest possible codimension, k--(j+ I). Then D~r ..., ~,; P) 508 K-THEORY AND STABLE ALGEBRA 25 so we can find xlj, ..., %; P) u ; thus f-ranka,(i,,)(al, ..., a,; P~/i.~)) =j<r, for o~j<r. By (6.9) we can find elements %,~A,(y.~) such that, if ~,---~+a,aq~eP,(i,~l , we have f'rankA,(j.~l(~:J~, "", ~i,-~i~; P*(i,*I)=-J" If we modify our choice of aqy modulo the radical of Ax(i.,) we can even choose the %,EA, and hence ~i,~P. This is permissible by (6.3). Now choose e~ -- 3(/,~/(h,~)mod x(h, ~), as in the proof of I, set ai---E~.~aij~ej, , and put ~=a-i-ka-,ai, x<i<r--I. Since ~---- ~,~. mod x(j, ~) we have, by (6.3) again, f-rankA~(i.,l(~l , ..., ~,_1; P~(J,~I) ----J, o<j<r, all ~. The form of the ~'s makes it evident that (~1, ..-, ~,-t, a,) ---- (ul, ..., a,-t, a'r), SO it follows from (6.8) that F/(~x, ..., ~,_~; P) cF~.+~(0h, ..., ~,; P). Since we have arranged that the former contains no irreducible components of lowest possible codimension, k--(j+ ~), of the latter, we conclude, as desired, that codim F~.(~, ..., ~_~; P) >k--(j ~ ~) + i =k--j, o<j<r. Counterexamples. -- ~) To see the necessity of the coherence condition in Serre's Theorem, we can let P = Q| where Q is the Z-submodule of the rationals generated by {P-~IP prime}. Here f-rankz(P ) = ~ and dim max(Z) = ~, but P has no projective direct summands. 2) The following example of Serre shows that P can even be made finitely generated and locally free. Let X be a Cantor set on the real line andy a point of X which is a limit point from both the left and right in X. Set D0={x~X[x~y}, D a ={x~XIx<y }. With A= C(X), the ring of continuous real valued functions on X, it is well known that max(A)-:-X and is hence of dimension zero. Moreover A is locally a field so that A-modules are locally free. Let al be the ideal of functions vanishing on D~, i=o, x, and let P= (A/ao)| Since supp A/ai=D~_, we see that P is locally free of rank one except at {y}---- D0r~D~, where it has rank two. Hence, if P had a free direct summand, the complementary summand would have support {y}. If (f0,f~) eP were in this summand, f~A/a~= C(Dt_~), f~ would be a function on D~_~ vanishing everywhere except, perhaps, aty. Sincey is a linfit in D~_i, fi has to vanish also at y, and hence fi----o. Thus the complement would have to be zero, and we have a contradiction. w 9" Cancellation. A, X, and A are as in (8.~). " Cancellation " refers to Theorem 9.3 below. Theorem (9-I). -- Let Q and P be projective A-modules, a a left ideal, and a-= ~q+ a-vEQ| an element such that o(a)+a=A. (See w 6 for definition of o(a-).) Suppose, moreover, that f-rankP>d=dim X. Then there is a homornorphism f: Q--->P such that o(f(~)+~)+a=A. 4 '26 H. BASS Proof. -- We induce on d, and the case d--o will be subsumed in the general induction step. Our hypothesis makes Serre's theorem available, to the effect that P=f~'A| for some unimodular ~'eP; say ~v=~'b=- Then o(~q) +Ab~o(~') +a=A. Let F1, ..., F o be the irreducible components of X, and choose x~EF~--kJi.~Fi, I <i<s. Modulo the product of the xi, A is a semi-local ring, so we can apply (6.4) to find a~ + a' + ceO(aq) -~- o(~') + a such that the image of b + aq + a' + c is a unit in A modulo the xi, and hence already in Axi , I <i<s. Let g be an endomorphism of P such that g(I~')=o,g(P')c~'A, and g(~')=~'a'. The existence of g follows from the definition of o(~') (w 6). If ~=ip--g then ~ is an automorphism (since gZ=o) and a~v=.3'(b+a') ~-e'. Setting 3--~-'(~') and ~I=a-~(~')ePI=G-'(P ') we have P = ~A| and ~p = ~(b + a') + ~. Now choose fl : Q-+~A cP such that fl(~q) = ~aq; again, ft exists by the definition of O(eq). Putting b t =aq+ b +a', then, we have (*) f~(~q) + ~p = ~bt + 0~1, and bl ~c is a unit in A~i , i<i<s. If dim X=o then X={xl,...,x,}, and we're done. In general, we can find a teA belonging to none of the x; such that tA cAb 1+ a. (E. g. semi-localize at xl,..., x~, solve (bl+C)Z= i, and clear denominators.) Let A*=A/At, A*=A/At, C=image of a in A*, etc. Then X*=max(A*) is a closed subset of X containing no xi, hence no Fi, so dim X*<dim X. Hence, using (7- i) we have f-rankA.P~>f-rankAP 1--f-rankaP-I>dim X--I ~dim X*. * * * 9 * * Consider "./=%+~teO)s174 Since A=o(~q)+Ab~+O(el)-~-n we have, over A, o(.v*) + A*b~ + a* -- A'. Putting this together we are in a position to apply our induction hypothesis to y*eQ*| and the left ideal A*b~_ C, the result being a homomorphism h* : Q* ~-P~ such that o(h*~+:~;) -!-A*b~+o*=A*. Since Qis projective we can cover h. with an h : Q-~-P~ cP. Now, for the theorem we take f=fa+h : Q-+~A| It remains to show that b+o=A, where b=o(fxQ+~p). By (*)above we have f~Q -- ~p = (h~q -~-f~eq) + ~p = h~q + (~b I ]-~) --- ,~b~ + (h~Q-t- ~(1) e~A| It is thus evident that Abtcb , and that o(h~Q+~l)cb. But since P~ is projective it follows that the image of o(h~q+~l) in A* is o(h*~+ ~). By construction of h* this together with A*b~ ~ a* generates A*. Back in A, then, o(h~q + ~) + Abt -t- a + At = A. But AtcAbt-~-a so we have b 4-aDo(ho%~-~)+Ab~+ct--A, as required. Let P be a A-module. Recall that an element eeP is unimodular if there is an feP*=Hom~(P, A) for which f~= i. We shall similarly call feP* unimodular if there is an 7eP for which f~ =; ~ (i.e. iffis surjective). Further, we shall find it useful sometimes to identify P with Homa(A , P) ; i.e. we identify ~eP with g~ : A~P defined by g~,(a)-=~.a. Thus, if f: P-+A we can compose, ,~f: P-+P; ~f is defined by (~f) ~ - ~(f~) for ~ e P. An endomorphism z of P will be called a transvection if = -- ~i, + 7f, where ~eP, feP',f~.=o, and either e orfis unimodular. -: is then necessarily an auto- 510 K-THEORY AND STABLE AI.GEBRA _o 7 morphism, since (a f)" = o. If q is an ideal in A we call -: a q-transvection if Ira(a f) cPq. Denote by GL(P) the group of A-automorphisms of P, and by E(P) the subgroup generated by the transvections. More generally, if q is an ideal, let E(P, q) be the subgroup generated by the q-transvections. Suppose z= i +af is a q-transvection and ~GL(P). Then -:~ = I + (~-l~)(f~r) is clearly again a q-transvection; hence E(P, q) is a normal subgroup of GL(P). The next result should be compared with Theorem 4.2 a). Theorem (9.~). -- Let M be a A-module which has a projective direct summand of f-rank>d+ I (d-dim X). Then for any ideal q in A, the orbits of E(M, q) operating on the unimodular elements of M are precisely the congruence classes rood q. In particular, E(M) is transitive on the unimodular elements. Proof. -- By hypothesis, M=P'| with P' projective and f-rank P'>d+ ~. By Serre's theorem (twice), P'---fiA|174 with ~ and y unimodular. Let q be an ideal in A. We will first show that if a is a unimodular element of M and if ~=~modMe, then there is a -r~E(M,q) such that ":~=9. With P=3A| we have M=yA|174 Write a----yq+~,+a~. Since ~-~sPmodq, we have qeq. There is an h:M-+A with h~----i, by assumption, so we can write (*) ~ ---- ha = (hv)q + hap + ha~. Let Y=yr+:(~+as with r=q(hy)q. To see that E is also unimodular, we first note that o(~) --Ar+o(a~) +o(~s). Left multiplying (*) by q, then, we find qeo(E), so that o(~) ~Aq + O(ap) + o(a~) ---- o(~) ---- A. Now we apply Theorem 9. ~ above to Q~yA, P, yr+~p~Q.| and a=o(a~). The result is an f: Q--+P such that o((fu +o(~s) --A. Let ~---- (fy)q(hy)ePq, and let gl : M~A by g~(y) ---- i and gl(P| =o. Then z~---- IM+8~gl~E(M , q), and vl~=yq-!-@+aNeyA|174 where @=(fy)r+ap. We have arranged above that o(:@ +o(~y) =A, so @q--~ is unimodular. Hence we can find an h' : M-+A with h'(a~,+~N) -- I, and such that h'(y)=o. Write a~=~b+xleP- -~AC~)P 1. Since vla=~ mod Mq (recall ":1- I~mod q) we have (**) +a, + and in particular, I--b~q. Thus, if g2----((I--b)--q)h':M~A, Im g2Cq, so v2= I~+yg,,eE(M, q). T2Ti0~=y(x--b) +o~+a N. Let cr= IM+ggleE(M); then o-.r2~lq = y(I --b) + (~,+ 9(I --b)) + a~ ---- y(i --b) + ~ + al + aNEvAQgA|174 Let ga:M~A by ga~=I and ga(yA|174 and let ~a----(y(~--b)+~l+a~), which, by (**), is in (kerga) q. Then Ta---- ~M+Sag3cE( M, q) and TaO~Tia=~. The presence of a, which need not belong to E(M, q), is harmless because ~-1~ = ~_9g1~ = 9, so that -:=o-ITa(~T2zI~E(M, q) solves our problem. 511 ,~8 H. BASS Now for the general case, we note first (taking q-= A) that we have shown E(M) to be transitive on unimodular elements. Suppose given arbitrary unimodular elements and ~' in M with ~-~' rood Mq. Choose geE(M) with ~'=~ (~ as above). Then ~a-ea'=~ mod Mq so the argument above produces a v~E(M, q) with ~-ee=~'. Finally, ~-lwa--0:' does the trick. Theorem (9.3) (" Cancellation "). -- Let M be a A-module which has a projective direct summand of f-rank>d=dim X, and let Q be a finitely generated projective module. Then, if M' is another A-module, Q|174 ~ M--~M'. Proof. -- Since Q| for some n and Q' we can reduce, by induction on n, to the case Q= A. Then using the given isomorphism to identify A| with A| we can write ~A|174 with ~ and ~ unimodular. ~A| satisfies the hypo- thesis of (9.2), so there is a zeGL(~A| with w=~. Hence M ~ (}A| ---- v(0cA@ M')/v(aA) ~ (:cA| M')/aA ~-- M'. Remarks. ~ I) If M satisfies the coherence condition in Serre's Theorem, and if f-rank M>2d, then M has a free direct summand ~A ~+1 (by Serre's Theorem), and hence M fulfills the hypothesis of (9.3) above. 2) If d=o then A is semi-local, and (6.6) gives the conclusion of (9-3) with no restrictions on M. If d= I and A----A is commutative, then for M projective no further hypothesis is needed. For one can reduce this case further so that spec(A) is connected and M is finitely generated (using [6]). Then one applies Serre's Theorem to make M and M' each a direct sum of a free module and a projective module of rank one. The desired isomorphism then follows by taking suitable exterior powers (see [29, no. 8]). However, for d= i and A non commutative, (9.3) gives the best possible result even for M projective and finitely generated (see Swan [33]). For d>I (9.3) is best possible even with A=A commutative. For if A--R[x,y, z], x2-4-y2+z 2= I, is the algebraic coordinate ring of the real 2-sphere, and if P =A3/(x,y, z)A is the projective A-module corresponding to the tangent bundle on S 2, then P is not free, whereas A|174 2 (see Swan [34, Example I]). w IO. Stable isomorphism type. Keep A, X, and A as above (see (8. i)). The Proposition which follows is merely a reformulation of a special case of the preceding results. We include it for the sake of putting in evidence the faithfulness of the analogy between the present theory, and its topological source (see Introduction). Proposition (I0.I). -- Let P,(A) denote the isomorphism types of finitely generated projective A-modules P, such that P~-A~, for all x~X. Let f, : P,(A) -+ P,_t(A) be the map induced by | Then, if dim X = d: 512 K-TIIEORY AND STABLE ALGEBRA 29 I. (Serre's Theorem) f, is surjective for n>d. 2. (Cancellation) f, is injective for n>d. Of course our theorems are much more general. For example, one could formulate a similar result for all finitely presented modules, relative to f-rank. Thus, calling M and N " stably isomorphic " if Q|174 for some finitely generated projective module Q., we see that stable isomorphism=isomorphism for semi-local rings (6.6), and more generally also for M projective of f-rank>d, or finitely presented off-rank> 2d. In a special case of some interest we can make a mild improvement in Serre's Theorem for non projective modules. Proposition (IO.2). -- Let A be a Dedekind ring of characteristic zero and 7: a finite group. If M is a finitely generated torsion free Arc-module of f-rank~ I, then M = P| with P a projective Arc-module locally free of rank one. Proof. -- With L the field of quotients of A let A be a maximal order of Lrc containing Arc, and let a be the annihilator of the A-module A/A~. Characteristic zero guarantees a+o. If we semi-localize at the maximal ideals containing a we obtain a free summand of M (Serre's Theorem in dimension zero) generated, say, by eeM. Let P be the A-pure submodule of M generated by cAre; this is automatically an A=-submodule. Then o-+P~M~M/P~o is an exact sequence of torsion free Arc-modules which splits at all maximal ideals containing a. At all others it splits auto- matically because Arc there agrees with the hereditary ring A. Hence the sequence splits -- its is an element of Ext],(M/P, P) which vanishes at all localizations. P, being locally projective, is projective. Finally, since L| is Lrc-free of rank one, a theorem of Swan [32] (see also [5]) guarantees that P is locally free of rank one. Remark. -- Short of the last sentence, and its special conclusion, it is clear that we have invoked only very general properties of Arc. Corollary (IO.3). -- With A and 7: as above, if M is a finitely generated torsion free Arc-module of f-rank> 2, and Q a finitely generated projective An-module, then Q|174 => M~M'. This Corollary responds to a question of Swan [32] and Swan himself has shown it to be best possible [33]. He shows, moreover [33, Theorem 2], using a result of Eichler [i9] , that if A is a maximal order in a semi-simple algebra over a number field, then one can always cancel projectives unless R| has a quaternion factor. w xz. A stable range for GL(A), and a conjecture. We keep A, X, and A fixed as in (8.r). Theorem (ix.x). -- If d=dimX, then dq-I d~nes a stable range for GL(A), in the sense of w 4. Hence, the conclusions of Theorem 4.2 are valid for A with n = d + I. Proof. -- We must show that if r>d+ I and if Z~=IAai=A , then there exist 9 9 ", -- i,-l~xkai -V bx, b r 1r such that Z r-IAI 'bias) A. 513 3o H. BASS Note first that 0~ = (al, ..., at) is unimodular in A~= Ar-I| .3 = (o, ..., o, i); = 0~' q- ~a,. Since f-rank A r - 1 = r-- I >-d + i > dim X we can apply (9. i) and obtain f: ~A~A ~-I such that r is unimodular. Then f~= (bl, ..., br_l, o) solves our problem. Let q be an ideal in A, and write, in the notation of Chapter I, w i, Gr(q) = GL(r, A, q)/E(r, A, q). Let f~: G,(q)-->Gr + l(q) be induced by the inclusions. We obtain a direct sequence of sets with base points, and )im Gr(q)=GL(A, q)/E(h, q), which, by (3. I) a) is an abelian group. With this notation we can now translate parts of (4.2) in such a way as to exhibit the (partial) analogy with Proposition IO. i. Proposition (xI.2). -- Suppose dim X=d. Then for all ideals q, we have: a) ((4.2) c)) G,(q) is a group for r>d+I. b) ((4-2)b)) fr:G~(q)~G,+t(q) is surjective for r>-d+I. c) ((4.2) f)) G,(q) is an abelian group for r>-2(d§ and >--3. The missing link here is an assertion that f~ becomes injective. Our prevailing topological analogy suggests quite explicitly in this regard, the following, Conjecture. -- Under the conditions of (ii.2) f~:G,(q)-+G~+l(q) is injective for r>d+ I. In terms of matrix groups this says, for r>d+ I, E(r+ I, A, q) nGL(r, A, q) =E(r, A, q). When A is a division ring (so d= o) it is the affirmative solution of essentially this problem which constitutes Dieudonnd's theory of non commutative determinants [i 7]. Klingenberg [24] has generalized his solution to local rings (still d=o), although Klingenberg's proof is not valid when q--A. On the other hand it works in any ring provided q Crad A. This procedure of axiomatically constructing determinants (see Artin [I, Chap. V], for a good exposition) runs into severe computational difficulties if one tries to generalize it naively. The interest in the conjecture above stems from more than a simple love of symmetry. One can consult w 20 below and [7, w 1] for some rather striking consequences of its affirmative solution. 514 CHAPTER III THE FUNCTORS K w x2. K~ and K~(A, q). Let A be a ring and ~ =,~(A) the category of finitely generated projective right A-modules, and A-homomorphisms. Let "; = TA : obj ~K~ solve the universal problem for maps into an abelian group which satisy (A) (Additivity) If o->P'-+P~P"---~o is ancxact sequence (asA-modules) then TP = yP' + TP". Uniqueness of (TA, K~ is the usual formality, and existence follows by reducing the free abelian group generated by isomorphism types of obj ~ by the relations dictated by (A). Let T be an infinite cyclic group with generator t. We build now a new category, ~[T] =~r t-1], whose objects are A-automorphisms, ~, of modules P = dom c~eobj ~. If ~'eAutA(P' ) is another, a morphism, ~-~e', is an A-homomorphism, f: P--+P', such that fe=~J. If ~eAutA(P ) then e defines an A-representation ofT on P, t acting as c~. In this sense we can think of ~[T] as a category of A[T] =A[t, t-l]-modules, and as such we may speak of " exact sequences " of ~'s. Let q be an ideal in A (possibly q=A) and let ~q[T] be the full subcategory of ~[T] whose objects are those e for which " ~-: modq"; i.e. if P----dome, we require that c~(~AA/q = i on P| We define the group K~(A, q) by letting W~ : obj .~q['F]-+K~(A, q) solve the universal problem for maps into an abelian group which satisfy (A) (Additivity) If o-,='-.~-~0d'~o is exact then Wq==W,~'+W,=". (M) (Multiplicativity) If dom = =dom [~ then W,=~ =Wq~ +W,~. Existence and uniqueness are clear by a remark analogous to that above for K ~ Although we have no need for this fact, the reader will be able to determine easily that K ~ is unaltered if we relax (A) to apply only to split exact sequences. When q=A we call K:(A)=K~(A, A) the " Whitehead group " of A, and W=W a the " Whitehead determinant " 515 3 2 H. BASS Let ~ be a automorphism in :~q[T] and Peobj ~. Then (M) implies WqIe:-=o, so (A) further implies that Wq(a|162162 If 0r A, q) r A) =AutA(A" ) then we can regard aceobj ~q[T]. With our convention, GL(n, A)r A), ~ is identified with ~| The last para- graph shows that Wq respects this convention, and hence we have a map, also denoted Wq, W, : GL(A, q) = U, GL(n, A, q) -+KI(A, q), which, by (M), is a homomorphism. Suppose aeGL(A, q) and ~EGL(A). Then ~-t:r so Wq:c----Wq(~-%r This means that [GL(A), GL(A, q)] ---- E(A, q)Cker Wq, so we have an induced homomorphism f: GL(A, q)/E(A, q)--->Kt(A, q). Proposition (i~,. I ). -- The inclusions GL(n, A, q)Cobj ~q[t, t -t] induce an isomorphism f: GL(A, q)/E(A, q)~Kt(A, q). Pro@ -- Let xeobj ~q[T], P-=dom=. We can find a Q such that P| (some n). This isomorphism induces an isomorphism a| A, q). With Q fixed % varies in its conjugacy class in GL(n, A), so its image in G=OL(A, q)/E(A, q) doesn't change. If we replace Q by Q| a, is replaced by a conjugate of %| , and again its image in G is unaffected. Finally, if P| then Q|174174174 so we see that the image of % in G is independent of Q. We define thus a map g : obj ~q[T]--~G, and we propose to show that g is additive and multiplicative. Once shown, the uni- versality of W, produces a unique homomorphism Kt(A, q)-->G, which is manifestly an inverse for f. g is multiplicative. For if dora a = P -- dom ~ an isomorphism P~)Q~A" induces O~| ~IQ~n , and O~@IQ~OCn~ n. g is additive. Let o-->~'-->~-->0d'-+o be an exact sequence, with domains P', P, P", respectively. Choose Q' and Q" with isomorphisms P'|174 Since the exact sequence induces P-'~P'| we can choose an isomorphism P|174 compatible with the direct sum of the given sequence with o-->iq,--->Iq,| With these isomorphisms, we have ~'@IQ,~Q(n, ~"@IQ,,~gn' , a~, c(: I :' o I (%~)-1q and a| oq ..... a2,=[o ,, :r 0 I I " Since the second factor is manifestly in E(2n, A, q), and since 0r n OCnCX n 0 -- mod E(2n, A, q), ] o 1 o (x o I by the Whitehead Lemma (I.7), we see that ae, and %~' do indeed agree in G, as required. 516 K-THEORY AND STABLE ALGEBRA Suppose H is a normal subgroup of GL(A). Then, by Theorem 3.i, E(A, q)r q) for a unique q, and H/E(A, q) is, via Proposition i2. I above, a subgroup of K~(A, q). Conversely any subgroup of KI(A, q) defines in this way a normal subgroup of GL(A). Thus we see that a determination of all normal subgroups of GL(A) is equivalent to a determination of all subgroups of KX(A, q), for all q. Finally we note that K ~ and K 1 are functors. If q0 : A-+B is a ring homomorphism, then | : ~(A)-+~(B) induces ~~ : K~176 If q is an ideal in A and q' an ideal of B containing q~(q), then | induces qr K~(A, q)~Kt(B, q'). Note that the isomorphism of Proposition I2. I is an isomorphism of functors. w 13. The exact sequence. Let q~ :A-+B be a ring homomorphism. If P andf are a right A-module and A-homomorphism, we shall abbreviate PB=P| , and fB-f| Our objective is to construct a relative group, K~ ~0), which will fit into an exact sequence (Theorem I3. i, below). To this end we manufacture the category c~(~) whose objects are triples, ~ = (P, a, Q.), with P, Qeobj ~(A) (i.e. finitely generated projective right A-modules) and ~ a B-isomorphism, PB-+QB. If ~'= (P', 0~', Q') is another such triple, a morphism ~r'-+, is a pair, (f, g), of A-homomorphisms, f: P'-+P and g : Q.'-~Q, making P'B > Q'B /B : gB PB ~, QB commutative. We call ,(I',r a(t"'g';I .... " exact if P'LP2P" and Q'-~Q.~Q" are both exact sequences of A-modules. Note that the objects of ,~(q~) are a groupoid. Thus, if a=(P, e, P') and '= (P', e'P"), then we write a'e = (P, ~'~, P"). Now we define K~ ~) by letting R : obj c~(?)~K0(A, q~) solve the universal problem for maps into an abelian group which satisfy: (A) (Additivity) If o -+ a' -+ a -+ a" --+ o is exact then Rc = R~' + R~". (M) (Multiplicativity) If ~'~ is defined then R~'a = R~' + R~. 5 34 H. BASS We need to know K~ ?) in some detail, and it will be convient to introduce some provisional terminology for this purpose. A triple -: = (P, ~, P) will be called an " automorphism ". Since ~eAutB(PB), WB~eK~(B) is defined, and we shall write WB-:=Ws,8 in this case. ve=(P, ip, P) will be called an " identity ". Since ~1,~v--'P, (M) implies (I) Rcp=o. If there is an exact sequence then (A) further implies R~=o. Let t3~GL(B)=U,GL(n,B); say ~3~GL(n,B)=Aut~(A"B). Then = (A", [~, An) eobj ~(~). Viewing ~eGL(n+m, B) replaces . by u| , so R~ is unaltcred. Hence we have a well defined map GL(B) ~K~ ~), (s) which is a homomorphism, by (M). Now if ~ is elementary, then , appears in an exact sequence of type (2), with P--A "-I, Q----A. Hence E(B) is in the kernel of (3), so (3) iuduccs a homomorphism 8 : K~(B) = GL(B)/E(B)-+K~ ~). Let 9 = (P, ~, P) be an automorphism. If WBx=o then, by Proposition I2. I, we can find a Oso that T| y, An), with TeE(n, B), and hence Rv--o. Now if ~, a'cobj (~(~) write if there exist identities, v and z', and an automorphism ~ with W~r such that (a|174 A tedious, but straightforward, exercise shows that --- is an equi- valence relation. Our earlier remarks show that a~--a':~Ra=Ra'. Hence, if R' :obj ~(?)--->G=obj cd(?)/~-- is the natural projection, then R=hR' for a unique h : G-+K~ ?). An easy check shows that .-~ rcspccts | so that | induces on G the structurc of a monoid, with neutral elcment the class of the idcntitics, and rclative to which h is clearly a homomorphism. We propose to show that a) G is a group, and that R' is b) additive and c) muhipli- cative. Once shown, the universality of R produces a homomorphism h' : K~ ?)--->G which is clearly an inverse for h. This isomorphism will permit us to conclude: (4) Every element of K~ ~) has the form R~, ~eobj cg(~). R~=o-~-.o. a) G is a group. Given a=(P, 0c, Q), let a'~-(Q,--a -t,P). Then ~|174 L 0~ P| 0r -t O' 818 K-THEORY AND STABLE ALGEBRA It follows from the Whitehead Lemma (i.7) (using the fact that PB~QB) that =o. b) R' is additive. Let o -+ a' ---> z --.'- a" -+ o be exact. If ~'=-(P', cr Q') and a"= (P", 0r Q"), then ~=~= (P'| 0% Q'| where = = o = 0r [O Ip,, B Since W~s=o we have (h,~(~'| as required. c) R' is multiplicative. Suppose (~ = (P, e, P') and (~'= (P', ~', P"). We must show (~'~(;| From the commutative diagram PB ~ > P'B 1pB / (-- 1p)B PB- P'B we see that e~.--~=(P, --e, P'). Hence it will suffice fbr us to show a'~| = (P| [3 -- o Ip,~ ' P"| and ~, P"| (--q)|176 (P| Y= t, -- where ~= (POP', y-t~, P| Since PB~P'B it follows from the Whitehead Lemma (1.7) that W~,=o, and this completes the proof of c), hence also of (4). Finally, we define a homomorphism d: K~ q~)-+K~ by d(R(P, ~, Q)) =yAP--yAQ. It is clear that d is well defined. Theorem (,3.I). -- Let (p :A~B be a ring homomorphism. Then the sequence ~1 8 0 o KI(A) --->KI(B) ~K (A, q~) s176 ~K~ is exact. If q~ : A~A/q is the natural map, then KI(A, q) ~KI(A) -~K*(A/q) ~K~ q) -~K~ ->K~ is exact. (In the second sequence, K~ q) =K~ 9), and dl is induced by the inclusion GL(A, q) cGL(A).) Pro@ -- Exactness at K~ v~ P, :% Q)) -~ q~~ = YB PB-'(BQ B -- o 519 3 6 H. BASS (since ~ : PB~QB). Suppose q0~ =yBPB--~,BQB=o. Then there exists an 0c : PB|174 for some n, and hence Y~P--'(AQ= d(R( P| 0~, Q| Exactness at K~ q~). If ~eGL(n, B) then dS(WB~ ) = d(R(A", ~, An)) = y~A"--yAA" = o. Suppose d(R(P,e,Q))=gaP--TaQ=o. Then P|174 for some P', and we can even arrange P@P'~-A". Hence (P| 0~| Q| f~, A") for some ~GL(n, B) and R(P, 0~, Q)-----=~(W~(~)). Exactness at Kt(B). If ~eGL(n, A), then ~qr = ~(W~q~) = R(A", 0~B, A") (note ~e=eB). But the commutative square A"B ~B> A"B ~B I l~nB A,B ~*"~ A,B shows that R(A n, eB, A n ) =o. Now suppose ~eGL(n, B) and 3(WB~ ) =o. Then, by (4) above, a = (A ~, ~, A "),-,o. This means that a|162 for some automorphism s, with WB~---o. We can add an identity to both sides and further assume P=A" and ~ = (A "+m, y, A~+"). Then the isomorphism a|162 is given by isomorphisms f, geGL(n+m, A) making g,+m f~ 1,~> B,+, ' gB fB gn+ m __ "( ~ Br~ +m commutative. Hence, in GL(n+m,B),~=(?g)-l"~(?f). Since WBy=o we have = = Now suppose q0 : A-+A/q. Exactness at KI(A). If eeGL(A, q) then q~4(Wqe) =q~l(WAa ) =WA!o(q)~ ) =% since q~e----I. Suppose aeGL(A) and qo~(Wxa)----WA!q(q~e)=o. Then q)aeE(A/q), so it follows by Homotopy Extension (I.I) that there is an seE(A) with q~,=q)e. Hence ~-~EGL(A, q) and WAe----WA(~s -~) =da(Wq(~s-1)). If we knew how to define, and extend the exact sequence to, K 2 the next result would be an immediate corollary. Proposition (x3.2). -- Suppose A=A0| q (as abelian group) with A 0 a subring and q an ideal. Then there are split exact sequences o-+K'(i, q)~K'(A)~K'(h0) -~o, i=o, I. 520 K-THEORY AND STABLE ALGEBRA Proofi -- Let r o be the retraction with kernel q. Since r has a right inverse, so does r i=o, :. Hence the Proposition follows from (I3.:) provided we show Ki(A, q)-+KI(A) has a left inverse. If a~GL(A), a----~oa: where ~o=q~ceGL(Ao), and a:-----(q~o~)-lo~EGL(A, q). If also ~eGL(A) then (0C~) 1=q)(0~)-16r = (~9~)--1(q)0r = (q9{~)--10~1~ = [q)~, 6r = [~0, ~ll]gl~l 9 Now [D0, ~7']e[GL(A), GL(A, q)] =E(A, q). Hence W,(~):=Wq~+Wq~:, and we have constructed a homomorphism GL(A)->KI(A, q) whose restriction to GL(A, q) is Wq. This clearly induces the required retraction K:(A)--~K~(A, q). Examples. -- :) If A is a local ring (e.g. a field) then K~ and K~(A) is the commutator quotient group of A*---- GL(:, A). The latter is due to Dieudonn6 [i 7] for division rings, and to Klingenberg [24] in general. 2) If qCrad A (Jacobson radical) and r : A~A/q, then q~l is easily seen to be surjective, and (p0 to be injective (see Lemma I8. i below). Hence K~ q) -----o. The methods of Klingenberg [~4] adapt without essential change to compute KI(A, q) also in this case. In case A is q-adic complete, or if T has a right inverse, then cp ~ is even an isomorphism. 3) The following remark is often useful. Let A be semi-local and P, Q eobj ~(A). Then yAP=yAQ=>P-~--Q. For yAP=YAQ implies p| " for some n, so our conclusion follows from (6.6) by induction on n. 4) In w I6 we describe in detail the exact sequence associated with the embedding of a Dedeking ring in its field of quotients. w 14. Algebras. Tensor products endow our functors with various ring and module structures, and it is convenient to record these circumstances now. Let A be a commutative ring, A and A' A-algebras, and q an ideal in A. If P~(A) and P'e~(A') then P'QAPe~(A'QAA), and this induces a pairing, (: ') K~ ') x K~ ~ K~174 If ae~(A)q[t, t-l], then :e,Qoce~(A'|174 t-t] and this induces a pairing (2') K~ ') x K:(A, q)-+KI(A'QAA, a'| Taking A'----A=A in (i'), K~ becomes a ring. Then with A'=A in (i') and (2'), K~ and KI(A, q) become K~ Moreover, the pairings are K~ so they define (i) K~ ') |176 --> K~ (A '| and K~174 q)--+KI(A'| A'Nq). (2) the obvious naturality properties with respect to A-algebra These structures have homomorphisms. 521 3 8 H. BASS In order to treat K ~ and K 1 simultaneously we shall sometimes consider the following situation. Let q be an ideal in A. For an A-algebra, A, write K*(A, qA)= K~174 qA). K*(A, qA) is, as noted above, a K~ Moreover, (~) above gives us K'(A, q)|176 qa)= KX(A| qQA). Hence, if we view K*(A, q) as a graded ring, zero in degrees 52, then K*(A, qA) is a graded K*(A, q)-module. Finally, if A-+B is a homomorphism of commutative rings, then B@ A induces K*(A, q)--~K*(B| , B| In case B is finitely generated and projective as an A-module, then there is an obvious " restriction " functor ~(B|174 , t -t] --->~(A)q[t, t-t], and this induces a homomorphism K*(B| , B| q) which, following the one above, gives the homothetie of K'(A, q) defined by v A(B)eK~ w 15, A filtration on K ~ There seem to be several " geometrically reasonable " filtrations on K ~ (under suitable circumstances). We choose one here with the properties needed for our applica tions. Let A be a commutative noetherian ring, and X=max(A). All modules and A-algebras will be understood finitely generated as A-modules. Let A be an A-algebra. We shall consider complexes, C : ...-+Pi->Pi_l-+... of right A-modules which are finite and projective (i.e. Pi is projective, and = o for almost all i). Then Z(C)=zA(C)=Y~i(--I)'yAP~eK~ is defined. Our finiteness conditions guarantee that H(C) is a finitely generated A-module, so supp H(C) is a closed subset of X. Since localization, being exact, commutes with homology, we see that supp H(C) ={xeX I H(C)x+ o} ={xeX]Cx is not acyclic}. K~ consists of all ~eK~ with the following property: Definition. -- Given Y closed in X, there is a complex C, as above, such that z(C) =4, and codimy(Yn supp H (C))~i. Proposition (x 5. x). -- The K~ are subgroups of K~ which satisfy: :) K~ = K~ K~ K~ 2) If P is another A-algebra the pairing T : K~174176176174 (see w I4) 522 K-THEORY AND STABLE ALGEBRA induces K~174 K~ K~ O~,F);, j. In particular, K~ is a filtered ring, and K~ a filtered K~ 3) The .filtration is natural with respect to homomorphisms of A-algebras. 4) K~ for />dim X. A very useful consequence of this Proposition is Corollary (I5.2). --If d=dimX, then K~ Proof of 15. i. -- K~ is a subgroup because the support of a direct sum is the union of the supports, and the codimension of a (finite) union is the infimum of the codimensions. I) follows from the definition. 2) Let 4~K~176 and let Y be closed in X. Choose C' over A with zA(C') =4 and codimy(Yrasupp H(C'))>i. Choose C" over F with )q.(C") ----~ and codimyqsuppu(c,)(Yr~suppH(C')r~suppH(C"))>~. Then, if C=C'| over A| , it is clear that ZA|174 Moreover the inequality above implies codimy(Yc~supp H(C')r~supp H(C"))>i+j. Hence we can conclude by showing that supp H(C) csupp H(C')nsupp H(C"). But if, say, C~ is acyclic, then it is homotopic to zero (a finite acyclic complex of projective modules), so likewise for = cx. 3) Let f: A-+r be a homomorphism of A-algebras, and ~eK " 0 (A)i , . we want f~176 Given Y, choose C over A with 7,A(C)=4 and codimy(Ynsupp H(C))>i. Since f~ ) it suffices to note that supp H(C| Csupp H(C). But if C~ is acyclic it is homotopic to zero, so likewise for C~| ~ = (C| 4) Take Y=X in the definition. Proposition (z5.3). - K~ n ker(K~ -,K~ xEX Proof.- If 4eK~ and Y--{x} then 4 = 7.A(C) with codimy(Yr supp H(C))>I. This implies xr H(C), so C~ is acyclic. But then the image of ~ in K~ is z (cx) = o. Conversely, given 4 = Y.~.P--YAQeK~ to belong to the right side of our equation means y&P~=y~Q, for all x, and then by (6.6) (see Example 3) in w I4) , P~--~Q~. Now, given Y closed in X, let Yi, 9 9 Y, be the irreducible components of Y, and choose xieYi--U~..~Y i. Then, if we reduce modulo the product of the xi, P and Q become isomorphic. Lift such an isomorphism to f: Q~P; then f*i is an isomorphism, i <i<s. Let C be the complex with f the differential in degree I, and zero in all degrees 4=0 or I. Then z(C)=(--i)lyaQ+(--I)~ Since Y~r codimy(Ynsupp H(C))~I, as required. Proposition 15 . 3 gives a description of the first term of the filtration which behaves well without any finiteness assumptions on A, as we shall see below. Let A be now an arbitrary commutative ring, and let spec(A) denote its prime 523 4 ~ H. BASS ideal spectrum (Zariski topology). If Peobj ~(A) and xespec(A), then P, is a flee A,-module of rank pp(x), pv : spec(A)->Z. It is easy to see that pp is continuous (for the discrete topology on Z). Let C(A) denote the ring of all continuous functions from spec(A) to Z. Since Pv is (clearly) additive and multiplicative in P it induces a ring homomorphism p : K~ p even has a right inverse q~. To define % suppose f: spec(A)~Z is continuous, and let X,=f-l(n). Since spec(A) is quasi-compact, X~=D for almost all n. Now disjoint decompositions spec(A)=O,X~, all X, open, correspond, bijectively, to decomposi- tions l--E,e, of I as a sum of orthogonal idempotents, almost all zero (see e.g. [3% Chap. I]). Define ~(f)=Z,f(n)yA(Ae,) ; ~? is clearly the desired right inverse to p. We shall use q~ to identify C(A) with a subring of K~ Thus, K~ = C(A)| where J(A)=kero. Suppose ~=y,P--yAQeJ(A). Then p~=pe--pq=o. Thus for all xespec(A). Conversely, if P~Q~ for all xemax(A), then we see by localizing in two steps that Px~Q~ for all xespec(A), so yAP--yAQeJ(A). Let us summarize: Proposition (x5.4). --- Let A be an arbitrary commutative ring and let J(A)= fl ker(K~ -+K~ x C max (A) Then K~ = C(A)| where C(A) is isomorphic to the ring of all continuous functions from spec(A) to Z. J(A) is both the nil and Jacobson radical of K~ Proof. -- It remains to prove the last assertion, and it clearly suffices to show that every ~ = yP-yQeJ(A) is nilpotent. If A were noetherian and of finite Krull dimension this would follow from (x5.2) and (15.3). In general, however, ~ is induced from a finitely generated subring of A, and such a subring is of the latter type. Hence our conclusion is a consequence of the following lemma: Lemma. -- If f: B-+A is a homomorphism (commutative rings, then ( fo)- ~ (J ( A ) ) zJ(B), with equality if f is injective. Proof. -- If ~eJ(B) we want f~ i.e. if xespec(A) we want f0(~) to go to o in K~ But B--->A-+A x is the same as B-+Bv-+Ax, where y =f-l(x) ~spec(B) and ~ goes to o in K~ Conversely, if ~K~ we want to show" f~ Thus, given 524 K-THEORY AND STABLE ALGEBRA yespec(B), we must show that ~, the image of ~ in K~ is zero. Let S =f(B--y); since f is injective, this multiplicative set in A does not contain zero. Moreover B t> A Bu s-,~ S_IA is commutative. If S-1~ is the image off~ in K~ then S-I~=(S-~)~ Since By is local, ~=neZ=K~ and hence also S-1~=nEK~ But our hypothesis implies that this n vanishes in K ~ of any localization of S-1A, and hence n=o as desired. Nowfor any ~eK~ we can write ~=p~§ (~--p~), with ~ee(A), ~--p~eJ(A). We shall call 9~ the rank of ~, and often identify it with a function from spec(A) to Z. In particular, P(YP) =Pv is the " rank" of the projective module, P. Expressions like " rank ~> r ", reZ, make sense now, where we think of r as a constant function. If P is a projective A-module, rank P>r is equivalent to "f-rank P>r " in the sense ofw 7- Proposition (I5.5). -- Let A be a commutative ring and P a finitely generated projective A-module. Then the following conditions are equivalent: a) P is faithful. b) rank P>I. c) Every K~ annihilated by yAP is torsion. d) 1| generates the unit ideal in Q|176 Proof. -- a).r because P is locally free, and a finitely generated projective module is faithful if, and only if, it is locally non zero. b) => c). Since pp is everywhere positive and bounded, there is a function feC(A) with fpp=n>o. Hence fyP=n+j, with jeJ(A) nilpotent. Thus, modulo yP.K~ n is nilpotent, so n'~eyP.K~ for some m>o. c) => b). If xespec(A) and P~ = o, then K~ is a K~ annihilated by ~,P. c)~d) is evident. Proposition (x5.6). -- Let A be a commutative ring for which max(A) is a noetherian space of dimension <d. Then, if ~eK~ has rank>d, ~=yP for some P. If rank P>d and gP=yQ, then P~-Q. Proof.-- We can write ~=yQ--yA'. Then pq=p~§ Hence, by Serre's Theorem (8.2), Q~P| so ~=yP. If yP=yQ for some P and Q, then P|174 for some projective P'. If rank P>d we can invoke the Cancellation Theorem (9.3) and conclude that P=~Q. tl CHAPTER IV APPLICATIONS w I6. Multiplicative inverses. Dedekind rings. Throughout this section all modules are finitely generated right modules. Proposition (xfi. I). -- Let A be a commutative ring with max(A) a noetherian space of dimension d<oo, and let A be a finite A-algebra. If P is a projective A-module such that I@yA1 ) generates Q|176 as a (QQzK~ then there is a projective A-module Q, such that QQAP is A-free. Pro@ -- i| n is a Q|176 multiple of I| , and by choosing n large we can solve INyAAn=I| P) with ~eK~ Hence yAAn--~.yAP is a torsion element in K~ If then we replace n by a muldple of n, we can achieve yA A" = "~. YaP" Let xEmax(A); K~ and the image there of ~ is (p~,)(x) (see w 15 for the definition of p). The equation above together with (6.6) (see also Example 3) in w i3) implies that A~- (p~) (x). Px, and this evidently implies (p~) (x) >o. Thus, if we replace n by a further multiple, if necessary, we can achieve p~>d. It then follows from (I 5.6) that ~ =YAQ, for some projective A-module Q. Our equation above then becomes ya(A')=yA(Q| Since, by this time, n>d, we can invoke the Cancellation Theorem (9.3) and conclude that QNAP~A", as desired. Corollary (I6.2). -- Let A be any commutative ring and P a faithful projective A-module. Then QQA P is A-free fi~r some projective A-module Q. Proof. -- Since P is induced from a finitely generated subring of A, it suffices to solve our problem there, so we may assume A a noetherian ring of finite Krull dimension. We can now invoke (16.1) provided I| generates QQK~ but the latter is a consequence of (I5.4[). Let A be a commutative ring and m the class of faithful projective A-modules. Write P~Q if PQAA"=~Q| "~ for some n, m>o. Then ~ is an equivalence relation respecting | so that M(A) =m/~ is an abelian monoid with neutral element the class of the free modules. Corollary 16.2 says that M(A) is even a group. Proposition (x6.3). -- Let A be a commutative ring for which max(A) is a finite dimensional noetherian space. Then M(A) = GL(I, QQzK~ Q). 526 K-THEORY AND STABI.E ALGEBRA 43 Proof. -- Let uswrite B-~Q|176 and B*~-GL(t, B). If P~m, let ~P denote its class in M(A). By (15.4) i| let ~zP denote its class in B*/O*. Evidently f: M(A)~B*/Q* by f(~P) ==P is a well defined homomorphism. Suppose ~zP= i ; i.e. I| Since K~ ----- C(A)| (i 5. 4), we have B ---- (Q|174 (Q| and writing yP=~p+(-(P--D) we see that IQyP= IQppeQ'. Hence n(~P--pt~ ) =o for some n>o, so nyP=y(A"| Choosing n sufficiently large we conclude from (I5.6) that A"QP~A", so ~P= i. This shows thatfis a monomorphism. To see that f is surjective consider an element of B*. Modulo Q* we can assume it has the form i| with ~_K~ of rank~dim max(A). Hence ~----yP for some P by (I5.6), and the class of ~ mod Q* is nP--f(~P). The classical Steinitz-Chevalley theory [I5] of modules over a Dedekind ring furnishes a familiar setting in which to illustrate the general shape of our theory. We consider finitely generated torsion free (hence projective) modules P, over a Dedekind ring A. First P~aGF, a an ideal and F free -- Serre's Theorem. If POA:-~-P'OA then P-'~-P'; if we required rank P~2 this would be the Cancellation Theorem. The stronger conclusion here is possible only because of commutativity. If a and b are non zero ideals, a|174 As an equation in K~ we recover this from the nilpotency of J(A) (Proposition I5.4): (I--ya)(i--yb) =o in K~ We indicate below (Proposition i6.4) how to recover the actual isomorphism, as well as a variety of similar identities. With these facts we see easily that, as a ring, K~174 with J(A) an ideal of square zero, additively isomorphic to the ideal class group G of A. Alternatively, if I is the augmentation ideal of the integral group ring ZG, then K~ 2 as an augmented ring. Let ? : A-+L be the inclusion of A in its field of quotients. We propose now to interpret the exact sequence KI(A) ~K~(L)~K~ ~) -~K~ K~ q~o dim The composite K~176 is the augmentation, " rank ", with kernel J(A)~--G. Next we recall that K~ ?) is built out of triples (P, a, O) with P and O A-projective, and ~ : L|174 an L-isomorphism. Using the description of K~ ?) given in w 13 one can show easily that every element is represented by a triple (a, u, A) with a an ideal and ueL*, viewed as a homothetie of L. Moreover, the fractional ideal au is an invariant which defines an isomorphism of K~ ?) with the group of fractional ideals in L. With this identification, K~ ?)-+G=~-ker c? ~ assigns to each ideal its class. Moreover, K ~(L) d~,.~ L*, and KI(L)-,K~ ?) assigns to ucL* the principal ideal Au. Thus, thekernelis A*=GL(I,A)--Imq~ ~. Finally, kerq~a----SL(A)/E(A) is the commu- tator quotient group of SL(A). We shall see in w 19 that ifA is the ring of integers in a (finite) algebraic number field, then this group is finite. 527 44 H. BASS Now let A be any commutative ring. If P is an A-module, and n a non negative integer, let nP denote a direct sum of n copies of P, and P| a tensor product of n copies of P. Also, if Qis another A-module, write P+Q=P| and PQ=P| With these conventions, if f(T1, ..., T.) ---- Zaq...i,T lq. . .T~"eZ[TI, . .., "In] is a polynomial with non negative coefficients, and if P1, -.., P, are A-modules, we can write 9 ., pO q f(P1, 9 P~) =Y, ai,. ,~ 1 .-.e~'~. Proposition (16.4). -- Let A be a commutative ring such that max(A) is a noetherian space of dimension <n. Then, if P1, ..., P, are invertible A-modules and if S,(T1, ..., "In) is the i th elementary symmetric function, we have Z Si(P1, ..., Pn)= ~oSr ..., P~). i even i Proof. -- I -- yPieJ (A) for all i, so II~= 1 ( i -- YPi) = o by (i 5.2). Thus the equation is valid after applying y to it. Since the ranks of the two sides of the equation exceed dim max(A), (15.6) permits us to remove the y. w z 7. Some remarks on algebras. For a ring A let A-mod and mod-A denote the categories of left, respectively, right, A-modules. A generator for such a category is a module whose homomorphic images suffice to generate any other module. Let E be a right A-module and put F----HOmA(E , E) and E* ---- HOmA(E , A). We are in the situation (tEA, AE~.), and there are natural bimodule homomorphisms: E*| (~) EQAE*--~ I" Moreover, we shall consider the functors: (2) E*| r : F-mod~A-mod. E| A : A-mod--~F-mod The following basic result is due essentially to Morita [26], although the best exposition of these, and other matters in this section, is in Gabriel [21, Chapter V, w i]. Theorem (Morita). -- Let A be a ring, E a right A-module, F=HomA(E , E), and E*= HOmh(E , A). Then the following conditions are equivalent: a) E is a finitely generated projective generator for mod-A. b) The homomorphisms (i) are isomorphisms. (It suffices that they be epimorphisms.) c) The composites of the functors (~) are each naturally equivalent to the identity functors. In this case q<-->Eq defines a b~iection between the two-sided ideals of A and the I'-A-submodules of E. Conversely, if A and F are rings and A-rood is equivalent to P-mod, then any such equivalence is isomorphic to one as in (2) above, with E determined up to F-A-isomorphism. Thus, the situation above is entirely symmetric with respect to E and E* and to A and F. 528 K-THEORY AND STABLE ALGEBRA Now let A be a commutative ring and A an A-algebra. Then, in the above setting, A-rood is an " A-category " (i.e. all the Hom's are A-modules), likewise for the A-algebra P, and the functors in (2) are " A-functors" (i.e. induce A-homomorphisms on the Horn's). The converse part of the Morita Theorem remains true also in this sense, provided the given equivalence from A-mod to P-mod is assumed to be an A-functor. If A-mod and F-mod are A-equivalent we shall call A and P Morita equivalent, denoted A~F. Let E be a finitely generated, projective right A-module. It is easy to see (e.g. using Proposition 15.5) that E is then a generator for mod-A r E is faithful. Using the Morita Theorem we then see that, if F=HomA(E , E), F|174 , EQAA)~-~MA. In this situation we shall write A~B(F| ) and call the two algebras Brauer equivalent. Suppose A,-~MA' , say A'=HOmA(P , P) with P a finitely generated projective generator for mod-A. Can we show that A~BA' , i.e. that the two equivalence relations are the same ? Suppose there is a faithful, finitely generated, projective A-module Q, such that Q| A. Then we have A',--BHom * (Q, Q) | =~ Hom A (Q| P, Q| P) Hom A (A n, A")--~ Hom A (A", A") | A--,~A. Thus, modulo the existence of O, ~M and ~B agree. On the other hand, Proposition 16. i gives a criterion for the existence of Q which we will verify under suitable conditions below. Let A be an A-algebra, A ~ the opposite algebra, and A*=ANA A~ To avoid confusion we shall use E to denote A viewed as a right A-module. A is called an Azumaya algebra (=central separable algebra in [4], see also [21, Chapter V, w I]) if it satisfies the following equivalent conditions: (i) A is a projective generator for A%mod. (ii) a) E is a faithful, finitely generated, projective A-module. b) Ae~Homa(E, E) as A-algebras. In this case every ideal of A has the form qA for some ideal q of A [4, Corollary 3.2]. We propose now to consider the K*(A, q)-module K*(A, qA), with A an Azumaya algebra. For this purpose we use the diagram of functors mod-A mod-A l' (x)AeE @(A ~ AAe) (A '?" AE) mod-A ~*~~ mod-A * mod_(A| in which, by the Morita Theorem and condition (ii) a) above, the vertical arrows are equivalences. We obtain thus K*(A, q)-homomorphisms (*) K*(A, qA)LK*(A, q)--->K " " (A, qA), .:~;AA> 4 6 H. BASS To compute them, let M be a right A-module. Then (M| A~ ) | | ') (A| --~ (M| | (AeNh ~E ) ~-- M| It follows that gf is the homothetie defined by yA(E)~K~ Next we note that (M|176174174174176174174 (as A-modules), recalling that E =A viewed as a left A e-, right A-module. Hence f is the " restriction " map obtained by viewing A-modules as A-modules. In particular, yA(E)EIm(f). Now by Proposition :5.5, :|165 is a unit in Q|176 for any faithful, finitely generated, projective module P. Hence, using (ii) a), we see that O| and QGgf are isomorphisms, and we have proved: Theorem (I7.X). --- Let A be a commutative ring and A an Azumaya A-algebra. Then for all ideals q in A, Q QzK*(A, qA) is a free Q QzK*(A, q)-module generated by :| for any finitely generated faithful projective A-module P. Using Proposition I6. : and the discussion above we further obtain: Corollary (x 7. 2). -- Let A be a commutative ring for which max(A) is a finite dimensional noetherian space, and let A be an Azumaya A-algebra. Then: :) If P is a faithful, finitely generated, projective A-module, there is an A-module Q. of the same type, for which P| Q is a free A-module. 2) The class of A in the Brauer group of A (see [4]) depends only on the A-category A-rood. Theorem :7. : is not very useful for computing K~ since, in number theoretic contexts, all the interesting invariants are torsion. On the other hand the theorem is no longer true if we remove the " Q| ". To see this, let F=O(%/p) with p a prime - -- : mod 4, let I denote the ideal group of F, P the subgroup of principal ideals, and P+cP those principal ideals generated by totally positive elements. If A is the ring of integers in F, then %//~A~P, | For if %/fiA had a totally positive generator there would be a unit in A of norm --:, and by [23, p. 288] there is no such unit. Hence [P : P+]):, in fact, =2. Let Z be the standard quaternion algebra over F and A a maximal order in Z. Then Z is unramified except at ~, so A is an Azumaya A-algebra. On the other hand, Eichler [:9] has completely determined K~ (see Swan [33, Theorem 2]). Namely, K~174 +. Hence K~ is not isomorphic to K~174 (see w :6). This example was pointed out by Serre. A further amusing example in this connection is the following: Let Y. be the quaternion algebra over Q ramified atp and 0% p a prime -=-- : mod 4 and sufficiently large (e.g. p=:: will do). Z has a basis :, i,j, k with z'2=--:,j~=k2=--p, and ij = k =--ji. The result of Eichler-Swan quoted above shows that, if A is a maximal order in Z, then M(n, A) is a principal ideal ring for all n>__2. On the other hand, according to Eichler [I8, Satz 2], the class number of A itself is ):. Finally suppose A is a field and A is a central simple A-algebra. Then K~ so if B is any field extension of A, K~176174 is a monomorphism. What 630 K-THEORY AND STABLE ALGEBRA 47 about Kt? Dieudonn~ [17] shows that KI(A) is the commutator quotient group of A*=GL(I, A). Therefore, if B is a splitting field for A, we have homomorphisms A* ~ K1 (A) ---> K~ (B| ~ B *, and it is easy to see that the resulting map A*--~B* is the reduced norm (see Bourbaki, [1I, w I2]) and has image, therefore, in A*. Combining these remarks with the fact that every field is contained in a splitting field, we have: Proposition (x 7 . 3 ). -- If A is a central separable algebra over afield A, K I(A) ~ K l(B| x A) is a monomorphism for all field extensions B if, and only if, the commutator subgroup of A* is the kernel of the reduced norm. By a theorem of Wang [35], this is the case if A is a number field. w x8. Finite generation of K. We propose to show (Theorem 18.6 below) that if A is a finite Z-algebra, then K*(A, q) is a finitely generated abelian group for all ideals q. Lemma (I8. x). -- a) Let A be a ring. If N is an ideal in rad A, then K~ ~K~ is a monomorphism, and even an isomorphism if A is N-adic complete (e.g. if N is nilpotent). b) If A is a semi-local ring, K~ is a free abelian group of finite rank. Proof. -- a) Suppose yAP---~AQ~ker(K~176 After adding a free module to P and Q we can assume that P/PN~----Q/QN. Such an isomorphism lifts to a homomorphism f : P~Q. f is surjective mod N, hence surjective, by Nakayama. Hence kerf, being a direct summand of P, is finitely generated. Since kerf is zero mod N, it too is zero, by Nakayama. Suppose now that A is N-adic complete, and let P be a projective (A/N)-module, say P| (A/N)". If P =Homa(A" , A n) =M(n, A), then r/Nr =M(n, A/N). Let eeP/NF be an idempotent projection onto P. Since F is (NF)-adic complete, e lifts to an idempotent e'er (see e.g., [16, Lemma 77.4]). Now P'=Ime' is a direct summand of A n covering P, so ~,A/.~P is the image of yA P'. b) If A is semi-local, A/rad A is a finite product of simple Artin rings. Hence K~ A) is free abelian, of rank equal to the number of simple factors of A/rad A. By a) K~176 A) is injective, so our conclusion follows. Proposition (z8.2). -- Let A be a noetherian integral domain of Krull dimension one with field of quotients L. Let A be a finite A-algebra, N the nil radical of A, and T the torsion A-submodule 0fA/N. Then N is nilpotent, T is a semi-simple Artin ring, and A/N - T � r (product of rings), where P is an A-order in the semi-simple L-algebra L| A P. Proof. -- L| is a nil, hence nilpotent, ideal in the finite dimensional L-algebra L| Hence some power of N lies in the torsion submodule of A and is therefore a nil ideal of finite length. Therefore some further power of N is zero. For the rest we may assume N=o, i.e. A = A/N. Regard T as an A-algebra, possibly without identity. If J =rad T, J can be described as the intersection of all 531 4 8 H. BASS kerr, wheref is a T-homomorphism into a simple right T-module S, such that ST = S. (Note that S must be an A-module, andf compatible with this structure, in particular.) Suppose g : T-+T is a right T-endomorphism and f : T-+S as above. Then fg : T-+S so Jckerfg; i.e. g(J)Ckerf. Letting f vary we see that g(J)cj. Now letting g be left multiplication by an element of A we see that J is a left A-ideal (using the obvious fact that T is an ideal of A). Similarly, J is a right ideal. However, T has finite length as an A-algebra, so J is nilpotent. But A now has no nilpotent ideals . o, so J = o. Hence T is semi-simple, so it has an identity element, e. If a~A, then ae~T so ae = eae. Similarly ea=eae, so e is central. Thus A=T� where I ~=(I-e) A. Since I'is torsion free with zero nil radical, I'r174 and L| is a semi-simple L-algebra. Lemma (x8.3). -- Let A be a ring and N a nilpotent ideal finitely generated as a Z-module. Then, for all n> I, every subgroup of GL(n, A, N) is a finitely generated group. Proof. -- Induction on m, where N m = o, reduces us immediately to the case N ~ = o. Then GL(n, A, N) consists of all 1 -~- a where a is an n � n matrix with coordinates in N. If I 4- a' is another, then (I 4- a) (I 4- a') = I 4- (a 4- a'), so GL(n, A, N) ~N (the additive group). Theorem 18. 7 below will be proved by a reduction to the following classical results: Lemma (I8.4). -- Let A be an order in a finite semi-simple Q-algebra E=Q| o) (Jordan-Zassenhaus, see [37]). If M is a finitely generated E-module, there are only finitely many isomorphism types of finitely generated A-submodules of M. i) (See Siegel [3 I] or Borel-Harish-Chandra [9]) GL(n, A) is finitely generated for all n~ I. Moreover, if Z is simple, the subgroup of elements of reduced norm I in GL(n, A) is likewise finitely generated. Proposition (x8.5). -- Let A be a finite Z-algebra, and let q be an ideal in A. Then GL(n, A, q) is finitely generated for all n> I. Proof. -- If N is the nil radical of A, then GL(n, A, q)-+GL(n, A/N, q(A/N)) is surjective, and GL(n, A, q)r~GL(n, A, N) is finitely generated by (18.3). Hence we can reduce to the case N=o. Then A=T� as in (18.2) and GL(n, A, q) splits likewise into a product. It suffices then to treat T and r separately, and T, being finite, causes no problem. The result for P is a consequence of Siegel's theorem above (see Lemma I9.4 below). Theorem (x8.6). -- Let A be a finite Z-algebra and q an ideal in A. Then K~ and KI(A, q) are finitely generated abelian groups. Proof. -- Since dimmax(Z)=I it follows from (11.2) b) that KI(A,q) is a homomorphic image of 13I.(2, A, q), and (18.5) says the latter is finitely generated. Now for K ~ : Let N be the nil radical ofA and write A/N=T� asin(i8.2). Then, by (18.I), K~176176176 is an isomorphism, and K~ free abelian of finite rank. It remains to show K~ finitely generated. Let I" be a maximal order in O| containing r. Then I" is hereditary [3], so every projective I"-module is 532 K-THEORY AND STABLE ALGEBRA isomorphic to a direct sum of right ideals [I4, Chap. I, Theorem 5.3]. By Jordan- Zassenhaus (18.4, o)) there are only finitely many of these, up to isomorphism, so K~ ') is finitely generated. Now P has finite index, say m, in F'. Let S be the multiplicative set of integers prime to m. Then S-1F is semi-local, so K~ is free abelian of finite rank (18. r, b)). It will be sufficient for the theorem, therefore, to show that the homomorphism K~176176 induced by the inclusions, has finite kernel H. Let ,(rP--yrFncH; we may assume n>2. Then vr,(r'| and ys_.r(S-1P)=ys_,r((S-1F)"). Since F' and S-lP are algebras over rings with maximal spectra of dimensions I and o, respectively, and since n>2, (15.6) tells us that P| and S-1P~(S-1F) ". Let xemax(Z). If mex then P~ is a localization of S-1P, hence free. If mr then Px is a localization of P| r', hence free. Thus P~ is free of rank n for all xemax(Z), so we can apply Serre's Theorem (8.2) and write P~Q| "-t, with Q locally free of rank one, by (6.6). But then Qis isomorphic to an ideal in P, and again, byJordan- Zassenhaus (i8.4, o)), there are only finitely many such Q up to isomorphism. Since u F, this proves H is finite, as claimed. w 19. A finiteness theorem for SL (n, A). Let Y,=IIi~ ~ with Z i a central simple algebra over a finite algebraic number field C i. The reduced norms (see [ii, w I2]) give homomorphisms GL(n, Y~)-+C*=GL(I, C~) (compatible with the inclusions GL(n)r and their product defines a homomorphism GL(n, Z)-+C*, where C = II~C~ = center Z. We shall call this also the reduced norm, and denote its kernel by SL(n, 11). If A is an order in X and q an ideal[ in A we shall write SL(n, A) =GL(n, A)nSL(n, Y,), and SL(n, A, q) = GL(n, A, q)nSL(n, I1). Theorem (i9.i). -- Let 11 be a semi-simple algebra finite over Q, let A be an order in Z, and let SL(n, A) denote the elements of reduced norm one in GL(n, A) (in the sense defined above). Then there is an integer no=n0(11 ) such that, for all n>n o and for all ideals q in A, SL(n, A, q)/E(n, A, q) is finite. We shall begin by deriving a reformulation of this theorem which will be useful in its proof. The next two sections are devoted to some of its applications. Lemma (I9.2). -- If q is an ideal in A, there is another ideal q' for which qraq'=o and A/(q -f- q') is finite. Proof. -- Z -= qZ| being semi-simple, and q' = Z' n A clearly serves our purpose. Lemma (19.3). -- If q and q' are ideals with qnq'=o, then GL(n, A, q-k q')=GL(n, A, q) � GL(n, A, q') (direct product), and similarly for SL(n, A, q + q') and E(n, A, q + q'). In particular, K~(A, q +q')= K~(A, q)| q'). g33 7 5 ~ H. BASS Pro@ -- If I +q+q'eGL(n, A, q4-q'), where q and q' have coordinates in q and q', respectively, then I +q+q'= (i +q)(I +q')eGL(n, A, q) � A, q'), since qq'= o = q'q. The conclusion tbr SL follows from the factorwise definition of SL, and for E it follows by applying the reasoning above to its generators. Lemma (I9.4). -- GL(n, A, q) and SL(n, A, q) are finitely generated groups for all n and q. Proof. -- Lemmas 19.2 and 19. 3 permit us to assume A/q is finite, in which case GL(n, A, q) has finite index in GL(n, A), so it suffices to show the latter finitely gene- rated. If I" is a maximal order containing A, then mFcA for some m>o, so GL(n, I', mr) cGL(n, A), showing that GL(n, A) has finite index in GL(n, 1"). Since P is a product of maximal orders in the simple factors of Z, finite generation of GL(n, P) follows from Siegel's theorem (Lemma I8.4, i)). Exactly the same proof applies to SL. Now consider the direct system. ...SL(n, A, q)/E(n, A, q)-->SL(n+ I, A, q)/E(n+ I, A, q)-~.., with limit SL(A, q)/E(A, q). Thanks to Theorem I I . I we can apply Theorem 4.2 to A with n--- 2. Hence we know from (4.2, b)) that the maps above are surjective for n>2, and, from (4.2, f)) and (19.4), that the terms are finitely generated abelian groups for n_> 4. Consequently, since finitely generated abelian groups are noetherian, the system stabilizes; i.e. the maps are eventually all isomorphisms. (Indeed, the conjecture of w Ii alleges they ate isomorphisms already fbr n_> 3. If true, one could take n0= 3 in Theorem 19.1 , as the proof will show.) By the theorem of Wang [35] (see Proposition 17.3) SL(n, Y~) is the commutator subgroup of GL(n, 2;), and, by Dieudonn~ [17] (see Proposition 5. i, b)) the latter is just E(n, N) for n~2. Thus the reduced norm induces a monomorphism KI(Z) = GL(Z)/E(E) ---> C*. Moreover, the inclusion GL(A, q)cGL(E) induces an exact sequence o-)-SL(A, q)/E(A, q)--->GL(A, q)/E(A, q)-+GL(E)/E(Z) ' ,I I] KI(A, q)- > Kt(E) The next corollary summarizes some of these remarks: Corollary (x9.5). -- The following conditions on A and q are equivalent: (i) There exists an n0>2 such that SL(n0, A, q)/E(no, A, q) is finite (resp. trivial). (ii) For the same no, SL(n, A, q)/E(n, A, q) is finite (resp. trivial) for all n>n o. (iii) SL(A, q)/E(A, q) is finite (resp. trivial). (iv) Kt(A, q)-~KI(Y~) has finite (resp. trivial) kernel. Remark. ~ As noted above, the conjecture of w I I asserts that no = 3 already suffices in the corollary. 534 K-THEORY AND STABI,E AI.GEBRA Except for the dependence ot n o only on Z, Theorem 19. I is now seen to be contained in the following result, of which the last assertion has already been noted above. Theorem (x9.6). -- Let A be an order a semi-simple algebra Z, finite, over Q, with center C. Then, for any ideal q in A, KI(A, q) ->KI(Z) has finite kernel, and its image is isomorphic to the image of GL(2, A, q) in C* under the reduced norm. The following sequence of lemmas will permit various reductions in the proof of this theorem. Lemma (x9.7). -- Let G be a group and R a normal subgroup. Then, if n=G/R and G/[G, G] are finite, so also is G/[G, R]. Proof. -- R/[G, R]->G/[G, G] has finite image, so it suffices to see that it has finite kernel. But this is just the second map in the exact sequence H2(~z) ->H0(~., Hi(R))->HI(G) which comes from the Hochschild-Serre spectral sequence in homology with integral coefficients (see [14, XVI, w 6, (4 a)]) for the group extension I --> R --> G -> n --> 1. Since is finite, so is H2(=), and this proves the lemma. Corollary (i9.8). --/f A/q is finite, then so also is E(n, A)/E(n, A, q) for all n>_3. Pro@ -- Put G----E(n, A) and R=GL(n, A, q)nE(n, A). Then G/R is finite, and G= [G, G], by (i .5, (i)). By (4.2, d)) E(n, A, q) = [G, R], so the corollary now follows from (i 9 . 7). Corollary (x9.9). -- If, for some n0_>3, SL(n,,, A)/E(n0, A) is finite, then SL(n, A, q)/E(n, A, q) is finite for all q and all n2no. Proof. --By (19 . 2) and (19 . 3) the conclusion above for all q follows once we know it for q with A/q finite, so we now assume this. If n>no, then the finiteness of SL(n, A)/E(n, A) follows from our hypothesis and (I9.5) , and that of E(n, A)/E(n, A, q) from (19.8) above. Hence SL(n, A)/I~.(n, A, q) is finite, and this proves the corollary. Lemma (x 9. xo). -- Let A and A' be two orders in Z, and let q be an ideal in A for which A/q is finite. Then there is an ideal q' in A' with A'/q' finite such that, for all n> 4, E(n, A', q') CE(n, A, q). Proof.- mA'cA for some m:>o, so qx=A'mqmA' is a A' ideal contained in q, and clearly A'/qx is finite. Let H =GL(n, A, ql) =GL(n, A', ql). Then from (I .3) (using n>3) and (4.2, f))(using n>4) we have : E(n, A', q~)c[E(n, A', q~), E(n, A', q~)] r H] cE(n, A, ql)CE(n, A, q). Hence q'= (ql) ~" serves our purpose. Corollary (x 9. xx). -- If, for some order A' in Z, KI(A')--->KI(Z) has finite kernel, then there is an n o such that, for all n>no, for all orders A, and for all ideals q in A, SL(n, A, q)/E(n, A, q) is finite. 535 52 H. BASS Proof. -- Our hypothesis and (i9.5) imply SL(no, A')/E(n0, A') is finite for some n0_>3, so our conclusions for A' follow from (I9.9). Taking no~4, we can apply Lemma 19. Io to any other A, and conclude that E(n0, A', q')r A) for some q' with A'/q' finite. Since SL(n0, A')/E(n0, A, q') is finite, E(n0, A)r~SL(n0, A') has finite index in gL(n0, A) nSL(no, A'). But the latter contains SL(n0, Ar~A'), which has finite index in SL(n0, A). Therefore SL(no, A)/E(n0, A) is finite, and the corollary now follows from (i 9.9). Corollary 19. i i reduces Theorems 19. I and 19.6 to showing that KI(A) -+KI(Z) has a finite kernel for some A. Since KI(A) is a finitely generated abelian group (Theorem 18.6), we need only show the kernel is torsion, and for this the following crite- rion is useful. It is here that K ~ effectively intervenes in the proof. Proposition (xg.x2). -- Let AcB be commutative rings with B finitely generated and projective as an A-module. Then, if A is a finite A-algebra, the kernel of KI(A)-+Ka(B| is torsion. Proof. -- The nature of B provides us with a homomorphism KI(B| (see w I4) whose composite with the one above is the homothetie of the K~ K~(A), defined by TA(B)eK~ The Proposition now results from (I5.6), which tells us that anything killed by y,~(B) is torsion. We come now to the proof that ker(K~(A)--->K~(Z)) is torsion. Passing to ANzACL| , where A is the ring of integers in a splitting field L for Z, we can reduce, thanks to (I 9. I2) above, to the case where Z is split. By (i 9. iI), moreover, we may take for A a maximal order. But then A is a product of maximal orders in the simple factors of Z, and everything decomposes accordingly, so we reduce further to the case Z=EndL(V), V a vector space over the number field L, and then (see [3] or [I5] ) A----EndA(P), with P a projective module over the ring A of integers in L. Now the Morita theorem (w I7) gives us equivalences from the categories of A-modules to A-modules (| and from L-modules to Z-modules (| which commute with the passages from A to L and A to Z, respectively. Thus we have KI(A) -+K*(L) Kt(A) --> K* (Z) commutative, and it suffices, finally, to show that ker(Kl(A)-+KI(L)) is torsion. Using (I 9. I2) again, we see tlhat this is a consequence of the following proposition: Proposition (*9. x3). -- Let A be the ring of integers in a finite extension L of Q, and let ~eker(Kl(A)--~Kl(L)). Then there is a finite solvable extension F of L, such that ~eker(Kl(A)~KI(B)), where B is the ring of integers in F. Proof. -- By (iI.2, b)) ~=WAe with ~ an automorphism of A 2, and ~eker(Kl(A)---~Kl(L)) simply means det ~----I. Passing to a quadratic extension F 0 of L, with integers B0, we can give % = IB.| an eigenvalue. As an automorphism of F0 2, q. thus has a one dimensional invariant subspace, and since B o is a Dedekind ring, the 536 K-THEORY AND STABLE ALGEBRA latter contracts to a direct summand P0 of Bo 2. Po is invariant under %, and, having rank one, P0~--a, a an ideal in B 0. Since the class group of Bo is torsion (even finite) ah=(a) is principal for some h>o. Let F=F0(~/a ) have integers B. Then P=B| and P is invariant under ~= Is| IB| If we choose a basis for B ~" the first member of which generates P, then ~ is represented by a matrix of the form iu x i__iu oil u tx :i I Io v I Io v!o ij" The second factor is manifestly in E(2, B). Since det ~ = 1, we have v----u -t, so the first factor lies also in E(2, B) by the Whitehead lemma (I.7). Therefore WB(IB| ) =WB(~) = O, as required. Remarks. -- I) Theorems 19 . i and 19 . 6 are probably valid also for semi-simple algebras over a function field in one variable over a finite field. The proof above has two ingredients which are not known, to my knowledge, in that case. One is Wang's theorem. However, this can be circumvented easily since the discrepancy between E(n, Z) and SL(n, )3) is easily shown to be a torsion group for semi-simple algebras over any field. The second point, which I don't know how to supply or outmaneuver, is the finite generation of $L(n, A), say for n>3 (1). Similarly, this is the only point requiring attention if one works throughout, say, with orders in )2 over a ring of the form Z[n-t], for some nEZ. 2) Theorem x9.6 suggests an obvious analogue for K ~ Namely, one can ask that K~176 have finite kernel. Jan Strooker (Utrecht thesis) has pointed out that a necessary and sufficient condition for this is that every projective A-module P, for which Q| P is Z-free, be locally free. He gives examples for which this fails. w 2o. Groups of simple homotopy types. Theorem (2o. x ). Let ~ be a finite semi-simple Q-algebra with q simple factors, and suppose R| has r simple factors. Then, if A is an order in Z and q is an ideal in A, KI(A, q) is a finitely generated abelian group of rank<r--q, and =r--q if A/q is finite. Theorem (20.2). -- In the above setting the following conditions are equivalent: i) KI(A) is finite. 2) KI(A, q) is finite for all q. 3) An irreducible Z-module remains irreducible under scalar extension from Q to R. 4) The center of each simple factor of Z is either Q or an imaginary quadratic extension Of Q. Proof of (2o.i). -- By (~9.2) and (I9.3) we can assume A/q is finite. Let 1TM be a maximal order containing A. Then GL(n, A, q) cGL(n, A)r P) are both (1) This has recently been established by O'Meara (On the finite generation of linear groups over Hasse domains, to appear) for commutative A. g37 54 H. BASS subgroups of finite index, for all n. Thus KI(A, q)-+Kl(r) has finite cokernel. By (I9.6), moreover, the maps KI(A, q)-->Kl(r)-+Kl(Z) both have finite kernel. Hence rank KI(A, q) ----rank K~(r). Now P is a product of maximal orders in the simple factors of ~, and KI(I ') splits accordingly. Since the function r--q likewise adds over the simple factors we can reduce to the case where 52 is simple (i.e. q = I), say with center L. ll| = center R| has the same number r of simple factors as R| and we want to show that rank Kl(P)=r--i. We know from (I9.6) that rank Kl(P) is the rank of the image UCL* of GL(F) under the reduced norm. If A denotes the integers in L, then P being integral over A implies UcA*. On the other hand A*cF*----GL(I, P), so (A*)"cU, where [X:L]=n z. Hence rank U = rank A*. By the Dirichlet Unit Theorem, rank A*=r--i, and this completes the proof. Proof of (20.2). --The equivalence of I), 2), and 3) is an immediate consequence of the above theorem, and that of 3) and 4) is trivial. If rc is a finite group then Z~ is an order in the semi-simple algebra O,~, so we may apply the preceding results. Viewing -t-r~cGL(I, ZT~)cGL(Z~), it makes sense to write Kl(Zr~)/4-rq with a minor abuse of notation. J. H. C. Whitehead showed [36] that if X and Y are finite simplicial complexes of the same homotopy type and funda- mental group ~, then the simplicial homotopy equivalences from X to Y, modulo the simple homotopy equiw~lences, are classified by invariants which live in KI(Z~)/ Herein lies the principal interest of the next result, which elaborates on some earlier work of G. Higman [38]: Corollary (20.3). -- Let r: be a finite group, r the number of irreducible real representations of re, and q the number of irreducible rational representations of =. Then the commutator quotient group of GL(Zr:) is a finitely generated abetian group of rank r--q. There are well known group theoretic interpretations of r and q : q is the number of conjugacy classes of cyclic subgroups of ~z (Artin). Write a~b in 7: if a is conjugate to b Then r is the number of--~ classes (Berman-Witt). Both of these results can be found in Curtis-Reiner [16, Theorem, 42.8]. Examples. -- I) If r: is abelian, then r = q if, and only if, 7~ has exponent 4 or 6. For each simple factor of O~ is a cyclotomic field of n *t' roots of unity, where n lex p r:. These fields are either Q itself or totally imaginary. They have degree <2 precisely when n 14 or 6. 2) The rationals are a splitting field for the symmetric groups and O_v(%/~----i-) for the quaternions. Hence the Whitehead group is finite in these cases. For groups with this property the results of the next section can be used to give a crude bound on its order. It is not inconceivable that it even be trivial. 3) If rc is cyclic of order n, then 7: has 3(n) irreducible Q representations, ~(n) = the number of divisors of n. 7: has In/2] + I irreducible R representations, where [x] = the integral part of x. Hence the Whitehead group has rank In/2]-~ t--3(n) in this case. 538 K-THEORY AND STABLE ALGEBRA 4) In [7] it is shown that the Whitehead group is trivial when n is free abelian. Milnor has asked whether it is always a finitely generated abelian group if rc is. It seems reasonable, though difficult to show, that KI(A) is finitely generated for A any finitely generated commutative ring over Z with no nilpotent elements. The same statement for K ~ would generalize the Mordell-Weil Theorem. w 2x. Subgroups of finite index in SL(n, A). Does every subgroup of finite index in SL(n, Z) contain a congruence subgroup, SL(n, Z, qZ), tbr some q>o? The answer is easily seen to be " no " for n=2, as was already known to Klein. For n>3 , however, the solution is affirmative; a proof is outlined in [4o]. The method consists of an application of the present results to reduce the problem to a (rather formidable) cohomological calculation. The latter, in turn, depends heavily on some recent results of Lazard on analytic groups over p-adic fields. I shall summarize here, in a form adapted to this method, the information provided by the present material. Let A be an order in a simple algebra E, finite over Q. We introduce the following abbreviations in our notation: S = SL(n, A) E= E(n, A); for each ideal q, Sq=SL(n, A, q), Eq=E(n, A, q), and Fq=Ec~Sq. Theorem (2x. x). -- For n_> 2 center S = center E is isomorphic to the (cyclic) .group of n ~ roots of unity in the center of A. For n>_3, a non central subgroup of S normalized by E contains E 0 for some q 4= o, and E/Eq is finite. Hence a normal subgroup oat" E is either finite or of finite index, and the same is true of S as soon as S/E is finite. The latter holds for all sufficiently large n. Proof. -- By (2.4) an element of GL(n, A), n>2, centralized by E, has the form u. I, with uecenterA. Being in S means U"=l, and it then follows from (1. 7) that u. IeE. Since center Accenter Z, a field, the n th roots of unity form a cyclic group. The rest of the theorem is an immediate consequence of (4- 2, e)), (19.8) and ( 19. ~)- To avoid some technical difficulties we shall henceforth assume A is commutative, i.e. E is a number field. We shall be speaking of " profinite " (=compact, totally disconnected) groups, and their cohomology, for which we give Serre's notes [39] as a general reference. If H is any group we denote by t2I its completion in the topology defined by all subgroups of finite index. Since each of the latter contains a normal subgroup of finite index we can describe I2I by I2I= limwH~f nit~.H/H~ 9 This defines a functor from groups (and homomorphisms) to profinite groups (and conti- nuous homomorphisms) which evidently preserves epimorphisms. 5.39 56 H. BASS On the other hand, S and E above can be completed also in the " congruence topology " defined by taking the Sq, resp. Fq, q 4= o, as a basis for neighborhoods of the identity. By Corollary 5.2 the inclusion EcS induces isomorphisms 9 E F,-S/S,=$L(n, A/q), for each q 4= o. Since this group splits uniquely according to the primary decomposition of q, and since lim m SL(n, A/q") = SL(n, ./~,) for q prime, we conclude that S and E have the same congruence completion, II = II SL(n, .X.q). We shall write q prime, * 0 C -- ker(S-+ H) and C0 = ker(l~-+ II ). The question discussed above asks whether the congruence and profinite topologies in S coincide, i. e. whether C = o. Theorem (21.2), -- (i) There is a commutative diagram with exact rows, I ----~ C ----~ g --~ l-I ----> I t t ki I -+ Co-+g-+ II -+ I. Here l I = limq, oE/F, = lim,, oS/S, = 1-I $I.(n, ~). q prime, ~ 0 (ii) For n>_3 Co = limq. oFq/E,, and C ---- libra,, o(S,/E,), and the maps in both of these projective systems are all surjective. (iii) For n>_3, CoCcentcr E. For n>4 , Cffccnter S, and E-+S is a monomorphism. (iv) Consider the following conditions: a) Sq/Eo = {~ } for all q. b) A non central normal subgroup of S contains Sq for some q 4= o. c) A subgroup of finite index in S contains SQ for some q 4= o. d) C={I}. c) (SJE,) ---- {i } for all q. For we have a) b) and for n24 theyareaUequivatent. Similarly, they are all equivalent for n>_3 if we substitue E, Fq, and C O for S, Sq, and C, respectively. (v) Writing S=S(n), C----C(n), etc., to denote their dependence on n, the inclu- sions S(n--i)CS(n) and E(n--i)CE(n) induce homomorphisms C(n--i)-+C(n) and C0(n-- i) -+Co(n) which are surjective for n>_3. Proof. -- Part (i) is contained in the remarks preceding the theorem. It is immediate from (21. i) above that (*) E ---- 0E/E q for n>3, and this makes it evident that Co----li~_m,,oFJE q. S-+S/Eq induces g-->(S~q)-->I, and hence S-->limq+o(S/Eo). It follows from (2I'.I) again that this is injective. It < . 9 1. li+_m_m,. K-THEORY AND STABLE ALGEBRA is surjective since the image is (clearly) dense, and S is compact. Since S/Sq is finite for q 4: o it is now evident that C --ker(fim q . 0(S/Eq) -+ 0S/Sq) admits the description in (ii). The last part of (ii) follows once we show that, if o + q C q', then F,/E,--+Fq,/Eq, and S,/E,--+Sq,/Eq, are surjective. This means simply that FoE q, = Fq, and SqEq, = Sq,. The first of these equations is a consequence of the second, and the second is contained in Corollary 5.2. For n> 3 we know from (4.2, d)) that [E, Fq] =Eq, so it follows from (ii) that C0ccenterE. If n> 4 then from (4.2,f)) we have [S, Sq] =Eq, so it follows similarly from (ii) that CCcenter S. To show that g-+g is a monomorphism it suffices, by (*) above, to show that S/Eq is separated in its profinite topology. But for n>4 , S/E, is a central extension of a finite group S/Sq by a finitely generated abelian group SJEq, using (4.2, f)) and (i9.4), and such a group is clearly separated. This proves (iii). Now for (iv). Assume n>3: a)~b) follows from (2I.I) and (ii). b) ~c) since a subgroup of finite index contains a normal subgroup of finite index, and the center of S is finite. c).,:~d) since c) asserts the coincidence of the profinite and congruence topologies. d)<>e) follows from (ii). If n>4, then Sq/Eq is a finitely generated abelian group, as already noted in the last paragraph, so d):~a). The proof for E is parallel, but the last point is simplified since Fq/EQ is even finite already for n>3. For part (v) it suffices, by compactness, to show that C(n--x)--~C(n) has a dense image for n>3, and similarly for Co. Denseness means that C(n--I) projects onto every finite quotient of C(n). But it follows from (ii) that every finite quotient of C(n) has the form S0(n)/H with Eo(n ) oH. By (4.2, b)), S0(n ) -: S0(n-- r)E0(n ) = S0(n-- I)H, so S,(n--I)/HnSo(n--I ) coming from C(n--~) maps onto S0(n)/H , as required. The proof for Co is identical, after replacing S and S 0 by E and Fq, respectively. In [4 o] it is shown that, when A=Z, H2(II(2), O/Z)=o (cohomology in the sense of [39]), and on this basis that C(n)= o for n=>3. (By virtue of (2I .2, (iv)) this is equivalent to E(q)-= S(q) for all q> o. For q<__ 5 this had been shown by Brenner [I 3] by direct calculation.) In the general case one knows only the following result, which Serre has proved using recent results of Lazard and of Steinberg (Colloque de Bruxelles, I962 ). Theorem (2x.3)(Serre, unpublished) H2(II, Q,/Z) is finite. Plugging this into the argument of [4o], and using the information in (2I.~) and (21.2) above, one obtains: Corollary (2x.4). -- C o /s a finite group for n>3, and C is finite for n large enough, so that S/E is finite. li+.m_mq. 5 8 H. BASS w 22. Some remarks on polynomial rings. Let A be commutative and noetherian, and let B=A[tl, ..., tn] with tl,..., t n (n~I) indeterminates. Grothendieck has shown that, if A is regular, the homomorphism K~176 is an isomorphism (see [29] or [7]). It follows that if P is a projective B-module, then -(BP=yB(B| ) for the projective A-module Q= P/(tl, ..., tn)P. Now this equation in K~ can be replaced by the isomorphism P~BQAQ , provided rank P>dim max(B) (Proposition 15.6 ). At this point the unpleasant fact emerges that dim max(B) = dim spec(B) = n + dim spec(A). Thus, for example, if A is local (so dim max(A) =o), dim max(A[t]) can be arbitrarily large. In any event, we can record the following conclusion, using the fact that: A is regular and dim spec(A) = d r global dim A = d. Theorem (22. i ). -- Let A be a commutative noetherian ring of global dimension d, and let B = A[t~, ..., t,], the t~ being indeterminates. Then a projective B-module of rank > d + n has the form B| for some projective A-module Q. If A is a field we see that projective B-modules of rank>n are free, but we can't conclude this if A is only local. However, we can make a very small compensation in this case (Corollary 22.3 below). For here, in the equation ~.BP---fB(BQAQ), Qwill be free, so we can conclude that P| r, for some r, s; we want P to be free. If we write (P|174174 and apply induction, we are reduced to showing, under suitable hypotheses, that P| r =>P~B r-1. It is easy to see (cf. proof of the Cancel- lation Theorem, 9-3) that this conclusion is equivalent to the assertion that AutB(B ~) is transitive on the unimodular elements of B r. Proposition (22.2). -- Let A be commutative and noetherian and suppose d= dim spec(A)> dim spec(A/rad A). Then, if B--=A[tl, ..., tn], t~ indeterminates, E(r, B) is transitive on the unimodular elements of B y for r>d+n. Remark. -- If we replace d + n above by d-r n -t- i then this Proposition is contained in Theorems 11. I and 4.1, a), since dim max(B) =d+n. Pro@ -- Suppose e= (a~, ..., ar)EB ~ is unimodular; we seek /~E(r, B) such that r = (i, o, ..., o). The remark above, together with our hypothesis, shows that this can be done if we replace A by A/rad A. It follows, using Lemma i. i, that we can find r B) so that zl0~= (al, ..., a;)- (I, o, ..., o)mod radA.g r. Since a' r ~ I mod rad A. B, a maximal ideal of B containing a'~ cannot contract to a maximal ideal of A. It follows that dim max(B/a~B)<n+d--i. Hence we can again apply the remark above, this time to the (unimodular) image of (a~, ..., a;) in (B/a[B) "-t, and transform this image into (I, o, . .., o) with ~E(r-- I, B/a;B). By Lemma i. i , I O I again, ~2 lifts to r B), and we set ~2= ,, sE(r, B). Then ,2~1c~ has O ~2 the form (a[, I + bza'~, b3a'l, ..., b,a[), and it is now clear how to finish with elementary transformations. 542 K-THEORY AND STABLE ALGEBRA From the discussion preceding this proposition we derive the following corollary: Corollary (22.3). -- If, in Proposition 22.2, A is a regular ring for which K~ ~Z, then a projective B-module of rank> d + n is free. As a special case, we have the following corollary: Corollary (22.4). -- If A is a semi-local principal ideal domain (so d~I) then a projective A[q, ..., t,]-module (t i indeterminates) of rank >n is free. This last corollary has recently been strengthened, for n=e, by S. End6 [~o], who shows in this case that all projective modules are free. This generalizes the theorem of Seshadri [27]. BIBLIOGRAPHY [i] E. ARTIN, Geometric Algebra, Interscience, n ~ 3 (I957). [2] M. ATIYAH and F. HmZEBRUCH, Vector bundles and homogeneous spaces, Proc. Sympos. Pure Math., Amer. Math. Soc., vol. 3 (I96I), 7-38. [3] M. AUSLANBER and O. GOLDMAN, Maximal orders, Trans. Am. Math. Soc., 97 (I960), 1-24. [4] --, The Brauer group of a commutative ring, Trans. Am. Math. Soc., 97 (i96o), 367-4o9 . [5] H. BAss, Projective modules over algebras, Ann. Math., 73 (I96I), 532-542. [6] --, Big projective modules are free, Ill. Journ. Math. (to appear). [7] --, A. HELL~R and R. SWAN, The Whitehead group of a polynomial extension, Publ. math. LH.E.S., n ~ 22, Paris (1964). [8] -- and S. SC~ANUEL, The homotopy theory of projective modules, Bull. Am. Math. Soe., 68 (I962), 425-428. [9] A. BOREL and HARIS~-CHANDRA, Arithmetic subgroups of algebraic groups, Ann. of Math., 75 (I 962), 485-535 . [IO] -- and J.-P. SERRE, Le th6or6me de Riemann-Roch (d'apr~s Grothendieck), Bull. Soe. Math. de France, 88 (1958), 97-136. [i 1] N. BOURBAKI, AlgObre, liv. II, chap. 8 : (~ Modules et anneaux semi-simples )~, Actualitds Sci. Ind., 1261, Hermann (i958). [i2] --, AlgObre commutative, chap. 1-2, Actualit6s Sci. Ind., 1290, Hermann (1961). [13] J. BRENNER, The linear homogeneous group, III, Ann. Math., 71 (I96O), 21o-223. [14] H. CARTAN and S. EIL~NBERG, Homological Algebra, Princeton (1956). [I5] C. CH~VALLEY, L'arithmdtique dans les algObres de matrices, Actualitds Sci. Ind., 323 (I936), Paris. [ 16] C.W. CURTIS and I. RmNER, Representation theory of finite groups and associative algebras, Wiley, New York (1962). [17] J. DIEUDONN~., Les d&erminants sur un corps non commutatif, Bull. Soc. Math. France, 71 (I943), 27-45. [18] M. ElCnLER, ~kJber die Idealklassenzahl total definiter Quaternionenalgebren, Math. Zeit., 43 (I937), IO2-IO 9, [19] --, f~lber die Idealklassenzahl hyperkomplexer Systeme, Math. Zeit., 43 (1937), 481-494. [2o] S. END6, Projective modules over polynomial rings (to appear). [21] P. GABRmL, Des categories abdliennes, Bull. Soc. Math. France, 90 (1962), 323-448. [22] A. GROTHENDmCK et J. DIEUDONN~, ]~Mments de g6om~trie algfibrique, I, Publ. math. LH.E.S., n ~ 4, Paris (196o) . [23] H. HASSE, Zahlentheorie, Berlin, Akademie-Verlag (1949). [24] W. KLINGENBERG, Die Struktur der linearen Gruppen fiber einem nichtkommutativen lokalen Ring, Archiv der Math., 13 (I962), 73-81. [25] --, Orthogonalen Gruppen fiber lokalen Ringen, Amer. Jour. Math., 83 (I961), 281-32o. [26] K. MORITA, Duality for modules .... Science Reports 2-ok. Kyoiku Daigaku, sect. A, 6 (1958). [27] C. S. S~SHAORI, Triviality of vector bundles over the affine space K 2, Proc. Nat. Acad. Sci. U.S.A., 44 (I958), 456-458 9 [28] J.-P. SERRE, Faisceaux alg~briques coh6rents, Ann. Math., 61 (I955), I97-278. [29] --, Modules projectifs et espaees fibres ~ fibre vectorieUe, S~m. Dubreil (1957-58), n ~ 23. [3 o] --, AlgObre locale; multiplicitds (r~dig6 par P. GABRmL), Coll. de France (i957-58). [3 I] C. L. SmoEL, Discontinuous groups, Ann. Math., 44 (I943), 674-689. 643 60 H. BASS [32] R. W. SWAN, Induced representations and projective modules, Ann. Math., 71 (I96o), 552-578. [33] --, Projective modules over group rings and maximal orders, Ann. Math., 76 (i962), 55-6i. [34] --, Vector bundles and projective modules, Trans. Am. Math. Soc., 105 (I962), 264-277. [35] S. WANO, On the commutator group of a simple algebra, Amer. 3our. Math., 72 (i95o), 323-334. [36] J. IcI. C. Wn-rr~n-ahD, Simple homotopy types, Amer. oTour. Math., '/2 (i95o), 1-57. [37] tL ZASSgNm~VS, Neue Beweis der Endlichkeit der Klassenzahl .... Abh. Math. Sere. Univ. Hamburg, 12 (I938), 276-288. [38] G. HIOMAN, The units of group rings, Proc. Lond. Math. Soc., 46 (I94O), 231-248. [39] J;'P. SEmi, Cohomologie galoisienne, tours au CollSge de France, ~962-63, notes polycopides. [4 o] H. BASS, M. LAZARD et J.-P. SEI~I~E, Sous-groupes d'indlce fini dans SL(n, Z), Bull. Am. Math. Soc. (to appear). Refu le 15 juin 1963.

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