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K. Kato (1982)
Galois cohomology of complete discrete valuation fieldsAlgebraic K-theory, Springer Lecture Notes in Math., 967
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by SPENCER BLOCH* and KAZtrYA KATO INDEX o. Global introduction ....................................................................... io 7 I. Local results ............................................................................. i Io ~. The differential symbol .................................................................... 113 3. A basic cohomological lemma ............................................................. I I8 4. Filtration on symbols ..................................................................... x 2 t 5. Galois cohomology ....................................................................... 124 6. The sheaf Man ........................................................................... x32 7. Ordinary varieties ........................................................................ x38 8. A vanishing theorem ..................................................................... 14 I 9- p-adic cohomology ........................................................................ 148 o. Global introduction The purpose of this paper is to apply the results of [i2], to the global study of p-adic etale cohomology and the associated p-adic Galois representations. We fix a field K of characteristic o which is complete with respect to a discrete valuation, with residue field k of characteristic p ~> o and valuation ring A. The generic (resp. special) point of S = SpA is denoted ~ (resp. s). We consider a diagram of schemes j i V=X~ > X < .... X~=Y SpK= ~ > S =SpA < s=Spk with all vertical arrows smooth and proper. A bar will either indicate algebraic or integral closure (viz. K, A) or base extension (X-----X; = X � S, V = V~, ...). Finally, G = GaI(I~/K) and Cp ~ K, the completion of K. * Partially supported by the NSF. 107 SPENCER BLOCH AND KAZUYA KATO ~o8 The basic global objects are the etale cohomology groups H:t(V , Q.p), which we study using the spectral sequence (o.2) E~" = H'(Y, i" R~j,(Z/p" Z)) =~ H,:" + '(V,-- Zip" Z). This spectral sequence induces a G-stable filtration (0.3) F'(n;,(V, Z,)) such that F ~ q=Hq and F"Hq= (o) for n>q+ I. We write (o. 4) gr" H ~ : k TM Hq/F n + t H q. Recall that one also has the de Rham-Witt cohomolog T [3], [io] H'(Y, and the crystalline cohomology [2] H0~,(Y/W(k)) (W(k) = Witt vectors over k), which depend only on the special fibre Y and are linked via the slope spectral sequence = s+t -- - (0.5) El,' H'(?, WY/') :~ H~,y, (Y/W(k)). H~,y, has a canonical endomorphism F (Frobenius) and we write (o.6) for the p~-eigenspace of F on H0,y, | Q.. Roughly speaking we will say Y is ordinary if the rank of (o.6) equals the rank of the k-vector space (Hodge group) Hq-'(V, fl ) for all i and q. (This definition is not quite correct in the presence of torsion in H:ry ,. For a more detailed discussion see w 7 below.) An abelian variety of dimension d is ordinary if and only if it has pd geometric points of order p. By Deligne (unpublished but cf. [No], p. I43), ordinary hypersurfaces of any given degree make up an open dense set in the moduli space. Theorem (0.7). -- Let notation be as above and assume Y is ordinary, Then there exist functorial G-module isomorphisms (i) grq-iH[,(V, Q,) ~ Hqo~,(Y/w(k))~)( - i) (ii) gr~-' Hqot(V, Q.p) | W(k) -~ Hq-'(Y, W~') (-- i)Q (iii) grq-'H~t(V, Q,p) | Cp ~ Hq-'(V, s | Cp(-- i). (The notation (-- i) means twist i times by the dual of the p-adic cyclotomic character on G. Also G acts in the natural way on C~.) Recall that a Q.~[G]-module M is said to admit a Hodge-Tate decomposition if the module M | Cp with semi-linear G-action is isomorphic to a direct sum 9 M.(n) 108 #-ADIC ETALE COHOMOLOGY xo9 with M, ~ C~" as a G-module. Assume now k is perfect. Tate has shown [I8] that the Tate module of a p-divisible group admits a Hodge-Tate decomposition. By using this fact, Tate and Raynaud proved that H~t(V , Q.p) has the Hodge-Tate decomposition H'.t(V, Qp) | Cp ~ [Hi(V, r | Cp] @ [H~ n~) | Cp(-- I)] for any smooth proper variety V over K. Corollary (o. 8). -- Assume Y ordinary and k perfect. Then for all q, Hqot(V, Q.p) ~ ? (Hq-'(V, n~) | C~(-- i)), so Hqt(v) has a I-[odge-Tate decomposition. The proof is straightforward from (o. 7) (iii) together with the result of Tate: (0.9) If n4:o, H~ Cp)(n) =o. If k is perfect and n4=o, then H'(G, Cp(n)) = o. We continue to assume now that Y is ordinary, and we suppose in addition that the residue field k is separably closed (not necessary perfect). There is some geometric interest in considering the extensions (o. ,o) o -* gr ~ + * H q --~ F i Hq/F i + 9 Hq --* gr ~ Hq -, o. If H;ry,(Y ) is torsion free, the isomorphisms of (o. 7) exist before being tensored by O (see (9.6)). Thus the extension class lies in (O. Ix) H~ (q-~), Hoq~,(Y) (q-~- ll) | Ht( G, Zp(I)). One has (cf. [19], prop. (2.2)) HX(G, Z,(,)) =~ lim K*]K "~' = ~', the p-adic completion of the multiplicative group of K. If a basis f~ for the Horn in (o. I I) is fixed, one gets (dual) functions { liftings of Y} ~,. f~: -~ over A This situation is understood in the case of abelian varieties (Katz [i3] ) and also K- 3 sur- faces (Deligne-Illusie [6]). It should be the case that the image offd lands in the group of principal units U ~ ~'. The f~* could then reasonably be thought of as p-adic modular functions, i.e. as p-adic functions on the moduli space (in fact, on the " period space" !?). Briefly, the content of the various sections of the paper are as follows. Sections I-6 are local. w i describe the local setup and states the main local results, (o.4) (o.5). In section 2 we identify the mod p Milnor K-theory of a field in characteristic p with the group of logarithmic Kahler differentials. w 3 contains a lemma about Galois 109 llO SPENCER BLOCH AND KAZUYA KATO cohomology which enables us to prove in w 5 that the Galois cohomology of a henselian field is expressed by Milnor K-theory. w 4 is preparatory, giving elementary properties of symbols which will be used in later sections. w 6 "sheafifies " these results. Sections 7, 8 and 9 are global. In w 7 we discuss ordinary varieties in characteristic p. We charac- terize these by the vanishing of all cohomology groups of sheaves of locally exact diffe- rentials. Finally, w 8 and w 9 are devoted to the proof of the main global result (o. 7). A summary of this work is published in [5]. The authors would like, first and foremost, to thank O. Gabber. His results on vanishing cycles [7], [8] played an essential role in this work. Further, he read the manuscript carefully giving us much valuable advice. The authors would also like to thank L. Illusie, N. Katz, N. Nygaard, A. Ogus, and M. Raynaud for helpful conversations and encouragement. x. Local results (t.x) Recall the situation of (o. x) V J ~X( -~Y Sp K > SpA ( Sp k but do not assume vertical arrows proper. In the local study w I- 3 6, we are principally interested in the structures of the ~tale sheaves on Y; Mq, = i~ (n, q> o). These are localizations on Y of the p-adic 6tale cohomology of V in suitably twisted coefficients. For y ~ Y, the stalk M~.o is isomorphic to the 6tale cohomology group H q Sp Ox, ~ , Z[p"Z(q) , where 0x, ~ denotes the strict henselization of ~x,~. In the case X is proper over A, the spectral sequence (0.2) relates the limit M~ = i* Rqj.(Z/p" Z(q)) of M~ to the p-adic 6tale cohomology H*(V, Z~) of V. We study M~ by using symbols and a natural filtration. We shall see that M~ is related to differential modules on X and Y, and to the De Rham-Witt complex on Y. (x.2) First, we define the symbols. The exact sequence of Kummer on V o ,r ,o induces an exact sequence on Y i*j. r ~, i*j. 0; , M~ , o. 110 p-ADIC ETALE COHOMOLOGY Ill For local sections Xl, ..., xq of/*]', tV~, let {xl, ..., xq} be the local section of M, q defined as the cup product of the images of x~ (1 < i < q) in M, 1. Then, {X, -- X} : O, {x,y}'-[- {y, X} : O, {g, I -- Z} : 0 for any local sections x, y, z such that i -- z is invertible. (The proofs of these iden- tities are essentially the same as Tate's proof of the existence of the cohomological symbol K~-+tVc~I(,Z/p"Z(2)) for fields. They also follow from Soul6's Chern class homomorphism Ka ~ H~( , Zip" Z(~)) for rings and the corresponding identities in K2 (cf. [I7] , [19]). ) Next we define the filtration of M, q. For m~ i, let U mM~ be the subsheaf of M~ generated locally by local sections of the form {Xx, ...,xq} such that x~ -- I e re" i* r where ~ is a prime element of K. It is possible to compute the subquotients I M~/U 1 M~ (m = o) gr~(M~) : u M /U (m >/ i) for those m such that o < m < e' -- ep , where e denotes the absolute ramification -- ~-- I index of K. If n= I, U '~M~--o for m>e' and thus we obtain a precise picture of M~. The result is very similar to the structure theorems of the K-theoretic sheaf SCKq(d~y) and of the De Rham-Witt complex of Y (cf. Bloch [3], Illusie [IO]). Indeed, if o<m<d, gr"(M, q) ~ gr'~(SCo~ K~(0y)/p ~ SCoo K~(0y)) for the filtration {U~SC~o Kq}m>l on SC| Kq which is defined by modifying the filtration ill* of [3] II, w 4, as U '~ SCo0 Kq = fil "-1SC~o Kq + {fiW -1SC~o Kq_l, T}. (Cf. also [II] w 2.) But this precise analogy holds only in this range of m, and the structure of grm(M~) for m > e' has rather different aspects which are not yet well understood. (x.3) Let ~ = g2~/z be the exterior algebra over r of the sheaf fl~/z of absolute differentials on Y. If k is perfect, this coincides with the usual D~/k, but is bigger than the latter in general. As in [3], [IO], define subsheaves BI and Z[ (i > o) of D~ such that o = BgC B~C... C Z~C Zg = D~ by the relations BI = Image(d : D~ -1 -> ~) Z~ = Ker(d : n~ -+ D~ +~) 0-, q q > Z~+I/B1 B~ => Bi+x/B1, Z~ c-'____ q q 111 11.7 SPENCER BLOCH AND KAZUYA KATO where C- ~ is the inverse Cartier operator: ~. dy, dy~ dy~ dy, f~-+Zt/B~, x--^ ... ^--.xP--^ ... ^-- =~ Y~ Yq Yt Yq (Yx, "" ",Ye invertible). Define a},~o, = Ker(~ -- C-~: n~ -+ a~/N). This is in fact the part of fl~ generated dtale locally by local sections of the forms dx x dx~ --- ^ ... ^ -- ([xo] Th. o~'.4.~). Let W,,f~c be the De Rham-Witt complex of Y~t xx x~ and let W, f~]r, ~og be the part of W, n~r generated 6tale locally by local sections of the form dlog(xx).., d log(x~). Note that, since all local rings of Y are inductive limits of smooth algebras over F~, the theory of the De Rham-Witt complex over a perfect base ([xo]) applies to W, f~. Our results are the following: Theorem (x .4). -- The sheaves M~ are generated locally by symbols, and ~ f~. ~. (i) gr~ q) = W,f~},~g| -~ (ii) For m > i, there is a surjective homomorphism p= : ~2~-x | n~-2 ~ grm(M~). (iii) Let I~< m<e'-- ep and let m= m xp', s ~ o, p C m a. Then, for p--I o < n < s (resp. n > s), the above homomorphism p,~ induces an isomorphism ~'~q-l/Tq--I (i) ~gq721zq-2 + grn,(Mq) Y IZ'Jn x t n ,., (resp. an exact sequence o -+ nV' L nV'/BI-' ~ a,-~n,-' .oy /~o -+ gr"(Mq,) -+ o, (s times)). where 0(~) -= (C-'(d~), (-- I)'mlC-~ C-'=C-'o... ,C -t Corollary (x. 4. I ). -- The sheaf M~ has the following structure. (i) gr~ ~ tat,,.~ ~ n~7, ~. (ii) If I < m < e' andre is prime top, gr'(M[) ~ n~-l. (iii) If I < m < e' and p [ m, gr'(M[) ~ ~-I/Z~-* | f~-2/Z[-2. (iv) For m >_ e', U" MI = o. The surjective homomorphism 112 p-ADIC ETALE COHOMOLOGY tx3 given by (i .4) (i) is a homomorphism such that {~, ..., ~'q} ~ (d log(xi) ... d log(xq), o) {~ .... , x'q-a, ~} i-~ (o, d log(xt) ... dlog(xq_a)), where ~ is a fixed prime element of K, Xa, ..., x a are any local sections of 0~, and are any liftings of x i (I < i < q) to i* ~. An analogous homomorphism is given by Theorem (x.5). -- There exists a unique homomorphism Mq, ~ ~m/p" f~s which satisfies d/x d f, ... A -~q { fl .... ,fq-l, c} F-+ o for any local sections f~, ...,f~ of i'~ and for any c zK ~ Here we regard as a sheaf on Y0t in the natural way. In conclusion, one might say that the p-adic dtale cohomology M, ~, the De Rham- Witt complex W, f~, and the De Rham complex f2x/s, live in completely different worlds, and there is no unified cohomology theory at present which combine them in an intrinsic manner. We must therefore use some presentation of them by symbols in the study of their relations. It becomes clear that the symbols play important roles in the algebraic geometry of mixed characteristic, though we do not know from what world the symbols come. 2. The differential symbol Let K~ be the Milnor K-theory of fields [I5]. For a field F of characteristic p > o, we write kq(F) = K~(F)/pK~(F), v~ -= Ker(f~ t-e-~ f~/df~_t) ' ...~ -------A ... A--. x I Xq The following result was proved independently by O. Gabber. Theorem (2.,). -- d~ is an isomorphism. We give here the proof of the injectivity of +. The proof of the surjectivity is similar to the proof of Proposition (~.4) below and is given in [I2], w I. We fix q so that Theorem (~. i) holds for all q' < q. We use the method in [4]. Lemma (2.2). -- If ~q is injective for F, it is injective for any #urely transcendental extension 15 SPENCER BLOCH AND KAZUYA KATO il4 This follows from the commutative diagram of exact sequences o ) kq(F) 9 kq(F(t)) ~ H kq_l(F[t]/m) > 0 where m ranges over all maximal ideals of F[t] and 0,, denotes the tame symbol for each m (Ix], Ch. I, w167 4 and 5)- The homomorphism i m is the compo- sition of kq_l(F[t]/m) _+ ~(r~0/m) ~-1 with the canonical injective homomorphism ~"]Ifft~lm)--~ oq I~q which is defined by xodx~ ^ ... A a~_~ ~ ~o d~ ^ ... A d~_~ ^ ~.~ d~., for any x0,...,xa__ 1 e F, any prime element 7rm at m, and for any lifting ~ of xi(1 </<q-- x). Corollary (2.2. x ). -- ,~o is injective for K if K is purely transcendental over a perfect field. (2.3) For a semi-local Dedekind domain R with field of fractions K such that char(K) =p> o, let kq(R) = Ker(kq(K) le')~ = H kq_ 1(R/m)), where m ranges over all maximal ideals of R. Let I be the radical of R, let k,(R) ~ kq(R/I) ~ Hkq(R/m) be the specialization map induced by the homomorphism in Lemma ('~.3.2) below, and let kq(R, I) be its kernel. Assume R has a p-base so that the Cartier and the inverse Cartier operators are defined, and let ~ = Ker(I -- C-1: ~ --~ ~Idd(~-l)), v~, I = Ker(v~ ~ v]~/x ). By Lemma (2.3.~) below, we obtain a diagram (commutative with exact rows) o > kq(R, I) > kq(R) > kq(R/I) ) 0 (2.3.x) o , ~.I , ~ ~" V~/I Lemma (2.3.2). -- Let R be a discrete valuation ring with quotient field K and with residue field F such that char(K) = p > o. 114 p-ADIC ETALE COHOMOLOGY (i) kq(R) is generated by symbols {xx, ..., xq} (xl, ..., xq ~ R*). (ii) There is a unique homomorphism kq(R) --> kq(F) such that {al, ..., ..., Z,}. (iii) If R has a p-base, there is a unique homomorphism + :kq(R) ---> ,~]~ such that { xx, xq } ~ dxl dxq ..., --A ... A --. x 1 xq Proof. .... (i) follows from [I], I (4.5) b) and (ii) is the ~ of (loc. cit.) (4-4)- The homomorphism in (iii) is induced by +:kq(R) ~ v~ by virtue of (i). For a finitely generated field F over Fq, we can find a discrete valuation ring R which is a local ring of a finitely generated algebra over Fv, such that Rim ~ F and such that the field of fractions K of R is purely transcendental over Fp. Since kq(R) -+ v~ is injective by Corollary (2.2.I), the diagram (~.3.I) shows that to prove (2.1) it suffices to prove Proposition (2.4). -- Let k be a perfect field of characteristic p > o, let R be a semi- local Dedekind domain which is obtained as a localization of a finitely generated k-algebra. Then, + : kq(R, I) is surjective. Proof of (2.4). -- To begin with, k~ has a norm compatible with the trace on vq and carrying kq(R', ~) to kr I) for R' the normalization of R in a finite exten- sion K' of K (cf. for example, [II], w (3.3), Lemma x3). The diagram Vi ) '1 ~ Norm t~ > ,o~, I kq(R, I) and the formula tr.f" = multiplication by [K' : K] reduce us to showing that for a given A E,~,I there cxists K' with [K': K] prime to p such that ffA eIm+. We now follow closely the argumcnts of [I2]. Choose a p-basis bl, ..., b, of K such that bl, 9 9 b,_ 1 ~ R" and these elements mod I form a p-basis for R/I and such that the valuation of b, at each maximal ideal is prime to p. Strictly increasing func- tions s : { I, ..., q} -+{ I, ..., n} are ordered lexicographically so s < t if for some i r I, . . ., q} we have s(i') =t(i') i'<i and s(i)<t(i). Write db so ) db s(q) 6~s = A ... A --. b so ) b s(q) 115 ~x6 SPENCER BLOCH AND KAZUYA KATO An element Za, cos lies in ,~ if and only if It lies in ,~,i if and only if, in addition, a, E I for all s. The notation ~,, (resp. ~2[(, <s) for s : { ~, ..., r} ~{ ~, ..., n} will mean the sub-K-vector space of ~[( spanned by o hfor t<__s (resp. t<s). Lemma (2.5). -- Let a ~ I and let s : { ~, ..., q} ~ { ~, ..., n } be strictly increasing. Assumg (a ~ -- a) (as ~ ~qK, <s + d~ -1 Then re#lacing K by some finite #rime to p extension K' which is a succession of Galois extensions and replacing R and I by R' and ~ as above, there exist Yl, ...,Ya ~ K such dyl dy~ ~ and a~o, -- ~ e ~2~,~ n ~, <s, that {y~,...,y~}~kq(R,I), 0~=--^... ^--e ~.s, where ~2~,~ = Ker(~2~ -+ f2~/z). Y~ Y~ Note that this lemma suffices to prove (~. 4) and (~. ~). In fact, given Y.a s o~ s ~ ~, we can by the lemma subtract ~ e lm(k~(R, I) -+ ~,~) and decrease the " size " of the maximal s with as * o. Proof of (~.5). -- Adjoining the (p -- ~)-st root of some element in R we obtain as in [I2] dc (~.6) a~ s ---- a' r ^ -- -F x where s': {I, ...,q-- I}-+{I, ...,n}, s'(i) = s(i + i), a' ~K, c a KP(ba, ..., b~it)), and (a'" - a') co., ~ ~7~, + a~ -2. We have dc i = s(1) db~ a t --= Z V~-~ (v~K), +a Define ~0l~I ~PtDI and let Rj, R L be localizations, so that JRj and LR L are the Jacobson radicals. Note that a' e LRL, so, by induction on q, we may assume (2.7) a'%,=~+v, [~=+{yl,...,yq_l}, {Yl,...,Yq-IIek~-I(R,L) q--1 Write T---- R sr3Kp(bl,...,b,(l)_l). Let H = RflJR a and P =T/J r~T. The image of --de in ~2~j/T dies in ~nm and is fixed under the Cartier operator. The diagram c 116 p-ADIC ETALE COHOMOLOGY IE7 (~ + JRs)/(T" c~ (~ + JRj)*) o > R]/T* a log fll > Rj/T o > H*/P* alog shows that there exists 8 e I + JR~ such that dc d~ -- + ~ in ~ with e R a. Im(n t -+ n~,) c n 1 -- K, < s(l} ~'~K, #(1)" By (2.6) and (2.7) we get a~o,=(~+v) ^(?+,)+, -- A ... ^ ^ (mod s <,). Yl Yq- 1 Note, quite generally, that if B 1 ~kt(Ra, JRa) and B~ e km(RL, LRL) the product Bx.B ~ belongs to kt+.,(R, I). This is a simple consequence of the fact that kt(Rm, mR,,).kl(K ) __ kt+l(Rm, mRm). In particular, {Yl, ...,Yq-a, 3} ekq(R, I), q.e.d. Corollary (2.8). -- Let F be a field of characteristic p > o. Then the p-primary torsion subgroup of K~(F) is infinitely divisible, and K~(V)/p" K~(F) ~ W. ~,,og. Here W. ~,log is the group of global sections of W. ~g on (Sv F), t. Proof. ~ For a discussion of W. ~,log see [IO]. In particular we have K~(V)/p 9 K~(V)/p" , K~(F)/p "-~ > o /1 o 9 ~', ,og , W. a~., ,o~ , W~_ 1 fl~, log 117 SPENCER BLOCH AND KAZUYA KATO ti8 where the bottom sequence is exact by op cit. (5.7.5). The left hand vertical arrow is an isomorphism by (2. i) and the right hand arrow is an isomorphism by induction. This establishes the isomorphism. The first assertion follows from the exact sequence of Tor(K~, -) applied to o ~ Zip -+ Z/p" -~, Zip "-1 ~ o. 3" A basic cohomological lemma Let K be a field, p a prime number prime to char(K). The cohomological symbol defined by Tate gives a map [19] K~(K)/p" K,~(K) -+ H;t(S p K, Z/p" Z(r)), which one conjectures to be an isomorphism quite generally. It is useful to formulate a relative conjecture. Let (Q/Z)' denote the prime to char(K) torsion in Q/Z, let x 6 HI(SP K, (Q/Z)') and let K' be the cyclic extension of K corresponding to Z- Conjecture (3- 9 -- The sequence K~_I(K, ) N K~_I(K) x~,> H'(Sp K, (Q/Z)' (r -- ,)) > H'(Sp K', (Q/Z)' (r - I)) Here N is the norm map in Milnor-K-theory [II], w (1.7), and "X 13 " is is exact. X-+zuh(x ) with the map h: K~_I(K ) -+ Hr-I(Sp K, Zp(r -- 1)), the cohomological symbol. See [I4] for definitive results on these conjectures in the K~ case. The following lemma is taken from [i'a]. It is the essential tool we will use in studying these questions. Lemma (3.2). -- Let notation be as above, but take [K': K] =p. Regard X as an element of Ht(Sp K, Z/pZ), and let G ----- GaI(K'/K) ~ Z/pZ. Then (i) The sequence (3.2.x) H q- t(Sp K, Z/pZ) zu Hq(S p K, Z/pZ) , Hq(Sp K', Z/p) is exact if and only if the sequence '~', Hq-l(Sp K', Z]pZ)o co,> Hq_t(S p K, Z]pZ) (3.2.2) Hq-l(Sp K, Z/pZ) is exact. (ii) The sequence 0or Hq_I(Sp K, Z/pZ) xv> Hq(S p K, Z/pZ) (3.2.3) Hq-l(Sp K', Z/pZ) is exact if and only if the sequence > Hq(Sp K', Z/pZ) G ~~ Hq(Sp K, Z/pZ) (3.2.4) Hq(Sp K, Z[pZ) is exact. 118 p-ADIC ETALE COHOMOLOGY x l 9 (For a G-module M, M ~ = invariants of G acting on M and M G = co-invariants = M/( X (, _ g) M).) gEG Proof. -- We will only prove (i). The proof of (ii) is similar, and it will not be used in the sequel. Adjoining a p-th root ~ of i involves an extension of degree prime to p, and hence induces injections on the homology of the complexes (3.2. I) and (3-2.2). Thus we may assume ~ e K. Sublemma (3.3). -- Assume ~ e K, and identify Z/pZ -% ~t v via I ~ ~. Thus HI(Sp K, Z/pZ) ~ K'/K "p and ~ ~ K" gives a class [~] e Ht(Sp K, Z]pZ). Let [3 : HI(Sp K, Z/pZ) --+ H2(Sp K, Z/pZ) be the Bockstein associated to the exact sequence (3.3. 9 ) o -+ Z/pZ -+ Z/p ~ Z ~ Z/pZ ~ o. Then = Z u Proof. --An element t e K* maps to the class a(.) : GaI(K~"P/K) --+Z[pZ where ~o(ol = (t,/p)o[t,/p. Let p~ = ~, 0P' = t. The cocycle w(a, v) associated to [3(t) is given by = 0 0/0 o 0". Note that 0 ~ = pAt~ A(~r) -- a(a) (modp). From this one gets easily = = The cohomology class represented by the right side is t u [~], q.e.d. Sublemma (3.4), -- Let S be a profinite group, p a prime number, Z a non-zero element of Hi(S, Z]pZ), and T = Ker(x:S -+Z[pZ). Let [3 :H*(S, Z[pZ) ~H*+~(S, Z[pZ) be the Bockstein. For X t y L Z a complex, call Ker(g)/Im(f) the homology. (i) Let q ~ 2. Then, the following two complexes have isomorphic homology groups. (3.4. ') Hq-'(S, Z/p) 9 Hq-2(S, Z/p)(5~'-~!x)-uA Hq(S, Z/p) "~> H~(T, Z/p). (3.4.2) H q -t(S, Zip) ~*'~ Hq-a(T, ZiP)SiT oor> Sq_t(S ' Zip). (ii) For q ~_ I, the following two complexes have isomorphic homology groups. (3.4-3) Hq-t(T, Z/p) oo,) Hq_,(S ' Z/p) r Hq(s, Z/p) | Zip) (3-4.4) Hq(S, Zip) ", Hq(T, Z/p) sIT ,or Hq( S, Z/p). 119 130 SPENCER BLOCH AND KAZUYA KATO Remark. -- These sequences are exact if p = 2, but need not be exact in the case p 4 ~ 2. For example, let p be an odd prime number, and let S be the semi-direct product Zp[~] � where ~p denotes a primitive p-th root of i and v is the homomorphism Zp ~ Aut(Zp[~p]) ; a ~ (x ~ ~ x). Let z:S ~ Zp be the homomorphism induced by the second projection S ~ Zp. Then, the sequence (3.4.2) is not exact in the case q = 2. Thus, though S is torsion free, S can not be isomorphic to Gal(ks/k ) for any field k. Proof of (3-4). -- Since the proofs of (i) and (ii) are rather similar, we present here only the proof of (i). Let X be the S-module of all functions S/T -+ Z/p, s an element of S such that Z(s) = I, and Y the image of s-- I:X-+X. Let g:X-+Y (resp. h : Y -+ X, resp. i : Zip -+ Y) be the map induced by s -- i (resp. the inclusion map, resp. the embedding as constant functions). Since there is a canonical isomor- phism Hq(S, X) ~ Hq(T, Z/p) for any q, the exact sequences of S-modules o ~Z/p ~'~X g ~Y >o, o ~Y h>X , Z/p , o is defined by j(f) = x~/Tf(x) for all f e X) induce a commutative diagram (J H'-2(S, Z/p) Z/p) Y) H,(S, Z/p) H,(T, Z/pZ) Z/p) , Z/p) Z/p) with two long exact sequences. Here 0 denote the connecting homomorphisms. (Note that the restriction maps and the corestriction maps are induced by h.i and j, respec- tively. The commutativity of the diagram follows from (3.5) below.) The assertion (i) follows from this diagram. This proves (3.2) and (3-4). Lemma (3.5). -- (i) The image of i eH~ under the composite map O 1 0 H~ Z/p) ~ H (S, Y) ~ H2(S, Z/p) coincides with [~(X). (ii) The image of I e H~ Zip) under the composite map i 0 0 -->. H~ Z/p) ~ H (S, Y) H~(S, Z/p) coincides with X. 120 IOI #-ADIC ETALE Proof. ~ (ii) is easy and so we give here the proof of (i). By functoriality, we may assume T = {I }. Let f 9 X be the function defined by f(1) = I and f(,) ---- o for ~. I. Then, j(f)= I. So, 0(I) 9 is represented by the cocycle S~Y, o~f~, where I if o4~ I and ,=6-t if cr 4= I and 9 = I f (T) = --, 0 otherwise. For ~ 9 define f~'eX by i I ifm+n>p or if m> I and n--o f#(s") = o otherwise 0 0(I) 9 H2(S, Z/p) is represented (o<m<p, o<n<p). Then, g(f")=f~. So, by the cocycle G � G -+ Zip C X, if m + n>p (s", s") ~J~" o s" --J,~§ +J;' = o if ra+n<p (o<m<p, o < n < p). But this cocycle also represents [3(X ) as is easily seen. 4" Filtration on Symbols In this section, A denotes a ring additively generated by A" (e.g. A local), and n denotes a non-zero divisor of A contained in the Jacobson radical of A. Let K~(q> o) be the group q times where J denotes the subgroup of the tensor product generated by elements of the form x l|174 such that x~+xj= I or o for some o<i<j<q. An element xa| | of K~ will be denoted by {xt, ...,xq}. One has of course ( [i].) {x,I--X}=O x,I--xeA , {x,--x}=o and also {x,y}=--{y,x}. In this section, we give some elementary lemmas concerning the structure of K~, which will be useful in later sections. The arguments are essentially the same as in [3], Ch. II, w 3, where Ouillen's K-functor is studied for A = R[[T]] and ~ = T. For m _> I, let U "~ K~ be the subgroup of K~ generated by symbols of the form {, + xTt",yl, . . .,yq_l} such that xeA and Yl,...,Yq-t eA[!]'. COHOMOLOGY SPENCER BLOCH AND KAZUYA KATO Lemma (4.t). -- {UiK~,UJK~}CU '+jKq+" ,. For a, b E A', we have (4.x.x) {z + a~ ~, x -+- b~ ~} -= {~ -- a~"(t -t- bn'), z + b~i}modU '~i = - { i + an~(I + b~0, -- a'~ ~} ~- -- I I -~ --. abn j aT~ i ! I -t- an" The lemma follows easily. For a ring R, f~ will denote the module of (absolute) Kahler differentials of R. We write f2~ = A~ f~. Define the homomorphism ~,: R | (R')| -~ ~, ay, ay, by ~,(x|174174 =x--^ ... ^--. Yt Y, Then ~, is Lemma (4.2). ~ Assume R is additively generated by R* (e.g. R local). surjective, and Ker ~, is generated by elements of the following types: (4"2" 9 X | | "'" | with Yi =.Yj for some I <_ i < j <_ r, (4.2.2) ~ xi|174174 | ~ x:|174174 | , ~. t xi, xi s R', ~-1 xi = ~-Z x~. Proof. - Straightforward and left to the reader. Let R = A/nA, and for any m_> x, define p~: ~-i| fl~-2 ~ gr ~ K~ = U ~ K~IU ~+t K~ (4.3) p,, (x-~tl ^ ... ^ O)y,_, d'yq-t, ={1 + ~'~,~, ...,.~_,} by ( dy, dy,_~ / p,, o,x--^ ... ^ ={~ + ~",y~, ...,y~_~,,~}, Yl Yq-2] cA, ~A* lifting xeR, .Yi ~R*. The fact that p,, is well defined is an easy consequence of (4. i) and (4.2). From now on, let p be a prime number and assume that R = A/hA is essentially smooth over a field of characteristic p. Note that (4" 4) ( I -~ 7~ n X) p ~ I -~ Tg rnp X p mod ='~P + t A if p ~"~P-tl+t A. 122 p-ADIC ETALE COHOMOLOGY ta 3 Lemma (4-5).-- Let m = m lp', s >__ o, p { m I and assume that pP e n'~v-t)+t A. Then (i) ~-1 q-2 )=(o) (ii) Define 0:n~ -2 -+ (f2h-'/B, q-l) | (n~-2/B q-2) by 0(~) = (C-'(dm), (-- I) qm 1 C-' (to)). Then p, o = (o). (See (I.3) for the notation B; and 13-1.) Proof. --- Let o <_ t < s. Part (i) follows from { x + x ~' rd ~1r x} = pt{ I + xrc'lr x} mod U '~+1, pt-t s-t p'(I +x~"~ ,x}=--{I +x~tP ,(--~)r =} -= --{, + xP'r~ ', -- I}--p'-tmi{x + xV'= ", r:} e U "~+'. (Use (4.4) with mp in place of m.) Part (ii) amounts to the assertion { ' + xr ~", x, Yl, .. ",Yq-2} - (-- I) q-1 ma{ ' + x v' ~",Ya .... ,Yq-~., ~}, i.e. { x + x v" n', x~ '~ } e U" + 1 This is again straightforward. Lemma (4.6). -- Let m, m 1 and s be as in (4.5) and let o <n <s. 7"hen p,.(zq-, | Z q 2) = (o) it, gf(K~/p" Kq M) = (U" Kq ~ + p" Kq~)/(U "+' K~ + p" Kq~). Proof. -- Let m'= mp -n. Note that p"{, + xr~",y} -- {x + xV"rd",y}modU "+1. Since Z, is generated by B, together with differentials x v" dyx -- ^ ..., the lemma follows. Yx Let m, mx andsbe as in (4-5) and let n>o. Define the group"G, qtobe (4.7) (f~- t/Z~- ~) | (f2~- 2/z,q-2) if n < s, Coker(ft~- z 0 (f2~- ~/B,) | (~-Z/B,)) if n > s. (4.~) We have established surjections "O q ~ gr'~(Kq/p~" Kq).M Remark (4.8). -- These surjective homomorphisms are in fact bijective. Indeed, by localization, the question of injectivity is reduced to the case where R is a field. If char(A[~]) =o, injectivity will be proved in w 5 and w 6 by using thecohomological symbol. If char (A [~] ) = p, injectivity follows from [3], Ch. II,w also [I I], w 2 ). 123 SPENCER BLOCH AND KAZUYA KATO x 24 Note that in the mixed characteristic case, the condition on m in (4.5) is actually restrictive. The structure of gr"(K~) for large m such that pP r (this is equivalent to m > e' = ep with the notation ofw 5, w 6) is not yet known. p--i 5- Galois cohomology In this section, K denotes a henselian discrete valuation field with residue field F such that char(K) = o and char(F) = p > o. In the next section, we shall apply the results of this section to the quotient field of the strict henselian discrete valuation ring Ox.; where v is the generic point of Y (not to the base field K of w o). Let kq(K) = K~(K)/pK~(K), hq(K) = H~(Sp K, Z/pZ(q)). The aim of this section is to determine the structures of these groups and to prove that the cohomological symbol gives an isomorphism K~(K)/p" K~(K) T Hq(Sp K, Z/p" Z(q)) for all q and n. We define the filtration U m K~(K) (m > I) as in w 4. Here we take the valuation ring 9 K of K as A and a prime element of K as n. Note that the homomorphism p,,: fl~-lOfl~ -z -+gr"K~(K) depends upon a choice of a prime element ~ of K, which, we will assume, has been fixed. Let U ~ = K~. Let U mkq(K) Ckq(K) (m> o) be the image of U m K~(K), and let U"*hq(K)C hq(K) be its image under the cohomological symbol map kq(K) ~hq(K). Let ord~ be the normalized additive discrete valuation of K, let U~"I ={x eK, ordK(x -- i)>m} for m> i, ep let e = OrdK(P) the absolute ramification index of K, and let e'-- p- I Lemma (5. 9 -- (i) U" k~(K) ----- o for m > e'. (ii) Assume that e' is an integer and let a be the residue class of pr:-~. Then, the surjective homomorphism (4-3) p,, : aV 1 9 a~-~ + u' k~(K) annihilates (i + aC) Z~- 1 | (i + aC) Z~- 3, where C is the Cartier operator. If F is separably closed, then U ~' kq(K) = o. Proof. ~ (i) follows from U~ '~) C (K*) p if m > e'. 124 p-ADIC ETALE COHOMOLOGY x25 The proof of (ii) is similar to the proof of (4.6) using (I + x~"lP) v - x + (x p + xftr-') 7r" mod~ ''+a Lemma (5.2). -- Let I <_ m < e' and let the group "G~ be as in (4.7) with R=F. Then, "~G~ =~ gr" kq(K) =~ gr" hq(K). Proof. -- By a limit argument we may assume that F is finitely generated over Fp of transcendence degree d. We may also suppose that K contains the p-th roots of x (a straightforward reduction using norms, which we leave for the reader). Then the group U~'ha+2(K) is non-zero by [II], w I, Th. 2 (cf. also Ix2], page 227). Note (5.~.x) ,,G~ ~ I f~-t o<m<e',p~m (B~ | -x o<m<e',p[m. We now consider a diagram of pairings Om � Or "G] � "-mGla+s-q > grmh q x gff'-" h d+2 (2) cup produet d d j_ ~F/B 1 (1) e,Gl~ + 2 Pc' Ue, ha+2 where arrow (1) is the natural surjection which exists because B~= (x +aC) B~C(I +aC) n~, and arrow (2) corresponds under the isomorphism (5.2. i) to wedge product of forms if p~'m (resp. to a,o ) � (ayl, ay,) . ,01 + ,o3 ay, if p Ira). It is a simple exercise with symbols (calculated as in (4. I)) to show that this diagram commutes upto an (Fp) � Also ~F/B1 d is a x-dimensional Yp vector space and (2) is a perfect pairing of Fp vector spaces. Injectivity of ~.~ follows. Since the arrows from left to right in the statement of (5- =) are already known to be surjective, we are done. Lemma (5-3). -- ~@v~-I ~ gr~ g gr~ 9 Proof. -- Results in [I] give an isomorphism gr ~ kq(K) ~ kq(F) | kq_ t(F) 125 x26 SPENCER BLOCH AND KAZUYA KATO so, from (2. i), we get a map ~o defined as the composition ~o : v~ | v~ -x ~ gr ~ kq(K) -~ gr ~ hq(K). Let K' D K be the quotient field of a henselian discrete valuation ring OK' D ~K with the property that K' is unramified over K, with residue field r' = where z is transcendental over F. Let ~' e d~, lift z. Multiplication by I + ~ gives - r ~ | v~, -~ -Y:* gr~ hq(K) -----+ grth q ~t(K') ~ g~,(,).q The composition is easily seen to be dA dL dA dL "'" ^ Z'z ^ "'" ^Z dfl dL-t dfl dL-1 Injectivity of ~0 is now immediate. Our next objective is to prove that hq(K) ~ hq(K). Let Shq(K) -.- U ~ be the image ofkq(K) in hq(K). We first prove Shq(K) = hq(K) in the case F is separably closed and K contains a primitive p-th root ~p of I. To apply the basic lemma of w 3, we devote ourselves in (5.4)'(5-I i) to proving Proposition (5.4). -- Assume that F is separably closed and ~p 9 K. Let b 9 O~ be such that the image ~ of b in F is not a p-th power. Let a = b I/p be a p-th root 0fb, L = K(a), Qt = ~vp, E--: F(0t) with G = Gal(L/K). Then, the sequences re, Shq(L) G eor> Shq(K) (5.4. 9 ) Sh~(K) '~', Shq(L)G ~~ Sh~(K) (5.4.~) Sh~(K) are exact for all q. Note that we already know the precise structure of Shq(K) and Shq(L), for gff'h q---o by (5.t) (ii). We begin with some lemmas concerning differentials. Let i : ~, ---> ~ be the canonical homomorphism, and let Tr : f~---> ~2[, be the trace map characterized by (i) Tr(E.i(~,)) = Tr(dE ^ i(g2~,-~)) ---- o (ii) For ~e~,-t andf 9 ~ Tr i(~)^ =co^ f--)-. A proof of the existence of Tr is that the norm on SI~IKq+I ([I3]) induces this homo- morphism Tr on its subquotient f~q. (The assumptions p 4= 2 and p > q in [3], II, w 4, Th. (4. I) are unnecessary by [tI], w 2, Prop. 2.) In (5-5)'(5.9), we need not assume F separably closed. 126 p-ADIC ETALE COHOMOLOGY t27 Lemma (5.5)-- (i) For ~ Efi}, the three conditions a) ~ A d[3 ---- o, b) o efl~-lA d~, c) i(o) ---- o, are equivalent. (ii) Let H = i(~) C f~. Then the map (E | I q) E) (E | Iq--~) ~ n~ defined by (x| o) ~ xo~ d~ (o, x | ~) ~ x~ ^ -- is an isomorphism. The proof is left to the reader. Lemma (5.6).- The sequence is exact. Proof. ~ By (2. I), the assertion is equivalent to the exactness of kq(E) s.,] kq(F) > kq(E) N.,~ kq(F). We use the fact that the composite is multiplication by p. This fact is reduced to the case where any finite extension of F is of degree a power ofp. In this case, K~(E) is generated by elements {x, yl, ... ,Yq-x} such that x EE', y,, ...,yq_x eF* ([I], Ch. I (5.3)). Now assume x e K~(E) and N(x) = py, y ~ K~(F). Then, px = i o N(x) = pi(y). Since the p-primary torsion part of K~(E) is divisible by (2.8), we have x- i(y) ~pK~(E). This shows the exactness of kq(F) -+kq(E) ~kq(F). The exact- ness of kq(E) ~ kq(F) -~ kq(E) is proved similarly. Now, we analyze the sequences (5-4. x) and (5.4.2) using the filtration on Sh q. Lemma (5.7).- (i) cor(U'hq(L)) C U'h~(K) for any m. (ii) The following diagrams commute. 9 4"4 -1 ~ g r~ h,(L) Tr le, or I 1 ~qc~,,q-1 ~ r 0kq(K) n~-le n~ -~ 9 gP h~(K) F~"F = g (m > i). Here in the diagram on the right, the horizontal arrows are induced by p,, defined using the same prime element ~. 127 x~8 SPENCER BLOCH AND KAZUYA KATO Proof. -- For m> i, let T,,be the image of U"ht(L) | Sh'-~(K) --> Sh'(L) x | ~ x to res(y). (ii), we can prove easily that By using (5.5) For any m > x, U~hq(L) is generated by rim and (s.7.x) res(U"hq-'(K)) to {a}, where {a} denotes the class of a in hZ(L). By using NL]K(I -- Xa i) = I -- X p h i for o < i < p and x e K, we have (5.7.=) NT,tK(U~ ~)) C U(K ~p) for I < m < -- -- --p--I N rU(")~ C U~" +e) for m > e --o L/Kk L ] -- p -- I I) and (5-7-2) prove (i). The commutativity of the diagrams in (ii) Note that (5.7. follows easily. Now, for m > o, let S,, be the homology group of the complex gr" kq(K) '"> gr '~ hq(L) - oor> gr" hq(K). By (5.6) and (5.7), we have Co~ottary (5.8). -- So = (o). /.emma (5-9). -- For x < m < e', we have an isomorphism (EIq-~/F -1) 9 (EI,-'/I,-') ~ s. characterized by x ^ ... ^@'--~',o ~{x +~",'y,, ...,Y~-~! \Ya Yq- 1 ( dYx dyq-21F_~{i_l_~xm,~t, " ~_2, ~} O~X--A ... A ..~ Yl Y~ - 2 / for x ~ E and Yx, ...,Yq-i ~F*, where tildas indicate liftings. Proof. -- Assume p lm. By (5.5) (ii), we have a commutative diagram for any q p--t ~q/Z q EF/(F n Z~,s) | Y~ =~Iq-~H-~I(H -~ c~ Z~,~ 1) ~ E, I,E pr= Tr iq-l/(i q-1 zq-~ t'3 ~q /z~ _,1 c , F, ,,F 128 #-ADIC ETALE COHOMOLOGY to 9 where the upper horizontal isomorphism is dot CO tt) Ca) It ) (co, co', ~-~ ,o + (co' + ^- ot d~ and the lower horizontal injection is i(o)v* co ^--. Now, (5.9) follows from (5.7) and (5.2) in this case. The proof for the case p { m is similar and is left to the reader. To proceed further, we need Lemma (5. Io). -- Assume F separably closed. Let ~ be a generator of Gal(L]K) and let e"-----. Then, p--I (i) U~ ''1 C (L*) ~ 1. K'. (ii) O~ I C (L*)V.K ". Proof. -- Note that p:U~"I--~U~ '1 is surjective (cf. [22], w (x.7)). Let x 9 ''). Then by (5.7.2), we have N(x)~U~ '1. Hence there is an element y of K" such that N(x) =yr. Since N(xy -t) = I, we have x 9 (L*)*-I.K" by Hilbert's theorem 90 . Next, let x 9 U'~ ~1. Then, x~149 U'~ ~'1 since o acts on OL]~ ~ 0 L trivially. Hence there is an elementy of U~ ''1 such that x ~ _yr. By (5.7.2), N(y) is a p-th root of I contained in U~ '1. Since ordx(~ v- I)- e", we have N(y) = x. By Hilbert's theorem 9 ~ , x ~ = (z~ v for some z 9 L*, and thus we obtain xz -v e K*. Let a (5. xx) Now we can prove (5.4). First we consider the sequence (5.4-I). and e" be as in (5-IO). We have (~ -- i) (U=h~(L)) C Um+''' hq(L) for m > o and this induces e-- I:S,,-+S~+,,,, m> o. We claim (5.IX.It) *-- I : Sm ---~Sm+e,, is an isomorphism for o < m < e. Indeed, for x 9 O K and letting ~p = ~r(a)/a, we have ( I + a' x,e") ~ = ( I + a' ~ x,e") / ( I + a' x,e') =- I + ia~xT:"(~,- x) mod= "+''+1 Our claim follows from this and from (5.9). Now assume x 9 Shq(L) ~ and cor(x) = o. We prove x 9 res(Shq(K)). We are reduced to the case x 9 by (5.8), and then to the case x 9 by T. C res(Sh~(K)) (5.II.I). Thenwehave xeT, by (5.9) (T,,is as in (5-7))- But by (5-x~ (ii). and cot(x) ---- o. Next we consider the sequence (5.4.2). Assume x 9 Shq(L) 17 t3o SPENCER BLOCH AND KAZUYA KATO We shall prove x eres(Shq(K))+ (o- i)Shq(L). We are reduced to the case xeU tSh~(L) by (5.8). We prove (5. xx.,,) If x~Tm+U~hq(L) with I<m<e" and (m--~)p<i<mp, and if cor(x) =o, then x~T,,+U ~ +thq(L). (5.ti.3) If xeT,,+U"Vh~(L) with I<m<e" and if cor(x)=o, then x ~ res(Umha(K)) + T,,+~ + U'V+!hq(L). Once we have these assertions, we are reduced to the case x ~ T~,, + U '''~ h~(L) = T,,, and then we have x ercs(Shq(K)) + (o -- ~) Shq(L) by (5.IO) (i). The assertion (5. I I. 2) follows from (5.7.2) and (5.9) easily. To prove (5. I 1.3), let (1) be a subset of F such that (I) u { I~ } form a p-base of F. For each V e (I), we fix a representative ~' of ~0 in O K. We endow 9 with a structure of totally order set. For q _> o, let (I)~ be the set of all strictly increasing functions { i, ..., q } -~- (1), and let E(I)r be the free E-module with base (l)q. We obtain a surjective homomorphism EOq_ 1(9 EOq_ 2 (9 Iq-~/(I q-~ n zq,~ 2) | Iq-3/(I q-3 n Z~,~ 3) -~ (Tin + U m~ hq(L))/(T,.+ t + U "v+! h*(L)) p--I p--I ~ ~r)q~EOq_t, 0, O, O) J--~ ~]{I -'~ X Xr.g ~ "/~m, ~(I), . ~(q-- I)}, ((r~0Xr, ~ ~ ~r "', = ~ r=0 p--I p--1 ~ N (o,( Y-, x,,,,,~'L~.~_,,o,o)~,%{~ + Z ~.,~,~- ~(~),..,~(q-2),,:}, r=O ~ r=O (o,o,~,,~')~,,, ~^--, ^ e -~' where x,., e F. The composite of this homomorphism with (T,. + U"Vhq(L))/(T,.+~ + U"p+thq(L)) oo~> gr,.Vhq(K ) ~ Bqt,F| ~ is given by 0q_ 1 | 0q_2, where 1Rq+l 0q : EOq| Iq-~/(I~-I c~ ZI.~ t) ~ _,,v, p--1 ((,X= 0 xr', ~'),, i(o) mod Zg~ t) r~' &(~) &(q)/:~ d~, ^ --. ~' ,~!~'% ~(i) ^ "'" ^ ~(q)/ p--! If (( Z0x,, ,= ~')~, i(co) ,nod zq~ 1) is contained in Ker(0q), since 9 is a part of a p-base Ot*{0~} ofE, wehave x,,~=o for o<r<p and for all q~eOq, and we also have i(to) modZ~.~ 1 = o. This proves (5.Ix.s). 130 p-ADIC ETALE COHOMOLOGY I3I Theorem (5. x2), -- Let K be a henselian discrete valuation field with residue field F such that char(K) = o and char(F) = p > o. Then, the cohomological symbol hpq.,K : K]'(K)/p" K~(K) -> Hq(Sp K, Z/p" Z(q)) is bijective for any q and any n. We are reduced to the case n = I and ~p e K by the following general lemma. Lemma (5. x3). -- Let k be a field and p a prime number which is invertible in k. Let E = k(~p) where ~p is a primitive p-th root of i. Fix q ~ o. (i) If the cohomological symbol h~, E is surjective, h~,,K /s surjective for any n. (ii) If h~,~ is injectiw, and h~,-~ ~ is surjective, then h~,,k is injective for any n. The proof is identical with the case q -=-- 2 treated in [I9]. (5. x4) We prove the surjectivity of h~, K in the case F is separably closed and ~p ~ K. Let C(K) be the quotient hq(K)/Shq(K). For the proof that C(K) = o, it is sufficient to show the injectivity of C(K) --+ C(L) for any extension L/K of the type of (5.4)- Indeed, as an inductive limit of successions of such extensions, one obtains a henselian discrete valuation field K with algebraically closed residue field. The cohomological dimension of K is one ([i6], Ch. II, (3.3)), whence hq(K) = o for q _> 2. Hence C(K) = o and this will imply C(K) = o if we prove the injectivity of C(K) ---> C(L). Let G = Gal(L/K) and consider the diagram with exact rows o ~ Shq(K) > hq(K) > C(K) > o o --+ Shq(L) a > hq(L) a > C(L) ~ (5" *4" I ) oor h (K) (note that cor ores = o). By induction on q, we may assume hq-'(K) = Sh~-*(K) and hq-l(L) = Shq-*(L). Then, by (5.4), the sequence hq_,(K) ,,~> hq_,(L) ~ ~o~> hq_,(K) is exact. Hence, by (3-2), the sequence (s. t4.2) Shq-l(K) hq(K) r0. hq(L) is exact. By the diagram (5-I4. i), the injectivity of C(K) -> C(L) follows from the exactness of (5-I4.2) and that of (5-4. I). (5.*5) Now we prove the bijectivity of h~,K assuming ~pcK. Note that we have already kq(K)/U" kq(K) -%- Shq(K)/U *' hq(K). 131 SPENCER BLOCH AND KAZUYA KATO I32 Let K~ denote the maximal unramified extension of K, F, the separable closure of F, and let Gr----Gal(F,/F) ~ Gal(K,,/K). One has ~* =f~| whence also = r, B* ~ B~ | F. ~- In particular by (5-2), grm hq(K~) ~ gr'~hq(K) | F~ for I < m < e', so H~ U ~ hq(K.,)) ~ U ~ hq(K)/U "' hq(K) H'(GF, U ~ hq(K~)) = o r > o. The exact sequences (cf. (5-3)) o , U ~ hq(K~,) > hq(K~) > o > -Fs ~ -Fs o >"4, ,o give H~ hq(K~)) = Shr '' hq(K) =~ kq(K)/U" kq(K) HI(GF, hq(K~,)) = (fl~,/(i -- C) Z~,F) | (n~,-~/(i -- C) Z~,~t,) H'(GF, hq(K~)) = o r> 2. The spectral sequence with Z/pZ coefficients H'(GF, hq(K.,)) ~ hq+'(K) yields exact sequences (s.,s.x) o (nVl/(, - c) r (nV I( - c) hq(K) -~kq(K)lU" kq(K) -+ o. As in (5-Q, let a be the residue class ofp~-*. The congruence (1 -- ~p)p-I __ p mod ~,+I gives a morphism shows that multiplication by the residue class of (i- ~p)P--" f]~/(, -- (3) Z; -+ f~/(I + aC) Z*~. and also So by (5- i), the exact sequence (5.15. I) shows that U*' kq(K) ~ U r hq(K) kq(K)/W' k~(K) ~ h~(K)IW' hq(K). 6. The sheaf M q Our objective in this section is to prove theorems (I .4) and (i .5) describing the sheaf M, q on Y~t. We first determine the structure of M, q. Let U'~MqC M~ (m_> i) be as in (I. 2) and let U ~ M~ = M/. Without loss of generality, we may assume that Y is connected. Let v be the generic point of Y, and let -~ : ~ --+ Y be the canonical map. Note that the structure of z* M q is known by the preceeding section, for the stalk M~q,~ is isomorphic to the Galois cohomology group of the quotient field of the strictly henselian discrete valuation ring 0,5. 132 p-ADIC ETALE COHOMOLOGY t33 Proposition (6. x). ~ Let the notation be as above. Then, (i) M~ -+ v. -~" M~ is infective. (ii) For any m > o, the inverse image of .:.. v'(U '~ MI) in M~ coincides with U" M~. (iii) The graded sheaves gr"~(M~) are described as in (i .4. I). structure of M~ using an injectivity In the first version of this paper, we proved this theorem of O. Gabber. For y ~ Y, let ~,~ be the strict henselization, and let K be the quotient field of the henselization of 0x, u at the generic point of the special fiber Gabber proved that of Sp Cx.~. -+ H,(Sp ~, z/p. Z(q)) is injective, as a consequence of his general results [7] [8]. In the case n = I, this is nearly (6. i) (i). It is possible to prove (6. I) using this injectivity, but in this paper we adopt another simpler method found later. Proof of (6. I). -- We first prove the injectivity of M~ -~ -r. -~* M~. Let T be its kernel. Since the problem is etale local, we may assume that X == P~ and k is separably closed. Furthermore, by induction on n, we may assume n > I and that the stalk of T at any non-closed point of P~ is zero. We may assume also that ~p e K, by a trace argument. Let G = Aut(P]) be the projective general linear group, Z[G] the group ring, and let I C Z[G] be the augmentation ideal. The ring Z[G] H (Pk, M~). Since T is a skyscraper sheaf, acts on the cohomology groups t , (o) 4=T :~I N .I'(P~',M~)4= (o), any N>_ I. On the other hand, by induction on q we may assume that (6. I) holds for Mqx -t for any t > i. In particular, M~-~ will have a filtration stable under G whose graded pieces are direct sums of sheaves like These are absolute differentials and not relative to k, but K)~ has a filtration stable q--i under G whose graded pieces are isomorphic to ~dk | ~)k , and there are exact sequences C-t o § ~q ___+ ~[d~q-1 _ > d~_ o -+ ~o~_ > ~q I-c-' -- > ~q/d~q -1 >'0. Thus we will have (since I kills H*(P~, fl~"/k)) N * ?t I H (Pk, Mi-') --- (o), t > i, N >~ o. This implies I ~. E~ q . (o) in the spectral sequence E~.' = H'(P~', Mi) :~ H"+t(P~, Z/pZ). 133 SPENCER BLOCH AND KAZUYA KATO But H~(PI~, Z/pZ) maps surjectively on E~ q and Hq(ed) | Ht(e ) | K) so IS.Hq(P;~) = (o) for N __> I. This contradiction implies T = (o). Let V"M[CMI(m> o) be the inverse image of v.v*U'~M~, and let fll q ~ VYtl grv(Ml) M~/V"+IM]. By the injectivity of M~-+-r,v'M~, we have using (5.I) (ii) V mM~=o for m> e'-- ep with e=e K. Furthermore, we have -- p-- I fl~Z, lo~ | ~;, log m = o f~}-I o<m<e' ra q p :f m i C*~ grv(Ml) Bq| -1 o<m<e' p Im/ 9 q * q--1 v. v OY, log| v. v f2Y, 10g gr~'(1V[]) ~ v. v* gr" M] = ~. v" f~-t v.v*B~| q-l, and we must show that the inclusion (.) is an equality for o .<_ m < e'. Indeed, for t rn , q o< re<e, the sheaves on the left map onto gr UM 1as in w Since f~.-+'r.v'l)], is injective, we see that (.) is an injection. For m = o, the inclusion q --1 ~]Y, log | ~'~, log ~ gr~'(M~) follows from the fact ([io], Th. (02.4.2)) that f~,lo.~ is generated etale locally by logarithmic forms. Now, let T be the cokernel of the map (,). We may assume again that T is a skyscraper sheaf. We proceed just as above. By downward induction on m, V"+1 M~ = U"+IM~ and it has the structure described in (6. i). Hence I'~.H'(P~, V "+1M]) ---- (o) for some N ~ o. From this, we have again (o) 4: T :~ I "~. F(P~, M~) 4: (o) any N > I. Now the same argument as above proves that T = (o). Next we study M~ for n> I. Corollary (6. I. x). -- For n > i, M~ is generated locally by symbols. This is reduced to the case n = i by induction on n using the exact sequence Mq._l-4 Mq.-+ M[, and the case n---- I follows from (6. I). 134 p-ADIC ETALE COHOMOLOGY I35 For o<m<e' and n,q>o, letG, q be the sheaf on Y defined to be the sheafi- fication of U ~"Gr~ for R = d~(U) defined in (4.7). Proposition (6.2). -- ~G q --5- grm Mq, for o < m < e'. Since the problem is to prove the injectivity, we are reduced to the case ~p ~ K. The exact sequence on V o , Z/p"-~Z(q) "v'; Z/p"Z(q) , Z/pZ(q) ~ o and the isomorphism Mq~ -t ~_ i* Rq-Xj.(Z/pZ(q)) induced by Z/pZ -~- Z/pZ(x); i ~ ~p yield an exact sequence > Mq,-a M, q, > M] > o. Here the surjectivity of Mq, ~ M~ follows from the fact that M~ is generated locally by symbols. Lemma (6.3). -- The boundary map ~ satisfies ~({x,, ..., x~_~}) = {~, x~, ..., x,_~}. In particular, ~(M] -j) C U'" Mq,_t where e .... . (.Note that OrdK(~ p -- i) = e".) p--I The proof is straightforward and left to the reader. Lemma (6.4). -- Let i < m < e' and let mo be the smallest integer such that m < pm o. Then the sequence U'* M.q_ ~ 22~ U" M. q > U" M[ ,o is exact. Proof. -- If t ~ e' is an integer, then as sheaves in the etale topology i +nti*OxC(I +,'zt-'i'Ox) p whence U t M, q s U t-' Mq t. Taking ~G, q = (o) if m is not an integer, the lemma will follow from the exactness of the sequences (6.4. x ) '~VG, q- 1 ''p'', "G, q > "G] > o, I ~< m < e', and from "G q ~ gr" M~ q (6. I) (iii). We show the exactness of (6.4. I). If p Jf m, then "~G.~ ~ ta1-1 =~ ~GI. Also mG~,-~"G q for n'> n so it suffices to consider the case n~> s, p[m. In this case, the map "p " is induced by the inverse Cartier operator G -1 and one finds ,..._1~ = (~i-1/zN ') o (~&'/zv ~) ~ ~G~,. mG.~/(,,p ,,~/p~ ~~ 135 t36 SPENCER BLOCH AND KAZUYA KATO Lemma (6. S). -- For I ~ m < e', the sequences o ) g r"/~' M, ~- 1 ''p'~ gr" M~. > gr~M~ 9 o are exact. (By convention gff = o /f x r Z.) this follows from If ptm, "G. ~ ~ "G~ gr" 1V[, q > gr"* M[. by (6.3) and (6.4) , we have exact sequences If Plm, M~_ 1 n> U,./p+XMq_ 1 "p" U,.+IM. q U"*+I M[ > o M~- 1 ~ " U~/P M. q- 1 -~ U~ M. q > U" M~ 9 o which prove the lemma. Now, the proof of (6.2) follows using (6.5) , (6.4.1) and induction on n. (6.6) Fix a prime element n of K. We prove that there is an isomorphism q--I (6.6.x) gr ~ M. q ~ W. ~{,1o~ @ W. f~Y,l~g such that {f~ .... ,fq}~ (dloga~ ... dlogj~, o) {f~, ...,L-i, n}~ (o, dlogJ~ ... dloga~_l) for any local sections fx,-..,fq of i*(O~). Here a~ is the image of f~ in O~ and dlog:O~--->W,~ 1 is the homomorphism of [IO], (3.23 x). We prove also the Y, log existence of the canonical homomorphism (6.6.2) M. q + f2~/s/P" ~/s stated in (x.5). By (5. x~), Mq,,; is isomorphic to the rood p"-Milnor K-group of the quotient field of Og:,~. This proves the existence of the vertical arrows having the desired properties in the following diagrams: 136 p-ADIC ETALE COHOMOLOGY x37 M~ > ~, v* M, ~ q--I w. ~, log ~ W. ~, ~og ~ r (W. fl~, ~og 9 w. M~ > v, ~* M~ The existence of the homomorphisms (6.6. x) and (6.6.2) follows from the injectivity of the lower horizontal arrows and from the fact that M, q is generated locally by symbols. The bijectivity of the homomorphism (6.6. i) is reduced to the case n = I by the diagram 4,pt, M~_JU ~ M~_~ ~ M~qlU ' M~ ~ M[IU' M] ~ o / 1 \ o 9 W._t~,,log~W. 1~ ~-1 ' q -1 Lastly, we give a description of a sheaf I_~ on Y,r closely related to M,% Let : Yet -+ Y,, be the canonical morphism of sites. Let L~ be the Zariski sheaf asso- ciated to the presheaf U ~, H~t(U, i* Rj,(Z[p" Z(q))), so, L~ = R q s, i* Rj,(Z/p" Z(q)), where the notations i* and Rj, are used in the sense of etale topology as before. Then, the etale sheafification of L~ is M~,. By Gabber [9], the stalk L~., of L~ at Y ~ Y 111-I\\ ~ S L~JI I A = (r i* Ox) v is the "henselization along Y" of ~x,~. Thus the study of L~ is a natural generalization of the study of the Galois cohomology of henselian discrete valuation fields in w 5. Define the filtration of L, q in the same way as in the case of M,% As in (5. I5), by using the spectral sequence E~ t = R 8 r M~ :~ L[+t assuming ~p e K, we can deduce from (1.4. I) a structure theorem of L]. The structures of gW(L~) with n ~ 2 and o < m < e' are obtained by the methods of (6.2) and (6.3). ~y,~-]og) 138 SPENCER BLOCH AND KAZUYA KATO Theorem (6.7)- -- (i) gr~ T W. f2{, ~g | W. t2{,1-o~ and gr*(L~) ~ =G. q for o < m < e'. Here W. fI;:,xog and "*Gq denote the restrictions to the Zariski site of their etale versions. (ii) Assume that e' is an integer and let e'=pSr, s > o, p { r. Take a prime element ~ of K and let a e k be the residue class of pzt-'. Then, for n < s, gr"(L, q) /s isomorphic to (oV /(I + aC) z.) | + aC) Z.) where the quotients are taken in the sense of Zariski topology. For n > s, gr*'(L~) /s isomorphic to the cokernel of f~{-2 _+ (n{-'/(i + aC) Bs) | (f~{-z/(i + aC) B,) o ~ ((x + aC) C-'(do~), (-- i) q r(x + aC) C-'(o~)). Remark {6.8).- Contrary to the case o <__m< e', gr"(L, q) and gr"(M, q) are not determined by only n, q and e. Their structures depend in a subtle way upon the nature of K. The structures of gr"(L~) and gr"(M, q) with m > e' seem to be closely related to the number of roots of x of p-primary orders contained in K. 7- OrdlnaryVa~e~es Throughout this paragraph, Y will denote a complete, smooth variety over a perfect field k of characteristic p > o. For simplicity we write Z q = Z~ = Ker(d : f2~ -+ O~-1), B ~ = B] = Im(d : f~],- ~ -+ f2~). Sheaves and cohomology will be taken with respect to the etale topology unless otherwise indicatcd. The results of this section ovcrlap with results of L. Illusie [2I, IV (4. I2), (4. x3)]. Fix an integer r ~ o, Lemma (7. 9 ). -- Assume the field k above is algebraically closed. and assume Hq(Y, B') ---- (o) for all q. Then: (i) The natural maps k | Hq( Y, f2~,,o,) ~ Hq(V, fY) are injective for all q. (ii) The homomorphisms in (i) are bijective for all q if and only if H~(B '+1) = (o) for all q. 138 p-ADIG ETALE COHOMOLOGY ~9 (iii) If the equivalent conditions in (ii) are satisfied, the map W.(k) | H'(W. fl},~o~) ~ H'(W. fl}) is an isomorphism for all q and n. (iv) Assume (,) (multiplicity . of slope r in Ho,.,s~ q+" (Y/W)) = dim~ H~(Y, f~) for all q. Then the groups Hq(Y, Wf~:.lo~) are torsion free for all q, and the equivalent conditions (ii) hold. In the absence of torsion in H'(W~..log), condition (*) is equivalent to (ii). Proof. -- For a later application, we prove that Hq(B ") = (o) implies Hq(f~,1og) | k ~ Hq(f~,) fixing q and r. Consider the sequence r r I--C-~ o > ~Y, lo~ ----+ f~Y -----" f~IB ...... -> o. By Hq(B ') = o, the homomorphism Hq-1(f~,) -->Hq-I(~,/B ") induced by the pro- jection ~} -+ f2}[B T is surjective, and hence I -- C-t: Hq-1(fl~) -) Hq-t(~2~,/B'). This shows that Hq(~,,Io~) ~ Hq(f~}). Let i : Hq(~]~) -+ Hq(D~,/W) be the injection induced by the projection f~} -+ f~}/W, and take a k-linear map t : Hq(~/B T) --> Hq(f~) such that t o i is the identity map. Then, the p-linear map t o C-I acts on Hq(~}). Since Hq(fl~,log) ~ Ker(1 -- t o C-1), we have Hq(~-~,log) | u.~ Hq(f/~) by p-linear algebra. Now fix r and assume Hq(B') ----- o for all q. Then, the above homomorphism t is bijective and o q r Ker(I --t C -1) =Ker(l --C -t) =H(f~y, log). By p-linear algebra, H~(a},log) | k ~ Hq(f~) if and only if t o C- 1 is bijective. This proves (ii). To prove (iii), the exact sequence ([Io], I (3-9. I)) r r r r--I r--1 o -,,- f~z[B," --> Ker(W. ny -+ W._ 1 f~}) ~ f~y [Z,, ---> o together with the isomorphisms (op. cit. (o.2.2)) Z'-I/Z'-I~B' /B' B''B" [B' n / n-;-I = n+l n, n ~ n+l 1 implies (under the hypotheses of (ii)) H (Ker(W. The result now follows by induction on n, using the exact sequence o ~ f~y, lo~ -~ W, f~,, lo~ ~ W, + t g~Y, ~o~ ~ o and the five lemma. To prove (iv), note the string of inequalities (multiplicity of slope r in H~'(Y/W)) = rank0~ Hq(Wg~[og)| Q,~ rankrv(H~(Wf~[og)/p) < rankv~ Hq(fl~., ~g) < rank~ H~(f~}). 139 SPENCER BLOCH AND KAZUYA KATO 14o Equality of the extremes forces (ii) to hold and the torsion subgroups of Hq(Wf~[o,) and Hq+t(W~lr~ to vanish. In the absence of torsion, (ii) is equivalent to this equality. Definition (7.a). -- Let Y be a smooth, proper variety over a perfect field k of characteristic p > o. We say Y is ordinary if H~(Y, B') = (o) for all m and r. Let k be the algebraic closure of k, Y = Y~. Proposition (7.3).- The foUowing conditions (I)-(5) are equivalent. (I) Y is ordinary. (2) Hq(Y__ f2~/~,,o,) | k -% Hq( Y- f~Yy,) for any q, r. (3) Hq( Y, W, ~,,og) | W,(h) ~ Hq(Y_ W, n~) for any q, r, n. (4) H'(Y__ Wf~Y, log) | W(k) ~ Hq(Y__ Wf~;) for any q, r. (5) F : Hq(Y, Wf~) -+ Hq(Y, Wf~:) is bijective for any q and r. Moreover, Y is ordinary and H~m(Y/W) is torsion free for all q if and only if the following condition holds: (6) For any q, the Newton polygon defined by the slopes of the action of frobenius on Hqy~(Y/W) coincides with the Hodge polygon defined by the numbers dim k Hq-~(Y, i ~Y/k). IfH'(Y, Wn~:) is torsion free for any q and r, these conditions (I)-(6) are also equivalent to (7) For any q, the slopes of frobenius on H~y~(Y/W) are all integers. Proof. -- The implications (x) -,~ (2) ~:- (3) =:" (4) =:" (5) are clear from (7. I). To prove (5) =~ (2) we may assume that the ground field is algebraically closed. Bijectivity of F implies vanishing of cohomology for the pro-sheaves Wfr/F, and hence the exact sequences o ~ W~'/F v Wa'/p -~ W~'/V ~ o o ~ Wa'- t/F ~ W~)~/V -+ a" -+ o ([io], I (3. I5), (3. I9)) yield (7.3. x) H'(WaTp) _-- H'(n~). In particular, I-I*(WCl'/p) are finite dimensional over k. Hence the exact sequences o ,a[o~ ,wa'/pl-~wa'/p ,o (op. cit., I (3.5), (5.7.2)) induce o > H,(f~[og ) > Hq(Wf~,/p) t-~ Hq(WfY/p) > o (exact) for all q and r. By the bijectivity of F, p-linear algebra gives k | H'(arog) =- ~,(Wa'/p). By (7.3.I), this proves (2). Assume now that Y is ordinary, k = k, and H~s,(Y/W ) is torsion free. Consider 140 p-ADIC ETALE COHOMOLOGY I41 the complex of pro-sheaves W~og | W(k) with o differentials. The map on hyper- cohomology (7-3. a) H*(Y, Wn;og) | W(k) -+ H*(Y, WfF) ---- HOT(Y/W ) gives rise by (4) to an isomorphism on the E x terms of the corresponding spectral sequences, and hence is itself an isomorphism. Thus Hq(Wf2[og) is torsion free for all q and r. Condition (6) now follows from (7. i) (iv). Assume that (6) holds. We apply (7. I) (iv) inductively starting with r = o to deduce that Y is ordinary and the Hq(Wf2[og) are torsion free. Using (7.3.2), we see that H:~(Y/W) is torsion free also. Finally, ifHq(Y, W~ ~) is torsion free for all q and r then the slope spectral sequence degenerates at E x [IO] and the slopes s of Hery q+' s (Y/W) with r < s < r + I are given by the slopes of p' F on Hq(Y, WY2'). Conditions (5) and (7) are then seen to be equivalent. The proof is complete. Example (7.4). -- Let Y be an abelian variety over k. Then, Y is ordinary in the sense (7.2) if and only if it is ordinary in the classical sense (i.e. pY(k) ~ (Z/pZ)dim~x)). Indeed, for an abelian variety Y, there are isomorphisms H~ Y, f~r.,og) =~ pPic(?), HI(Y --, Z/pZ) ~ Hom(pY(k), Z/pZ). The orders of these groups are pat~cY) if and only if the equivalent conditions k|176 ?, ~"~,,og) -'~ H~ ?, ~-~1), k | H'(Y, ZlpZ) = Ha(Y, r are satisfied. Assume these conditions are satisfied. Then, Hq(Y, n~r ) =~ A H'(Y, d~) | A H~ Y, f~) shows that By induction on r using (7. I) (i) (ii), it follows that Y is ordinary in our sense. 8. A vanishing theorem Let the notation be as in (o. I). In particular, X is smooth proper over S = Sp A, V = X~ is the generic fiber and Y = X, is the closed fiber. By base change, we obtain diagrams V >X< Y V| , j'> X| , <i' y| , SpK' ~ 9 S' < Spk' Sp K 9 g 9 Spk 141 zr SPENCER BLOCH AND KAZUYA KATO where K' is a finite extension of K, S' = Sp A' is the integral closure of S = Sp A in K' and k' is the residue field of K'. Let Mq,.K , = i" R~j:(Z[p" Z(q)), ~/I~ = ? RqL(Z/p" Z(q)) ----- lim M q n,K', K' UM q =limU 1M qK,CMq.. K' By (6.6), we get an exact sequence (8.0..) o ~ u~ ~ ~ -+ w, a~,~o~ ~ o, for the symbols {xl, ..., x,l_t, 7r} die in the limit M~. Since Mq, is generated locally by symbols (I .4), the long exact sequence ... ~ ~-~(i) , ~,,, M~, -+ ~I~, -+ ~+~(- 1) ~ ... breaks up. So in the diagram o 0 o l 1 1 ~- UM, ~, -----> UMq,+I > UM~ -- > o (8.0.2) o -- > Mq, > Mq,+l ~ M~ ---> o W. t~., log > W. + 1 f~,,log ~ f/~,lo~ -- " o O O O the bottom two rows are exact as are all three columns. It follows that the top row is also exact. The aim of this section is to prove Theorem (8. z ). I Fix q and r, and assume Hq(Y, B~,) = (o). Then, Hq(Y, UM~,) = (o) for any n. Corollary (8.2). -- If Y is ordinary, we have Hq(Y, UM~,) = (o) for any q, r, n. The proof of (8. i) is rather long and complicated. There is a shorter proof of (8.2), and we give it first. Proof of (8.2). -- We may assume that the residue field k of A is algebraically closed. By (8.o.2), it suffices to show that Hq(Y, UM[) = (o). For this it is enough 142 p-ADIC ETALE COHOMOLOGY to show that the map Hq(Y, U 1 M[) ~ Hq(Y, UM[) is zero. We proceed by induction on r. When r=o, M~ (Z/pZ)y and U 1M ~ (o). Assume the result for all t < r. In particular, after ramifying, we may assume that the maps H'(Y, Mi) -+ H*(Y, n~.,og) are onto for all t < r. For any integer m ~ i and any u ~ W(k) consider the diagram H'(Y, U ~ +~ M~) > H'(Y, U" ME) 9 H" Y, ~{-I t B~,- ' e B~ H'(Y, ME-') H'(Y, ,-1 " f~Y, log) where the left hand vertical arrow is 0~ ~ (1 + un").o~ and the right one is induced by the natural inclusion fl"-~ s ~-t followed by multiplication by the residue class Y, log ofu in k. Note that U mME = (o) for m >> o. Proceeding by downward induction on m and using the hypothesis that Y is ordinary, we see that any class in H'(Y, U 1 M[) can be written as a finite product with u~EW(k) and ~EH'(ME-1). Since (I +uir , we concludethatthe map H'(Y, U 1 ME) ~ H'(Y, UM~) is zero as claimed. Now we give the proof of (8.i). In (8.3)-(8.6) below, we do not assume H~(B ') = (o). However, we shall always assume k = k and ~ ~ K without loss of generality. Definition (8.3). -- Fix r _> o. For an element b of K*, let Ug' be the kernel " ~>} M't ~t. TTmlTTm+I of U'M 1 Let gr~'= ~b/~b Lemma (8.4). ~ Assume that OrdK(b ) is prime to p. > i, U'~ is generated locally by local sections of the forms (i) For m { x, -- b } with x ~ U" M'I- 1, (8.4.,) { I --fP k s c p, gx, "", g,-1} (8.4.2) where f, ga, ..., g,-t E i" 0~, c e K', ands is an integer such that p { s and OrdK(b'c ~) _> m. (ii) For o < m < e', we have an exact sequence o > gr~' > gr~(M~) P;"> B~ > o where p~ is the homomorphism induced by ( I . 4. I). (Irrespective of whetl~r or not p [ m, (t. 4. I ) gives a surjection gr'(M[) -~ ~'-I[Z E- t =~ Br.) 14.7 SPENCER BLOCH AND KAZUYA KATO x44 Proof. -- Let o < m < e'. It is clear that the local sections (8.4. i) and (8.4.2) are annihilated by u ( b }. Furthermore, the class of these local sections fill out Ker(p,~). So we obtain diagrams 4- ordx(b).inc] ordK(b) fl gr"(M~)/gr~" ~'(~ gr"(M~ +1) gr~(M~)]gr~ , u(b>) gr~(M~+l) (P t m) (P I m) which prove gr~ = Ker(p~,). Definition (8.5). -- Let L be a finite extension of K. For o< m< eLp eLp p -- x (e L =OrdL(p)), let n = m, and define p--x U,~ M' U" ~ r" 1,L = M1,L, gr,,(M~,L) = g (Mxd,)- We have thus an increasing filtration (o) =U 0M" CU1M" C 1,L 1,L " " * Since e L varies with L, it is not true that reSL/K(U m M~,K) C U~ M~, L. However, we have the following result. Lemma (8.6). -- Let b be an element of K* such that OrdK(b ) /s prime to p. Let o< m< e'-- eKp and let U b,.=U~, grb,,,=gr~ where n=e'--m. Consider p--i' the following two cases. Case I. ~ Let L=K(a) where (-- a) p=-b, and t = e'. Case 2. ~ Assume o<orda(b )<e'. Let L=K(a) where (i--a) p= I --b and let t = e' -- ordr(b ). Then, in both cases, we have the following (i) (ii) (iii). (i) For m <_. t, res~/K(Ub,,~ ) C U m M ~ 1,L ~ (ii) If m< t and p [ m, M ~ resLm(Ub,m) C Urn_ t t,L" #-ADIC ETALE COHOMOLOGY (iii) If m < t and p ~ m, we have the following commutative diagram: Z~_ l o > tl}_ ~ res grb,~ gr.(MI,L) where C is the Cartier operator, and 9., and +.` are isomorphisms defined below. (iv) In Case 2, we have the following commutative diagram: 0--1 Z~ -i ___~ tl} -1 r~s grba , grj(M~,L) where % and ~ are isomorphisms defined below. In (iii), the definitions of 9,. and +. are as follows. Fix an integer s and an element c of K* such that ordK(b' c p) = e'--m. Then, 9., is the homomorphism induced by --> gr,.(M~,x) dyl dy,_, - b" c~ x--A ... ^ ,{i ~,Y,, ...,Y,-1}, Yl Y,- a and +" is the homomorphism r_l -~ ~y -+ gr.~(M~.r,) X--1 A ... A --t~(I -JFpaSc'~yl, ...,~-1}. Yl Y,- I The homomorphisms 9t and +t in (iv) are defined in the same way for the particular choices m--t, s= x and c== I. Proof. -- Let o<m<e'== ei:___PP b,c~K', and a~L" be as above. Note p-- I' that L is a totally ramified extension of K of degree p and ordL(a ) -- ordK(b) both in Case x and Case 2. In Case I (resp. Case 2), (8.6) will be the consequence of the following (8.6. i) 19 146 SPENCER BLOCH AND KAZUYA KATO and (8.6.5) (resp. (8.6.3) , (8.6.4) , (8.6.7) , (8.6.8)) and of the fact that Z ~-1 is generated by B *-1 and elements of the form x p-^...^- dyx dy,_l Yl Yr- 1 Let f e {* ~x and let m L hc the maximal ideal of L. By the binomial theorem, we have with ordL(Pa ~c) = e'p -- m OrdL(a *p d) = e' p -- rap. In Case i, this shows that (8.6.I) (1 --bSd'fP}={I +pa'cf}modU"V-m~iM], L in M[L. In Case 2, the equation (x --a) v-~ I --b shows that a T a mLp- 1)t (8.6.2) b - i --p~mod +t with ordL (P ~) = (P -- I) t. Hence we have (I -- asr :~ ---- I --pa'cf-- I --p b'dff - I --pa"c.f-- b*c~f p + spabS-lcPfPmodm~'P-m+li" ~x with ordL(b' c p) = e'p -- rap ordL(pab s-t c p) = e' p -- m + (p -- I) (t -- m). This shows that, in M[L , (8.6.3) {I b~cVf p} - {I +paScf}modU "'v-"+l M 1 -- 1, L for o<m< t, and (put m----t, s---- I and c= i) (8.6.,1) {~ -- bff} - {i +pa(f--ff)}modOeP-'+lMI.L . On the other hand, in Case I, (8.6.5) res({M[K, -- b}) = o in M[L. In Case 2, by (8.6.2) { ~ (8.6.6) {b} --- {-- b} -- i +p~ modUl~-l/t+tMl, L. Hence we have in this case, for o < m < t, (8.6.7) res({ U~ Nit,K, -- b}) C {U ~''-m' M[L , U I'-11' M1,L} C U "'p-mp+(p-I/t M~, L by (4. x) C U e'~-m+l M 2 1,L ~ 146 p-AI)IC ETALE COHOMOLOGY t47 Also (8.6.6) shows in Case (8.6.8) {I--bf,--b}--[I--bf, I +p~} -{I +paf, p ~}modU"'-'+tM~,L by the proof of (4. I). Let b be an element of K* such that Lemma (8.7). -- Assume Hq(Y, B') = (o). OrdK(b ) /s prime to p, and let o < m < e', p { m. Then, in the commutative diagram H'(Y, U~') - . H'(Y, U" M~) (s.,) (~) He(y, Z ,-1) J, Hq(Y, fl~-t) the homomorphisms i and j are surjective and Ker(i) -+ Ker(j) is also surjective. Proof. -- By (8.4) (ii), U" M~/U~' has a filtration whose successive quotients are all isomorphic to B'. tience He(Y, U" M~/U~") = (o) and Hq-~(Y, U" M~/U~') § H~-t(Y, B'). The lemma follows from the diagram HvI(U "M~/U~) , Hq(U~) -~ H~(U'M[) , Hq(U "M[/U~) = (o) H,-a(B ') , Hq(Z '-t) , H*(~2~ -t) , H*(B')-~ (o) (8.8) Now we are ready to prove Theorem (8. i). We prove the following fact by induction on m > o. (8.8. 9 ) For any finite extension K' of K such that m < ew p , the map p--i H'(Y, U,. M~,K,) -+ H'(Y, UM~) i$ zero, Let First assume p]m and m> I. Wereplace K' by K. xeH*(Y,U,,M~.x) I <<. m < eKP p-- x' plm. 147 z4 8 SPENCER BLOCH AND KAZUYA KATO Let b be an element of K* such that ordK(b ) is prime to p, and let L = K((-- b)l/v). By (8.7), x comes from Hq(Y, Ub.,~). By (8.6), we have reSL/K(Ub.m) CUm_IM[, L. Thus, the image of x in Hq(Y, UM[) is contained in the image of Hq(Y, U,~_t M ~ t,L), and this completes the induction in this case. Next, we consider the case p J( m. Fix a k-linear section s of the surjection Hq(Y, Z "-t) --+ Hq(Y, ~y ,-r ). Then, Cos acts on Hq(Y, ~-i). We say that an element co of Hq(Y, ~-1) is of order <__ i, if there are elements ~z, ..., ~i of k such that (c o s - o ... o (c o s - = o. (Cos--J3 means co~Clos(~) --[~c0.) Then, any element of Hq(Y,a~c -t) is of finite order. For b e K* such that ordK(b) = e' -- m, the isomorphism Pb : aF t ~ gr~(M[); dyl dy,_, x-- ^ ... A x - Zb,y , Yt Y~- 1 induces a homomorphism p; : Hq(Y, U m M~) ~ Hq(Y, n~r-t). It is easily seen that for an element x of ]-Iq(Y, U,, M[), the order of p~(x) is independent of the choice of b. We call this independent order of p~(x) the order of x. We prove the following assertion by induction on i. (8.8.2) For any K' such that m< eK, p and for any xeHq(Y, UmM~,K ,) of p--I order <__ i, the image of x in Hq(Y, UM~) is zero. We replace K' by K and let x be an element of Hq(Y, U,, M],K) of order i > x. By easy pdinear algebra, we can find an element b of K* such that ordK(b) = e' -- m and such that one of C o S(p'b(X)) and (G o s -- x) (p~(x)) is of order < i -- I. By (8.7), there is an element y of Hq(Y, Ub,,, ) whose image in Hq(Y, U. MD is x and whose image in H~(Y, Z '-t) under by Ob Ub,,~ --* grb, ~ ~ Z,-Z is s(p'b(X)). In the case C o s(p~(x)) (resp. (C o s -- I) (p;(x))) is of order < i -- I, let L = K((--b) a/v) (resp. L = K((I -- b)a/P)). The image ofy in Hq(Y, Um Mr.L)" is of order <__i-- i by (8.6), and it has the same image asxin Ha(Y, UM~). This completes the induction on i and hence proves (8. I). 9" p-adic cohomology Keep the notations of w 8. We consider the spectral sequences E~"' = Hs(Y, M'.) (-- t) =~ H:+t(V, Z/p" Z) 148 p-ADIC ETALE COHOMOLOGY ~49 and the associated filtration. Let H~m(Y/W(k))(') = Ker(F -- p' : H~s~ -~ Ho~,). Then, we Theorem (9.I). -- Fix q> o and assume H~-i(Y, B~) ----o for all i. have canonical isomorphisms of Gal(K[K)-modules (9. x- 9 gr ~-' H~,(V, O,) = H~m(?/W(k))~ ) (-- i) (9. I. ~) gr ~-~ H~,t(V, Q,) | W(k) ~ Hq-i(Y, W~'), (-- i) for all i, Proof. -- By (8. ~), the exact sequences o UM'. W. o give injections ~,~) is finite by the proof of (7.x) (i) and hence Furthermore, Hq-~(Y, H~-I(Y, W. ~',10,) are finite. Since passing to the inverse limit preserves the exactness for systems of finite groups, gr~-'Hq(V, Z,) ~ ~j_~g~-'H~V, Z/.O" Z) is canonically isomorphic to a subquotient of Hq-'( ~, Wn fl'2.10,)" Thus we have dimQ, O..p) < dimQp H~-'(Y, W~.~)Q ~ dimw<aQ Hq-r --, W~2~.) q = dimw(;I Q H~m(?/W(k)), = dim~H*(V, Qp). Hence all the inequalities are in fact equalities. Thi~ proves our theorem. Corollary (9- *- ')- -- If k is p~rfect and H*-'(Y, B~) = o for all i, Hq.,(V, O,v) admit~ a Hodge-Tate decomposition (of. w o). This follows from (9. ~) and the result of Tate (o. 9). We prove some integral statements assuming Y ordinary. Theorem (9.2). ~ Assume Y ordinary. Let -Sn ~- Spec (ALP" A) and X. = X � -Sn. Then, we have, for all r and n, (9.~,.x) Hq(Y, M~) ~ H'(Y, W,, ~Z, lo,) (9. ~. 2) Hq(Y, M~) | W.(k) ~ H'(Y--, W, fl~) (9.a.3) H*(Y, M~) |162 A/p"A = H~(X,, flx~,). Proof. -- The first two isomorphisms are clear from (8. ~) and (7-3). To give of (9.2.3), we use the map M~ 149 SPENCER BLOCH AND KAZUYA KATO t5o n--- x we consider the diagram (A. = A.[p"A) For H'(~4~) | ~ H'(Y. q ('). .% r q H (nv,,o,) | k .... ,- H'(Y, n') By passing to the limit over discrete valuation rings contained in .~, one sees that (,) n ~ I. We now consider the diagram on is an isomorphism, proving (9-2.3) for cohomology associated to o 9 M,q| I l Xn Xn + i Xl Q.E.D. and apply the 5-1emma. Corollary (9.3). -- Assume V ordinary and let D = limA. be the ring of integers in C v. Then, (limHq(V, M~)) | D ~ Hq(XD, ~,n~) (9.3.x) ,| If k is perfect, we have for all q, for all q and r. H.*t(V , Q.,) | = @ Hv-'(V. ' ' ~V/K) | C~(-- i). (9.3.2) Then the spectral sequences Corollary (9.'t). -- Assume that Y is ordinary. E~ ,~ = lim tP(Y, Mr,)(-- t) ~ H~+t(V, Zp) p--I the spectral sequences degenerate modulo torsion at E~. If dim(V)< (eK, P -- O' E~,' = H'(Y__ UriC)(-- t) ~ Hlt+'(V, Zip" Z) degenerate at E 2 for all n. Indeed, the last assertion in (9.4) follows from the facts that I-Iomc,.t(gtK./,(Z/p(i) (K), Zip(j)(K)) = (o) p-- I if o< li--j[< (eK, p- i) and that M. q=o if q>dim(V). 150 p-ADIC ETALE COHOMOLOGY x5 t Lemma (9.5). -- Assume k perfect and Y ordinary, and fix q > o. Then, the following four conditions are equivalent: (i) H~m(Y/W(k)) is torsion free; (ii) Hq-~(Y, Wfl~:) are torsion free for all i; (iii) H~a(X/S ) = Hq(X, ~x/s) /s torsion free; (iv) H~-i(X, ~x/s) are torsion free for all i. Proof. -- The equivalence of (i) and (ii) are proved in the proof of (7.3) by showing that H~q,(Y/W(k)) ~ ? Hq-'(Y ' = , wn0. Similarly, consider the complex of sheaves on Y M.| D with zero differentials. The map of complexes which induces an M~, | D ~ ~x./s., isomorphism of Et-terms, shows that H~a(X./S.) ~= 0 H'-'(X., ag./g.).' Hence, (iii) and (iv) are equivalent. The equivalence of (ii) and (iv) follows from (9.2). The following result is now deduced from (9.2). Theorem (9.6). -- Assume that Y is ordinary and that H~m(Y/W(k)) and H~+m 1 (Y/W(k)) are torsion free. Then, for all i, we have (9.6.x) grq-'H~t(V , Zv) =~ H~m(Y/W(L,))I') (- i), (9.6.~,) gr'-' H[t(V , Zp) | W(k) : H[w,(V , Wf~') (-- i), (9.6.3) gr q- ~ H~ot (V, Zp) | r~ D = H q-'(xv, ~XD/D) ( -- i). REFERENCES [I] BA~, H., and TAm, J., The M.ilnor ring of a global field, in Algebraic K-theory, II, Springer L~tva'e Notes in Math., 34.2 (t972), 349"442. [2] BERTHELO% P., Cohomo/og~ cristalline des acMraas de charaatristiqu# p > o, Springer Lecture Notes in Math., 4.07 (x974). [3] BLOeH, S., Algebraic K-theory and cry3talline cohomology, Publ. Math. LH.E.S., 47 (x974), 187-268. [4] BLoex, S., Torsion algebraic cycles, Kt, and Brauer groul~ of function fields, in Groupr de Brauer, springer Lecture Notes /n Math., 844 (z98i), 75-IO2. [5] BLOCH, S., p-adic ~tale cohomology, in Aritbraaic and Geometry, vol. I, 18-27, Birkl~u~er (1983). [6] DBuo~m, P. (with ILLtrSm, L.), Cr~taux ordlnaire~ et coordorm~en eanoniques, in S~facea al#brlquea, Springer Lecture Notes in Math., 868 (t98t), 8o-x37. [7] GABBER, O., An injectivify property for aale cohomology, preprint, 1981. [8] G~Bm% O., Gersten's conjeaure for some complexes of vanishing eyries, preprlnt, 1 98x. [9] GAnBEa, O., Affne analog of proper base change theorem, preprlnt, 1981. [io] ILLUSm, L., Gomplexe de De Rham-Witt et cohomologie cristalline, Ann. Sd. Ec. Norm. Sup., 4 9 ~r., 12 (1979), 5ox-66i. 151 I52 SPENCER BLOCH AND KAZUYA KATO Eli] KATO, K., A generalization of local class field theory by using K-groups, Ii, J. Fa,. sci. Univ. Tokyo, 27 (i98o), 6o3-683 . [12] KATO, K., Galoia eohomology of complete discrete valuation fields, in Algebraic K.theory, Springer Lecture Notes in Math., 967 (I982), 2x5-238. [t3] KATz, N., Serre-Tate local moduli, in Surfaces alg~briques, Springer Lecture Notes in Math., 868 (t98x), x38-2o2. [x4] M~P.xtrajEv, A. S., and SvsuN, A. A., K-cohomology of Severi-Brauer varieties and norm residue homo- morphism, Math. U.&&R. Izvestiya, 21 (x983), 3o7-34 o. [I5] MXLNOR, J., Algebraic K-theory and quadratic forms, Inv. Math., 9 (I97O), 3x8-344. Ix6] SEP.a~, J.-P., Cohomologie Galoisienne, Springer Lecture Notes in Math., 5 (1965). It7] SOUL~, C., K-th6orie des anneaux d'entiers de corps de hombres et cohomologie 6tale, Inv. Math., 55 (1979), 251-295 . [18] TATE, J., p-divisible groups, in Proceedings of a Conference on localfields, Springer-Verlag, I967, I58-183 . [19] TATS, J., Relation between K 2 Galois cohomology, Inv. Math., 36 (I976), 257-274. [2o] DELIG~S, P., La conjecture de Weil, II, Publ. Math. LH.E.S., 52 (t98o), 3x3-428. [2x] ILLUSI'~, L., and RAYSAtm, M., Les suites spectrales associ~es au complexe de De Rham-Witt, Publ. Math. LH.E.&, 57 (I983), 73-212. [22] SERm$, J.-P., Sur les corps locaux ~ corps r~siduel alg~briquement clos, Ball. Soc. Math. France, g (i 96 x), t o 5-154. Department of Mathematics, University of Chicago, Chicago, Illinois 6o637 , Etats-Unis. Department of Mathematics, University of Tokyo, Hongo I x3, Tokyo, Japan. Manuscrit re~u le x 4 janvier x98 4. Added in proof. Since this paper was written, there has been considerable progress. Faltings has shown that the cohomology of any smooth proper variety over K has a Hodge-Tate decomposition, and work of Fontaine and Messing has thrown much light on the structure of these representations in the non-ordinary case.
Publications mathématiques de l'IHÉS – Springer Journals
Published: Feb 6, 2008
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