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A rank theorem for analytic maps between power series spaces

A rank theorem for analytic maps between power series spaces A RANK THEOREM FOR ANALYTIC MAPS BETWEEN POWER SERIES SPACES by HERWIG HAUSER and GERD MI~ILLER Introduction ................................................................................ 95 1. Constant rank and flatness ................................................................ 97 2. The Rank Theorem in Banach scales ...................................................... 100 3. The Rank Theorem in power series spaces ................................................. 101 4. Application: Stabilizer groups ............................................................. 103 5. Scissions of 0n-linear maps ................................................................. 106 6. Banach scales ............................................................................ 110 7. Analytic maps between power series spaces ................................................. 113 References ................................................................................. 115 Introduction Purpose of this paper is to give a criterion for linearizing analytic mappings f: C{ x }" ~ C{ x }~ between spaces of convergent power series in n variables. Such mappings often occur as defining equations of specific subsets of power series spaces relevant to singularity theory and local analytic geometry. One wants to show that these subsets actually are submanifolds. To this end the defining map has to be linearized locally by analytic automorphisms of source and target. For maps between finite-dimensional or Banach spaces the appropriate tool is the Rank Theorem: maps of constant rank have smooth fibers. It is established as a direct consequence of the Inverse Mapping Theorem. But this is known to fail for spaces more general than Banach spaces. The same happens in our situation: we can indicate simple examples of analytic maps between power series spaces which are not local analytic isomorphisms at a given point although the tangent map at this point is an isomorphism. Extra assumptions will be necessary. There are, in fact, extensions of the Inverse Mapping Theorem beyond the frame of Banach spaces, see [C, Ham, L, P3]. They involve technical conditions which it seems hard to verify in our situation. Instead, we shall develop an Inverse Mapping Theorem which is adapted to the HERWIG HAUSER AND GERD MOLLER very concrete context we are working in. It allows, using a detailed version of the Division Theorem for modules of power series, to reach our main objective, the Rank Theorem. Let E = (~v, F ~- ~a and I'~ ={a 9 ord 0a>/ c} for c 9 where C, denotes the ring of convergent power series in n variables with complex coefficients. Rank Theorem. -- Let g(x,y) 9 0,,+~ q be a vector of convergent power series in two sets of variables x and y satisfying g(x, O) = O. There is a c o c N such that for any c >t Co for which the induced map f:E c ~F:a~g(x,a(x)) has constant rank at 0 there are local analytic isomorphisms u of Ec at 0 and v ofF at 0 linearizing f, i.e. such that vfu -1= Tof.. The number c o is related to the highest order occurring in a standard basis of the image of the tangent map Tof Constant rank is equivalent to saying that the kernels of the tangent maps T~f form a flat family of modules. The linearizing automorphisms will in general not be given by substitution. The above Rank Theorem allows to apply methods of differential calculus in the infinite-dimensional context. For instance, we shall prove that for the infinite-dimensional Lie group G = Aut(C", 0) of local analytic automorphisms of (s ", 0) the stabilizer group Gh of a given power series h under the natural action of G on 0, is a submanifold and thus a Lie subgroup (Theorem 4.1). ~ts Lie algebra consists of the vector fields which annihilate h. Similar results hold for the contact group K and show that hyper- surfaces X C ((l", 0) are determined up to isomorphism by their Lie group of embedded automorphisms (Theorem 4.2). Example. -- For h(x,y) = x.y one has to investigate the equationy.a + x.b + a.b = 0 with unknown series a and b. The corresponding map f: (x,y).C{ x,y } 2 -+C{x,y}: (a, b) ~--~ y.a + x.b + a.b is of constant rank at 0. The tangent maps off are given by r,a.b,f(v , w) -- v.(y + b) + w.(x + a). Their kernels form a flat family. The Rank Theorem applies and f can be linearized at 0 into (a, b) ~y.a + x.b. Let us briefly indicate how the Rank Theorem above is proven. As in all theorems of this type o~e estimates the size of the terms of order 1> 2 in comparison to the tangent map at 0. In the present situation, the spaces are filtered by Banach spaces. The mapf is shown to respect these filtrations and thus induces by restriction Banach analytic maps. Each of those can be linearized locally by the Rank Theorem for Banach spaces. A RANK THEOREM FOR ANALYTIC MAPS BETWEEN POWER SERIES SPACES 97 The point is to show that the linearizing automorphisms of each level glue together to well-defined analytic automorphisms u and v of E, and F at 0. It is here that the restriction off to E, C E is used. Moreover, in the construction of u and v, scissions of O,-linear maps E ~ F are needed and have to be estimated. This relies on the Division Theorem for power series in the version of Grauert and Hironaka with norm estimates. The first three sections present the notion of constant rank and the main results. This is applied in section 4 to show the smoothness of stabiliTer groups. Sections 5 to 7 are auxiliary and collect technical tools needed in the proofs. The results of the present paper were announced in [HM1]. The first named author thanks the members of the Max-Planck-Institut in Bonn for their hospitality during part of the work on this article. 1. Constant r~nk and flatness For our purposes a definition of constant rank weaker than the one given in Bourbaki [B] is convenient. For an analytic f: U ~ F with U C E open we do not assume that there is some closed J such that Im T,f~J = F holds pointwise for all a near ao. We only require that this equality holds analytically in a, namely that any analytic curve in F decomposes uniquely as a sum of curves in Im Tof and J. Let us give this sentence a precise meaning. For a topological vector space F (Hausdorff, locally convex and sequentially complete), consider the tangent bundle TF ----- F � F and subbundles Jx* = F � J where J c F is a closed subspace, together with the tangent bundle map Tf: TE It~ -+ TF : (a, b) ~-* (f(a), T,,f(b)). For the germ of an analytic curve y : (C, 0) ~ E let I'v(TE) denote the vector space of germs of analytic sections of TE over y, i.e. germs of analytic maps (C, 0) ~ TE : t ~-* (y(t), b(t)), with induced map Tf: r~(rE) -+ r,~(TF) : (V, b) ~. (fit, (Tvf) b). Define rtv(Jx, ) in an obvious way as space of sections over fit with values in J. Thenfis said to have constant rank at a 0 ~ U if the image of T~e f has a topological direct complement J Im T,~f~J ----- F such that for all analytic germs y:(C, 0) ~ E with y(0) = a 0 Tf(Pr(TE)) | I~(J~) -- I'sv (TF) lS 98 HERWIG HAUSER AND GERD MOLLER (as an algebraic direct sum). This means that any analytic section o of TF over fy can be written uniquely as a sum of a section z of JF over f~ and a composition (Tf) p where 0 is a section of TE over y o= (Tf) p We sometimes say thatfhas constant rank at a 0 w.r.t.J. On the other hand, f is called flat at a0 if for all analytic germs y : (C, 0) ~ E with y(0) = a 0 the evaluation map Ker(Tf: P,(TE) ~ rtv(TF)) -+ Ker(T,of: E ~ F) is surjective. This means that for any b ~ E with T~0f(b ) = 0 there is an analytic germ t ~ b t in E with b o = b and Tvc,f(b,) = 0 for all t (" firing of relations "). In the special case where the tangent map T~o f off at a o is injective, f is automatically flat at a 0. Proposition 1.1. ~ Let f: U ~ F, U r E open, be analytic. a) If f has constant rank at a o ~ U it #flat at a o. b) Assume that f is flat at ao ~ U and that the image of T~o f admits a topological direct complement J in F. Then for every analytic germ y : (C,, O) -+ E with y(0) = ao one has Tf(r.r(TE)) n = 0. Proof. -- a) Let 7 : (C, 0) -+ E with y(0) = a o be analytic, and fix b E Ker T~o f. The analytic germ t ~ Tv,~f(b ) can be written Tv(.f(b ) = t.'b(t) with an analytic germ ~: (G, 0) ~ F using power series expansion and T~of(b ) = O. Then t~ (f(y(t)), b'(t)) is contained in Ptv(TF). Suppose that f has constant rank at a o w.r.t.J. The existence part of the constant rank condition allows to write = + d, with analytic germs t ~ c~ in E and t ~-, d, in J. Then the analytic germ t w-, b t = b -- t. c t in E satisfies b o = b and Tv,,j;(b,) = t.d~ ~J for all t. The uniqueness part of the constant rank condition implies Tvl,f(b~) = 0 for all t. b) Let b t be an analytic curve in E with c t = Tvc,f(b, ) ~J for all t. In particular, Co = T~f(bo) = 0. By assumption there is an analytic curve ~, in E with ~o = bo and Tv(,,f(~) = 0 for all t. This yields Tv(,,f(b , -- ~,) = c,. Since b 0 -- o b -= 0 one can factor out t on both sides and then prove by induction on the order that c t = 0 for all t. This proves the Proposition. In the cases we shall be concerned with the existence part of the constant rank condition will be automatic (and thus, roughly speaking, constant rank will be equivalent to flatness). A RANK THEOREM FOR ANALYTIC MAPS BETWEEN POWER SERIES SPACES 99 Let I:E--> F be a continuous linear map. A scission of l is a continuous linear map a:F-~E with lal=l. Then id--a/:E-~E and la:F-,F are continuous projectors onto Ker al = Ker l and lm la := Im l inducing decomposi- tions E = Ker l | Im(id -- al) and F =Im l | Ker la. If E and F are Banach scales (cf. sec. 6) l admits a scission ff and only ff Im l and Ker l admit topological direct complements in F, respectively E. If Im l@J = F and Ker l | L = E then l]L : L ---> Im I is bijective with continuous inverse (the Closed Graph Theorem holds in Banach scales). If n : F -~ Im l is the projector with kernel J then a = (ILL)-1 7t is a scission of l. Proposition 1.2. -- Let E and F be Banach scales, U C E be open and f: U --> F be an analytic map which is compatible at a o e U. Assume that l = T~o f admits a compatible scission : F ~ E with corresponding decomposition F=I| I=Iml, J=Kerla. a) For every analytic curve y : (C, 0) ~ E with y(0) = a 0 one has I'tv(TF ) = Tf(Fv(TE)) + I',v(JF). b) If f is flat at a o (e.g. if f has constant rank at a o w.r.t, some other complement J" of I) f has constant rank at a o w.r.t.J. Proof. -- As b) is immediate from a) by Proposition 1.1 b) we have only to prove part a). Let b(t) be an analytic curve in F. By [P2, Proposition 2] the germs y : (C, 0) ~ E and b : (C, 0) ~ F have values in E,, respectively F, for sufficiently small s and are analytic as E~ respectively F,-valued map germs. We are thus reduced to the case where E and F are Banach spaces. Let L(E, F) be the Banach space of continuous linear maps E -+ F. By [U, 1.8] the map U -+ L(E, F) : b v--, T~f is analytic. Moreover, the invertible elements of L(F, F) form an open subset, and the inversion mapping ~-* ~?- ~ is analytic, see [U, 2.7]. For a E U consider ~a = Tof~ln + idy -- la E L(F, F). As ~o ---- id we may assume after shrinking U that all ~o are invertible. We see that a ~-~ q);1 is analytic. Then c(t) = ~la*?~ b(t) and d(t) = (id~ -- l~) q)v~l b(t) define analytic curves in E, respectively J satisfying b(t) = + d(t). This proves the Proposition. In section 5, remark b), we deduce: Corollary. -- Let f: U § ~, be an analytic map given in a neighborhood U s OP, of 0 by substitution in a power series g(x,y). Then f has constant rank at 0 if and only if f is flat at O. Examples. -- a) f: Oa + Ox : a ~-* xa + a ~ has G~teaux-differentials T~f : 01 ~ O x : b ~ (x + 2a) b. I00 HER.WIG HAUSER AND GERD M~LLER Since Tof is injective, f is flat at 0, hence of constant rank. But Im T,f = 0x for a(0) + 0 and Im To f-= (x). Thus for a close to 0 the images of the tangent maps T,f do not have a simultaneous direct complement. b) f: 01 ~ 0~ : a ~ a q- (l/x) (a 2 -- a"(0)) has G~teaux-differential To f= id. Againfhas constant rank at 0 and for a close to 0 the kernels of the tangent maps T,f do not have a simultaneous direct complement, see the example of section 6. 2. The Rank Theorem in Banach scales For the notion of Banach scales and compatible maps see section 6. Rank Theorem 2.1. -- Let E = U, E,, F = U, F, be Banack scales and f: U -+ F be an analytic map in a neighborhood U C E of 0 with f(O) = O. Set l = T0f, I = Im l, f = l + h. Assume that l admits a scission ~ : F -+ E, lol ~ l, with the following properties: (i) f has constant rank at 0 w.r.t, to the topological direct complement J = Ker lo of I. (ii) ah:U -~ E is compatible at 0 and its restrictions satisfy I (~h), 1,<~ c.r for some constants 0 < c < l[e(e + 1) and 0 < r, and all small s > O. Then there are local analytic isomorphisms u of E at 0 and v of F at 0 linearizingf, i.e. such that vfu-' =l. Iff and n are compatible with the filtrations, assumption (i) can be replaced by the condition thatfhas constant rank at 0 (w.r.t. some other complement J'), see Pro- position 1.2. If the image of l admits a topological direct complement J then constant rank is necessary forf to be linearizable near 0. In fact, l has constant rank at 0 w.r.t. J since T, l = l for all a ~ E. Observe that (i) plus (ii) imply in particular Bourbaki's rank condition. Proof. - Set u = id a + oh:U ~E. Then u is analytic and compatible at 0 with u(0)-= 0 and To u = id a . By the norm estimate in (ii) the Inverse Mapping Theorem 6.2 applies: u is a local analytic automorphism of E at 0. Note that l~f =- l~l + l~h = l + l~h = lu, hence l~fu-1 = l near 0. Replacing f by fu-~ we may assume that l~f----- l. The map v = id F -- (ida, -- l~)f~la is a local analytic automor- phism of F at 0 with inverse id F + (id F --l~)f~l~. Assume for the moment that f =f~l near 0. Then vf = f -- (idF -- 1o) f~rlaf = f -- (id, -- &r) fal = f -- (idF -- I~) f =l f= t A RANK THEOREM FOR ANALYTIC MAPS BETWEEN POWER SERIES SPACES 10l near 0 as required. The equality f=firl follows from the constant rank condition: a being a scission of l, ~l is a linear projection onto a topological direct complement of K=Kerl in E. It suffices to show that T~f[K =0 for all a~E close to 0. But laf= l from above implies f= l + (id F -- l~)f, hence T~f= l + (idF -- l~) Tof. Fix a point a in an absolutely convex open neighborhood of 0 in U, and consider the analytic curve ~- : (C, 0) ~ E, u = t.a. For b ~ K = Ker I we get Tv,,f(b ) = (idp -- l,) Tv,,f(b), so that (fv, Tvf(b)) ~ Tf(r~(TE)) n p,,(J,). By assumption (i) Tv,~f(b ) must be 0 for t close to 0. Then T,f(b) = 0 by analytic continuation. This proves T,f]~ = 0 and the Theorem. Corollary. -- Assume that f is compatible at 0 and that l admits a compatible sdssion a : F ~ E such that I, for some constaats 0 < c < 1]e(e + 1) and 0 < r, and all small s > O. a) If I is injective f admits locally at 0 an analyt# left inverse g. In particular, f is locally injective. b) Ill is surjective f admits locally at 0 an analytic right inverse g. In particular, f is open at O. Proof. -- a) Since I ks injectivefks flat at 0 and thus has constant rank at 0. There- fore f can be linearized locally. From the equality &rl = l we obtain an analytic map g with fgf = f. Moreover, injectivity of l implies that f ks locally injective. Thus gf = idr. b) In case l is surjective one has l~r = ida. The beghming of the proof of the Rank Theorem shows that lafu-t = l, hence fu-~ ~ = l~ = id r. 3. The Rank Theorem in power series spaces Fix a weight vector Z = ()'0,)") ~ R1 +" with Z-linearly independent components ),, t> 1. It induces a Banach scale structure on O, = [.J O,(s) where O,(s) are the sub- spaces on which [bl,=Z~[b~ls x'~ for b=Z~b,x~O, is finite. Moreover, d~q, becomes a Banach scale by I b[, = Z,s xo''-t' Ib, l~ for b = (bx, ...,b,) ~O~. For subsets M C 0q, we set M, = M c~ (0q,)0. For a submodule I of Oe, denote by [~(I) the maximal weighted order of a minimal standard basis of I, cf. section 5. For 0t ~N" set I ~] = ~1 + .-. + ~,. Let m, denote the maximal ideal of 0,. 102 HERWIG HAUSER AND GERD MI~I,LER e p q Rank Theorem 3.1. -- Let E = d~, ~, E~=m,.~, and F = 0,. Let g(x,y) ~+p be a vector of convergent power series in two sets of variables x and y with g(x, O) -- O. Let I = (0 v g(x, 0)) C g~, be the submodule generated by the partial derivatives w.r.t, y, and write g(x,y) = ~m,(x)y ~. For any c ~N with [3(m*. I) < ord(m~) + c. I a I for all ] ~ ] >>. 2 and suck that f: E, ~ F : a ~ g(x, a(x)) has constant rank at 0 there are local analytic isomorphisms u and v of Eo and F at 0 linearizing f vfu- ' = ToY Remark. -- There always exists a c o with [~(m[ I) ~< ord(m~) + c. [ ~ [ for c >t Co, see Remark c), section 5. Proof. -- We reduce to the Rank Theorem for Banach scales. Equip E~ = m[. O,P with [[a[I. =s-~Z, la,[, for a= (ax, ...,a,) EEl. The pseudonorm is contractive, [[ a [1, ~< 1[ a l I,, for a e E= and s < s', because X~ >I 1. Set f= l + h with l = Tof and h : E~ ---> F of order >/ 2. Let h, = ~h,, be the power series expansion of h, in continuous homogeneous polynomials and set ] h, [, = Y~ [ h0k [ r k for r > 0, cf. [U]. We prove: a) l : E~ ~ F admits a compatible scission a : F ~ E, satisfying [ a, [ ~< c x.s- ~ for some constant c I > 0 and all small s ~> 0 where ~ = ~(m~ I). b) For any constant c2 > 0 there is an r > 0 such that, for all small s > 0, l h, [,< c~.s~.r. c) There are constants 0 < c3< l[e(e + 1) and r> 0 such that, for all small s> O, I (ah), 1,~ < c3.r. Let us show a): Let N be the number of monic monomials of degree c in the r 19 variables xx,..., x,. Let p:O~ 'v ~E, = m,.O, be the canonical surjective O,-linear map. Provide 0,~'~ with the Banach scale structure given by lal,=Y~,la, l,, fora=(a,...,a~.v) eO, ~'v. Then p is compatible and satisfies [ P~ [ ~< 1 for all small s > 0, since ~. i> 1 and thus Ix ~ 1, ~< s ~ for monic monomials x ~ of degree c. By Theorem 5.2 the O,-linear map lp : 0~'~ -+ F = ~. admits a compatible scission x : d~, -+ 0~, "~ with I "~, I < q.s-~ A RANK THEOREM FOR ANALYTIC MAPS BETWEEN POWER SERIES SPACES 103 for some constant c t > 0 and all small s > 0. Then ~r := Oz : F ~ E, is a compatible scission of l w~th i o, i~ q.s-~, proving a). To see b), note first that one has, by definition of [h0 ],, Ih, l,=Ek~>z sup [El,t= ~m.a sl~ k IlaI[,~<l ~< ~k~>z sup El,l= klm~l,[a s[,.r~< Zl,l~> zlm,],.s'l'l.r I~1. Now set d~(s) ---Im= I~ -~. It follows from X,~> 1 and the assumption on ~ that d,(s) decreases as s goes to 0. Moreover Y~l,l>~z d,(so).r I'1 ~< sff ~.Zl,l~> 2 [ m, [,o.r j=l converges for small s o and r since g(x,y) converges. Therefore for any c2 > 0 there is an r > 0 such that [ h, 1, ~ s ~. Zl=l~z d~(so).rill ~< sf~.c~.r for all s <~ s o. This is b), and c) follows from a) and b). Thus 2.1 applies and the Theorem is proven. Examples. -- a) Consider g(x,y) = xy -k y 9" where I, = (x) '+t. As f: (~)" ~ ~ : a ~ xa + a ~ has constant rank at 0 it can be linearized locally near 0 for c/> I. b) The map f: Oz -+ 0~ : a F--~ xya + (x ~ + y~) a ~ has constant rank at 0. Further- more, I = (xy), and ~(I) > 2. Hence Theorem 3. I cannot be applied to linearize f on 0 z. But one can use the Rank Theorem for Banach scales directly. Provide d~z with the Banach scale structure associated to Z = (Zl, Z2) = (1, 1). These weights are not permitted in 3.1. The natural projection 02 ~ I and division by xy induce a compatible scission o. One calculates I%l~< s-2 and [h,[,~< 2s er z for the quadratic part h(a) = (x ~ -t-.y ~) a ~. Thus the norm estimate (ii) of Theorem 2.1 is satisfied and f: 9 z ~ d~ z can be linearized locally at 0. 4. Application: stabilizer groups We apply the preceding results to prove the smoothness of stabilizer groups. Let G = Aut(C", 0) be the group of germs of analytic automorphisms of (C", 0) and p be a natural number. The contact group K is the semidirect product K = Aut(C ", 0) ~< GL~(0,) 104 HERWIG HAUSER AND GERD MI~LLER defined through the right action of G on GLp(0,). It is an open subset of k = m, ~," | 0. ~2. By Proposition 6.1, one can prove that K and G are infinite-dimensional Lie groups in the sense of [Mi, sec. 5]. The tangent space k of K has a natural Lie algebra structure [v, w] = ad(v) w where k ~k : v ~ ad(v) w is the tangent map of K -+k :g ~ Ad(g) w and Ad(g) :k ~ k is the tangent map of conjugation with g. Identifying elements of O~ with derivations of 0, the Lie algebra of G is g = ( D 9 Der 0,, D(m,) C m, } with the usual bracket [D, E] = DE -- ED. The bracket on k equals [(D, v), (E, w)] ---- ([D, E], [v, w] -k Dw -- Ev). The contact group acts analytically on the space 0~, of rows from the right by h(% a) = (h o ~).a. Theorem 4.1. -- For h 9 O~ the stabilizer group Ka=((%a) 9 is a submanifold of K, hence a Lie subgroup. Its Lie algebra is k n={(D,v) 9 Dh+hv----0}. Proof. -- a) For fixed c E N consider the finite-dimensional algebra V c ---- O,/m~ and the finite-dimensional algebraic group Ac = Aut V, ~< GLp(V,). There is a natural surjective homomorphism of Lie groups ~ : K -+ A c. Let Kc ----- Ker ~,, H = K n and H, ---- H n K c. We apply the Rank Theorem to show that for c sufficiently large, H, is a submanifold of K~. Writing ~(x) = x + +(x), a(x) = 1 + b(x) we identify K~ as a manifold with the space Ec = m~. 0~, + ~2. Then He equals the zero-fiber of the analytic map f: Eo ~ r b) = h(:, + +).(1 + b) -- h. Its G~teaux-differential is given by T,+,b,f(D, v) = Dh(x + +).(1 + b) + h(x + +).v. If T0f(D , v) = 0 then r(+,b,f(D(x + +), v(x + +).(1 + b)) = 0. Hence f is flat at 0. The Rank Theorem implies that f can be linearized locally. b) Similarly as in [Mti 1, see. 2] it is shown using Artin's Approximation Theorem [A] that the image B, = :r,(H) is an algebraic subgroup of A0. Below we shall see that ~, : K ~ A, admits locally at 1 9 Ao an analytic section , : U ~ K which restricts to a section U n Bc ~ H of the restriction H ~ B,. Then %-x(U) --* U x K, :g ~ (n~(g), (a~,(g))-' g) is an analytic isomorphism with inverse U � K c -+ ~-~(U) : (g,,g) ~(gc) g. A RANK THEOREM FOR ANALYTIC MAPS BETWEEN POWER SERIES SPACES 105 It restricts to a bijective map rc~-'(U) c~ H ~ (U c~ Be) � He. Therefore H is a submanifold of K. r To prove the existence of a let a~ be the Lie algebra of A~ and ~c :k ~ a~ the tangent map of r~. Moreover let h = k h and bc = ~(la). Using again the Approxi- mation Theorem it is checked that b e is the Lie algebra of B~. By solving an appropriate initial value problem one can show that the Lie group K admits an exponential map exp: k ~ K, i.e., an analytic map such that for all (D, v) e k the map C ~ K : t ~ exp t(D, v) is a one-parameter subgroup with initial velocity vector (D, v), see also [P1, sec. 1]. Since the exponential map of the finite-dimensional Lie group A, is a local analytic isomorphism near 0 ~ a e and since the exponential map of K maps la into H, see step d) below, the existence of the section ~ follows from the corresponding infinitesimal assertion: rk e :k ~ a e admits a continuous linear section .r:a~--*k which restricts to a section b e ~ It of the restriction h - ~ b c. For this take the canonical section % : ac -+ k which maps (De, re) c a. onto the unique polynomial of degree ~< c -- 1 in the fiber of (D e, v~). Then adjust "% by defining = (id k-p~)oT 0:a~k where r :k ~ 0~ is given by ~(D, v) -- Dh q- hv and p : 0~ ~k e is a scission of r res- tricted to k c - Ker 9 e. This -r is a continuous linear section of ~,. To show that it maps b e into h, take (D~, v~) ~ bc and let (D, v) ----- "r0(D~, v~) q- (D', ~') be an ele- ment of h mapped onto (D,, v~). Then (D', v') ek~ and e~0(D~, v,) : -- r v'). This implies r v,) = 0 and ~(D,, re) e h. d) It remains to show that the Lie algebra of K h is k h and that exp maps k~ into K~. Clearly { (D, v) ~ k, exp t(D, v) e K h for all t } C 7": K h C k~. Conversely, if (D, v) e kh, calculation shows that (hoqh).a,=0 for (qh,a,) =expt(D,v). Therefore exp t(D, v) ~ K h. This proves the Theorem. Let now X C (C", 0) be a reduced hypersurface defined as zero-fiber X = h-X(0) of some h e 0,. Let to, = x) be the group of embedded automorphisms of X. Thus G x ={~ ~G, ho ~ e 0~.h}. 14 106 HERWIGHAUSER AND GERD M~LLER For q~ ~ G x the factor a E 0, such that h o q~ = h. a is a unit which is uniquely determined by q0. Therefore the projection K -+ G : (9, a) ~-~ 9 induces an isomorphism of groups K h -% G x. Thus we may view G x as a Lie group. If X is isomorphic to another hypersurface Y C (C", 0), say defined by g ~ 0,, then h and g are in the same K-orbit: g = (h o 9). a for some (% a) ~ K. Conjugation with (q~, a) induces an isomorphism K h -% K o of Lie groups. In particular, the Lie group structure on G x is independent of the chosen equation h, and isomorphic hypersurfaces X, Y have isomorphic Lie groups Gx, Gy. The converse is also true: Theorem 4.9.. -- Two reduced hypersurfaces X, YC (C", 0), n/> 3, are isomorphic if and only if the# Lie groups Gx, Gy of embedded automorphisms are isomorphic. Proof. -- Let Dx, 0 = { D ~ g, Dh e h. O. }. The projection k = g | O, ~ g induces a continuous isomorphism of Lie algebras k h --% Dx, 0. Both k h and Dx, 0 are r modules of k, respectively g, hence closed subspaces. By the Closed Graph Theorem [Gr, chapter 4.1.5, Theorem 2] we conclude that k h -% Dx. 0 is even an isomorphism of topological vector spaces. Therefore, if f: G x ~ Gy is an isomorphism of Lie groups, the tangent map off at the unit induces an isomorphism Dx. 0 -% Dy, 0 of topological Lie algebras. By [HM2, part II, Theorem and Comments d), e) in section 1] the topo- logical Lie algebra Dx, o determines X up to analytic isomorphism. Therefore X and Y are isomorphic. 5. Scissions of O.-linear maps In order to construct and control scissions of 0,-linear maps we need a version of the Grauert-Hironaka Division Theorem with special attention paid to norm estimates [Ga], [Hau]. For fixed X = (X0, K) E R a+" + with Z-linearly independent components X~, ele- ments ofN a +" are ordered by ix < j~ ifk(ix) < X(j~), where X(ix) = X0(i -- 1) + 2~ ),~ x k. Define a total order on the set of monic monomial vectors (0, ..., O, x ~, O, ..., O) of dF, by setting (0, ..., 0, x ~, 0, ..., 0) < (0, ..., 0, x ~, 0, ..., 0) if j~ < ix where i and j denote the position of x ~ and x ~. Denote by in(a) and in(I) the initial monomial vector and initial module of elements and submodules of O~,. Set A(I) =(b no monomial of b belongs to in(I)}. For a standard basis ml, ..., m~ of I with initial terms ~tx, ..., ~%, partition the support of in(I) as a disjoint union U~= 1 M~ with M~ C supp(O,, bq)- Then set V(I) = { a ~ O. ~, supp(a~. ~q) C M~ for all i }. Any weight vector X as above induces on 0, a Banach scale structure [hi,= Z=Ib=I s~'= forb=~,b,x'sO,,. A RANK THEOREM FOR ANALYTIC MAPS BETWEEN POWER SERIES SPACES 107 In the sequel when dealing with O,-linear maps 1: 0, ~ ~ d7~ we consider on the domain space O, ~ the Banach scale structure defined by Ibl0 = ~,1 b~[~ and on the target space ~. the one given by I b l, = Z, s~,,-,I b, [~ Division Theorem 5.1. -- Let l : O~, -+ Oq. be an O,-linear map, say l( a) = a. m = Y~, a~ m, . Assume that the raps form a standard basis of I = Im l. Set K = Ker l and let A(I) and V(I) be defined as above. a) I~ A(I) ----- d~,, K@V(I) = dT~, and these decompositions are compatible. b) There is a constant c > 0 such that for all small s the following holds: For any e e (O~,)~ the unique elements a ~ V(I), and b e A(I)~ with e = Y~ a i m~ + b satisfy (min, I m, I,).l a I~ + Ibl~ Z [ a, I, 1 m,t. + Ibl, < c[Za, m, + bl~ Proof. -- We may assume that all m~ are 4: 0. For s > 0 with ml, ..., m~ the continuous linear map u, : V(I),| A(I). -+ (d~,), : (a, b) ~-. a.m + b will be shown to be bijective for small s. To this end supply the Banach space V(I)~ | A(I), with the norm II(a,b)l]~ = Z, I a,l.]~,l, + I b I~ By definition of V(I) the map v, : 7(I), @ A(I), ~ (O~,)~ : (a, b) ~ a. Vt + b is bijective, bicontinuous of norm 1, and its inverse v: ~ has norm 1 as well. Decompose u, into u,=v,+w~ where w~ =a.m' with m; =m~--~q. There are constants c>0 and r such that I~', I, ~< ]m~ l, <c [~., [, and I m, h < s'l~, ], for all i and all sufficiently small s. This yields tw, l<~s* and ] wsv~-I ]~ st<~ 1 for small s, say s ,< s 0. Using the geometric series one sees that u, v~ -1 = id + w, v~ -~ is invertible with I(u, v~-1)-1] ~ < -- =: c'. l-s; 108 HERWIG HAUSER AND GERD MOLLER Consequently u, is invertible and l u~ l l <~ c ' fors~<s 0. This implies part b) of the Theorem. Since trivially I, n A(I), = 0 we obtain moreover I, | A(I), = (0~,), and K, | V(I), = (0~),. This gives a). Example. -- Consider l: 0~-+0 2 given by m i = x +y, m,. = x--y. Assume X i<x*. Then 1.mr-- 1.m,=2y, but there is no constant c>0 such that for all small s I 1 I, I re, l, +1- l l, I re, l, = 2( sx' + sx') is bounded by c12y I, = The reason is that mr, m 2 are not a standard basis. We now use the Division Theorem to estimate the norms of projections onto sub- modules I C 0~ and of scissions of 0,-linear maps l : ~,~ -+ 0~,. Consider the Banach scale structures on 0,~ and 0~, as defined above. Let ~i = (0, ..., 0, x% 0, ..., 0) with entry in the jcth place the unique minimal monomial generator system of in(I). The number ~(I) = max~ X(j i ~,) is called the weighted order of I w.r.t, the weight X. It equals the maximal weighted order of the elements of a minimal standard basis of I. Theorem 5.2. -- Let l : 0, ~ -~ O", be an O,-linear map with image I =Im l and [5 = [5(I). a) The projection ~i : Oq, ~ I induced by the decomposition 0q, = I 9 A(I) is compatible and satisfies for some constant c > 0 and all small s > 0 I (~,), I~ c. b) l admits a compatible continuous linear scission ~ : t~, ~ 0~, satisfying for some constant c > 0 and all small s > 0 Proof. -- By the choice of the norms, a) follows from part b) of the Division Theorem. For b) choose a minimal standard basis mi,..., m~, ~ I with ~ = in m~. Consider the 0,-linear map l' : 0, ~' ~ 0~, : a ~ ]~ a~ m~. The Division Theorem gives a compatible decomposition Ker l' | V'(I) = 0~. '. Then 7' = (l' Iv,) -1 : I -+ O. r is a compatible continuous linear section of l'. Moreover there is a constant c' > 0 such that c'l A RANK THEOREM FOR ANALYTIC MAPS BETWEEN POWER SERIES SPACES 109 for a e V'(I) and all small s. By definition of ~ we have s ~ = min~ [ ~, 1, and thus Choose an 0,-linear map p : 0~, ' -+ 0~, with commuting diagram Observe that p is compatible with I P0 ] ~< c" for some constant c" > 0 and small s. Then a = p-r' ~I is a compatible continuous linear scission of l satisfying a norm estimate as claimed. Remarks. -- a) For an 0,-linear map l : 0, ~ ~ 0a, with image I one trivially has Im l~163 I,. The existence of the compatible scission ~ implies that Im l, = I, for small s. This is false if instead of the Banach scale structure defined by Z-linearly inde- pendent weights X0, ),~, ..., X, as above we would have taken different Banach scale structures. For example define [ a [, = Y~ ] a,, I s'+' for a = Y_, a~, x~y ' 9 C{ x,y } = 0,. Let l:02--->02:a~a.(x-y) with I=Iml= (x--y). For fixed s>0 set I k--1 a = Z ~oX'~ -1-/ tt-o ~ I We then have la.(x_y)[ = ~ 2 ,_o~ <~176 lal.= =~176 Thus a.(x --y) ~ I, but r l,. b) We can now prove the corollary of section 1: Let f:U---0~ be an analytic map in a neighborhood U C 0~ of 0 given by substitution in some g(x,y). Thenf has constant rank at 0 if and only if f is fiat at 0. In this casefis of constant rank at 0 w.r.t. A(I), where I = Im I. Indeed, let a be the scission constructed above. Then Ker/a = Kern I = A(I). Now our claim follows from the results of section 1. c) For any submodule I of 0~ the weighted order ~(m~ I) increases at most linearly with coefficient 1 in c. To see this, consider the inclusions m~.in(I) (2 in(m~, I) C in(I). Let ~ be an element of the minimal monomial generator system of in(m~ I). Then = x'v with u an element of the minimal monomial generator system of in(I) and I10 HERWIG HAUSER AND GERD M1DLLER x" E ~,. If I ~ ] > c, 7 belongs to m~.in(I) but is not a minimal monomial vector of this module w.r.t, the component-wise order on N". The first inclusion implies that it neither is minimal in in(m~I) which is a contradiction. Therefore [a[~ c and I) .< + c. 6. B*o*ch scales For details and the basic notions of this and the next section we refer the reader to [BS, He, Gr, U]. A Banach scale is a topological vector space E, Hausdorff, locally convex, and sequentially complete together with pseudonorms I I,, s c R and s> 0, such that (i) if s < s' then I1,-< I I,', (ii) E, := { a ~ E, I a [, < oo } is a Banach space with norm ] I,, (iii) E = [J, E, as topological vector space with the final topology. We often say that E is a Banach scale. But the reader is reminded that the Banach spaces E, with their fixed norms ] ], are part of the structure. Remarks. -- a) For r eR, r > 0, let B,(0, r) = { a a E,, ] a ], < r} be the open ball of radius r. Then B(0, r) := [3, B,(0, r) is open in E. Indeed, for s < s' and a E E,, C E, we have I a l, ~< I a I,', hence B,.(0, r) C B,(0, r). Therefore B(0, r) n E,. = B,.(0, r) u [.J,<,,(B,(0, r) n E,,) is open in E,. as E,, C E, is continuous for s < s'. b) Let F be a closed linear subspace of a Banach scale E = U s E,. Then F = U, F, with F, = E, n F is a Banach scale. In fact, it is immediately seen that a subset A C F is relatively closed if and only if A n F, is closed in F, for all s. Thus the final topology of F with respect to the inclusions F, C F coincides with the relative topology, which proves (iii). c) Let F be a Banach scale. Every decomposition F = I | J as an algebraic direct sum of closed linear subspaces is in fact topological. To show this it is enough to prove that the projection ~ : F ~ I with kernel J is continuous. By the Closed Graph Theorem [Gr, chapter 4.1.5, Theorem 2] it suffices to show that graph n is closed. But this is obvious from graph ~ ={(a, b) eF| a -- b e J}. A continuous linear map I:E ~ F between Banach scales is called compatible if, for small s, l restricts to a map l0 : E, -+ F,. Then l, is necessarily a continuous linear map. Since the relative topology on F, induced from F is a Hausdorff topology coarser than its Banach space topology this follows from the Closed Graph Theorem, see [Gr, chapter 1.14, Corollary to Theorem 10]. A decomposition F = I | as a direct sum of closed linear subspaces is called compatible if F, = Io | with I0 = F, n I, J, -- F, n J for small s. Then clearly the projections onto I and J are compatible. A RANK THEOREM FOR ANALYTIC MAPS BETWEEN POWER SERIES SPACES 111 An analytic (i.e. continuous and G~teaux-analytic) map f: U ~F, U CE open is called compatible at a e U if, for small s, f restricts to an analytic map f,: U,-+F, where U, is a suitable neighborhood of a contained in U n E,. In that case the chain rule gives that Tar , = 7"ofl, for the G~tteaux-differentials Taf:E~F and Taf,:E,~F ,. Proposition 6.1. -- Let E and F be Banach scales, U C E an open set and f: U ~ F a map. Then f is analytic if and only if, for all analytic curves "~ : T --~ E with T C C open and ~,(T) C U, the composition f o "( : T -+ F # analytic. Proof. -- [He, Theorem 3.2.7 d)]. Inverse Mapping Theorem 6.2. -- Let E = U, E,, F -= [.J, Fo be Banach scales, and let f: U ~ F be an analytic map in a neighborhood U C E of 0 with f(O) = O. Assume: (i) The Gdteaux-differential l -~ To f: E -+ F is an isomorphism of topological vector spaces. (ii) The composition l -~ of is compatible at 0 with restrictions (l -~ of), : U, -~ F,. (iJi) There are constants 0 ( c ( 1/e(e § 1) and r > 0 such that for all small s > 0 ](1-1 of), -- idE, I,~< c.r. Then f is a local analytic isomorphism at O. Proof. -- We proceed exactly as in the usual proof of the Inverse Mapping Theorem in Banach spaces. Upon replacingfby l -~ of we may assume F = E and I = To f= ids. Write f = ids § h and define rccursively g0=0, g" +l _id s_hog.. As ordh >/ 2 it is shown by induction (using Proposition 6.3 below) that ord(g "+a -- g") > n. Let ~ be the unique formal power series satisfying ord(g -- g") ~ n for all n. Then also ord(fo ~ --fog") > n. Since ord(fo g" -- ids) = ord(g" -- g" + ~) > n we conclude thatfo ~ = ida., i.e. ~ is a formal right inverse off We shall show that converges on the open set B(0, r') = [J, B,(0, r') for r' -~ ((l/e) -- c).r. Note that r' > 0 since c < 1/e. Let us first prove by induction that for small s. Indeed, using assumption (iii) we have I g~ ],' = ](idE -- h o g"-~), I,' r' + ]h, ],.10"-'J,, r'§ I, <~ r' § e 112 HERWIG HAUSER AND GERD MI~ILLER But then, since ord(~- g")> n, we also have ILl,,<. -. This implies that ~ induces analytic maps g, : B,(0, r') ~ E, with [g, I,' <` r/e as well as an analytic map g:B(0, r') -+ E. Thus the following holds: fog = idB(0.,,~, g(0) = 0, Tog : ida. ce [g, -- idE, 1,' <` c'. r' with c'-- 1 -- r ~ Using c < I/e(e + 1) we see that c'< 1/e. Hence the whole argument above can be applied to g with r and c replaced by r' and c'. Setting r" = ((l/e) - c').r' we find an analytic map ft." B(0, r")~ E with g off= idB~0.,,, ~. Uniqueness of the inverse implies f=f],,o,,",. Thus f: B(0, r") ~f(B(0, r")) is an analytic isomorphism. Example. -- We indicate what can happen if the norm estimate (iii) in the Inverse Mapping Theorem is violated. View E = 01 = G { x } as a Banach scale as in section 5. Then ~ : E -~ E : a ~ a -- a(0) is a compatible continuous linear map. Define f: E -~ E by - a (0) f(a) =a+ --a=--- x x It is analytic and compatible at O. Its G~tteaux-differential is given by ~(2ab) raf(b ) = b + -- In particular, To f=-id z. Nevertheless f is not a local analytic isomorphism at 0. In fact, if a E E is invertible, a(0)4: 0, we have for b := a(O).(x q-2a) -1 that Taf(b ) :- (l/x) 7~((x + 2a) b) ----- 0. Hence there are points a s E arbitrarily close to 0 such that T,,f is not injective. This phenomenon can be explained by comparing the size of the terms of order/> 2 at 0 with Tof = ida. For this let us calculate If, -- ida, ],. We have f(a)= a + P(a) where P(a)= (1/x).n(a 2) is a continuous homogeneous polynomial of degree 2. For fixed s we have ] 7z(a 2) I, --=- [ a2 -- a2(0) 18 <` ] a2 Is <` [ a I~. If a = x.s -~ then [ a l, = 1 and I n(a2) I, = 1. This impfies I P, [ = s-~ and If. -- ida, l, = I P, l, = re's-" But we cannot find a constant c such that r2.s -~ <~ c.r for all small s. Note that T.fis injective for a ~ (x), say of order /> 1. Moreover, Taf: 0 x ~ 01 is surjective for all a close to 0. Indeed, x + 2a is not in (x) * for a close to 0. tIence given c ~ 9 1 we can solve cx = (x + 2a) b for b. Then Taf(b) = c. A RANK "tHEOREM FOR ANALYTIC MAPS BETWEEN POWER SERIES SPACES 113 Proposition 6.3. -- Let f: U---> F, g:V ~ G be ana~tic with U C E, V C F open and f(U) C V. Then g of: U ~ G is analytic and for a ~ U T,(g of) -~ T1,~,g o T,f. Moreover, if 0 ~ U, f(O) ~ 0 ~ V and g(O) -~ O, the expansion of g of at 0 is given by the composition formula for formal power series applied to the expansions of g and f at O. Proof. -- The first part is standard. For the second, one can assume by the Hahn- Banach-Theorem that E = G = C. Let g ---- 2~ g~ be the expansion of g in homogeneous polynomials and gi be the corresponding/-linear symmetric maps. Let then i be fixed. As ~ is continuous there are a constant c > 0 and a continuous seminorm p on F such that [~(ba .... , b~) I ~< c.p(b~) ... p(b,) for all b~, ..., b, ~ F. Fix a near 0. By [BS, Proposition 4.1] there are constants M > 0 and 0 ~ ]0, 1 [ such that p(f~(a)) <. M.0 ~ for all k. Consequently ~>, ~k~+ ... *~.=k I g~(fk~(a), " .,f~.(a)) I ~< ~ ~ cM ~ 0 ~ 0~( oo. As ~, is /-linear and continuous we obtain gdf(a)) -= ~,(f(a), ...,f(a)) = + ... g,(L,(a), This is the power series expansion of g, of. For sufficiently small r > 0 we have g(f(a)) == X;,g,(f(a)) for I ~ I < r, the series converging uniformly on [a[< r, see [BS, proof of Theorem 6.4]. By Weier- strass' Theorem on (locally) uniform sequences of analytic functions = g,(L,(a), is the power series expansion on [ a[ < r. 7. Analytic maps between power series spaces We now specialize to power series spaces. Equip O, p with pseudonorms defined through weight vectors as in section 5. Then O](s) ---- { a ~ d~, ~, I a 1, < oo } is a Banaeh space and d~ = O, 0,~(s) with the final topology a Banach scale [GR]. Partial differen- tiation and integration O, ~ O, as well as algebra homomorphisms 6, ~ 6,, and On-module homomorphisms are continuous linear maps, hence analytic. The next result characterizes analytic maps with finite-dimensional domain [Mfi2, sec. 6]: 15 114 HERWIG HAUSER AND GERD MI[II.LER Proposition 7.1. -- a) For a map f: T-+ d~,, T CC ~ open, the following conditions are equivalent: (i) f is analytic. (ii) f is continuous and for all jet maps ~ :~W, ~ (O,/m~,+') ' the composition =,of is analytic. (iii) For every t o 9 T there are open neighborhoods T'C T oft0, VC C" of 0 and an ana- lyt# map g : V � T' ~ C ~ such that for t 9 T' :fit) (x) = g(x, t 1. (iv) For every t o 9 T there are an open neighborhood T' C T of t o and an s ~ 0 such that f(T') C d~,(s) and f: T' ~ O~,(s) is analytic. b) Assume that (i) to (iv) hold. Let g(x, t o + t)= Z,,~g~ x ~ t ~ be the power series expansion of g near (0, to). For t 9 C ~ let f~(t) = Yq~r ~ ~g~ x~ t~ 9 0~. 9 Then f(t o + t) = Y~kfk(t) is the power series expansion off near t o. c) For t e T' as in (ill) above, the Gdteaux-differential Ttf : C ~ --> Oq. is given by T,f(ba, .... b~) = 2~, Oq g(x, t).b,. The main examples of analytic maps between power series spaces are given by substitution: Proposition 7.2. -- Let g:V � W-+C" be analytic, where V x W C C ~ x C ~ is an open neighborhood of O. Let U = { a 9 ~,, a(O) 9 W }, and define f: U ~ ~ by substitution in g : f(a) (x) : g(x, a). a) f is analytic. For a c U its Gdteaux-differential To f: 0~ -~- O~ is given by Toy(b,, ..., b,) = ]E, Ovi g(x , a).b,. b) Let g(x,y) = ~,~g~ x~ y ~ be the expansion of g. For a 9 0~, let fk(a) = Z I ,! =~ Y~ g~ x ~ a ~ ~ 0~. Then f(a) = Zkf~(a) # the power series expansion off near O. c) f is compatible at O. Proof. - We may assume q = 1. a) Propositions 6.1 and 7.1 imply thatfis ana- lyric. For ~(:U and b 9 ~, let h(t)=f(a+tb), t 9 close to 0. Then h(t) (x) ---- g(x, a + tb) =: h'(x, t). By Proposition 7.1 c) we have Tof(b) = ~ h(t) I,- o = 0, ~(x, t)l, =o = }-], Ou, g(x, a).b,. b) For small s the fk are continuous homogeneous polynomials O.~(s) ---> O,(s) of degree k. For a e B,(0, r), r sufficiently small, the series ]~kf~(a) converges to f(a) in O,,(s). Hence f restricts to an analytic map B,(0, r) ~ O.(s). This proves b) and c). A RANK THEOREM FOR ANALYTIC MAPS BETWEEN POWER SERIES SPACES 115 REFERENCES ARTIN, M., On the solutions of analytic equations, Invent. Math., ,5 (1968), 277-291. C A ] [BS] BOCHNAX, J., SicI~a~, J., Analytic functions in topological vector spaces, Studia Math., 39 (1971 ), 77-112. BOURBAKI, N., Variltts diff~rentielles et analytiques, Hermann, I%7. [BI COLO~mEAU, J.-F., Differentiation et bornologie, Th6se, Universit6 de Bordeaux, 1973. COl GALLmO, A., Th6or6me de division et stabilit6 en g~omdtrie analytique locale, Ann. Inst. Fourier, 29, 2 [Gal (1979), 107-184. [Gr] GROTHBNDmCK, A., Topological vector spaces, Gordon and Breach, 1973. [GR] GRAU~nT, H., REMMBRT, R., Analytisehe Stellenalgebren, Springer, 1971. t-IAMILTON, R. S., The inverse function theorem of Nash and Moser, Bull. Amer. Math. Sor 7 (1982), 65-222. [Ham] I-L~usER, H., La construction de la ddformation semi-universelle d'un germe de vari~tfi analytique complexe, [Haul Ann. Sci. fie. Norm. Sup. Paris (4), 18 (1985), 1-56. t-IAusER, H., MOLLER, G., Automorphism groups in local analytic geometry, infinite dimensional Rank [HMI] Theorem and Lie groups, C. R. Acad. Soi. Paris, 818 (1991), 751-756. HAUSER, H., MOLLER, G., Affine varieties and Lie algebras of vector fields, Manuscr. Math., 80 (1993), [HM2] 309-337. [He] t-~RV~., M., Analytieity in infinite dimensional spaces, Studies in Math., 10, De Gruyter, 1989. LESLIE, J., On the group of real analytic diffeomorphisms of a compact real analytic manifold, Trans. [El Amer. Math. Sot., 274 (1982), 651-669. [Mi] MXLNOR, J., Remarks on infinite-dimensional Lie groups, in Relativitg, groupes et topologie II, Les Houches, Session XL, 1983 (eds : DE Wrr'r et STox~), p. 1007-1057, Elsevier, 1984. MOLLER, G., Reduktive Automorphismengruppen analytischer C-Algebren, J. Reine Angew. Math., 364 [Mall (1986), 26-34. MOLLER, G., Deformations of reductive group actions, Proe. Camb. Philos. Soc., 108 (1989), 77-88. [Mii2l PISANELta, D., An extension of the exponential of a matrix and a counter example to the inversion theorem [Vll ofa holomorphlc mapping in a space H(K), Rend. Mat. Appl. (6), 9 (1976), 465-475. PISANELH, D., The proof of Frobenius Theorem in a Banach scale, in Functional analysis, holomorphy and [P21 approximation theory (ed. G. I. ZAPATA), p. 379-389, Marcel Dekker, 1983. PIS,~ELH, D., The proof of the Inversion Mapping Theorem in a Banach scale, in Complex analysis, func- [P31 tional analysis and approximation theory (ed. J. MujmA), p. 281-285, North-Holland, 1986. UPMEXER, H., Symmetric Banach manifolds and Jordan C*-algebras, North-Holland, 1985. [Ul H~ H~ Institut ft~r Mathematik Universit/it Innsbruck A-6020 Innsbruck Austria G~ M~ Fachbereich Mathematik Universit~it Mainz D-55099 Mainz Germany Manuscrit refu le 7 juillet 1992. Rgvisg le ler april 1994. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

A rank theorem for analytic maps between power series spaces

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Springer Journals
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Copyright © 1994 by Publications Mathématiques de L’I.É.E.S.
Subject
Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
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0073-8301
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1618-1913
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10.1007/BF02698897
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Abstract

A RANK THEOREM FOR ANALYTIC MAPS BETWEEN POWER SERIES SPACES by HERWIG HAUSER and GERD MI~ILLER Introduction ................................................................................ 95 1. Constant rank and flatness ................................................................ 97 2. The Rank Theorem in Banach scales ...................................................... 100 3. The Rank Theorem in power series spaces ................................................. 101 4. Application: Stabilizer groups ............................................................. 103 5. Scissions of 0n-linear maps ................................................................. 106 6. Banach scales ............................................................................ 110 7. Analytic maps between power series spaces ................................................. 113 References ................................................................................. 115 Introduction Purpose of this paper is to give a criterion for linearizing analytic mappings f: C{ x }" ~ C{ x }~ between spaces of convergent power series in n variables. Such mappings often occur as defining equations of specific subsets of power series spaces relevant to singularity theory and local analytic geometry. One wants to show that these subsets actually are submanifolds. To this end the defining map has to be linearized locally by analytic automorphisms of source and target. For maps between finite-dimensional or Banach spaces the appropriate tool is the Rank Theorem: maps of constant rank have smooth fibers. It is established as a direct consequence of the Inverse Mapping Theorem. But this is known to fail for spaces more general than Banach spaces. The same happens in our situation: we can indicate simple examples of analytic maps between power series spaces which are not local analytic isomorphisms at a given point although the tangent map at this point is an isomorphism. Extra assumptions will be necessary. There are, in fact, extensions of the Inverse Mapping Theorem beyond the frame of Banach spaces, see [C, Ham, L, P3]. They involve technical conditions which it seems hard to verify in our situation. Instead, we shall develop an Inverse Mapping Theorem which is adapted to the HERWIG HAUSER AND GERD MOLLER very concrete context we are working in. It allows, using a detailed version of the Division Theorem for modules of power series, to reach our main objective, the Rank Theorem. Let E = (~v, F ~- ~a and I'~ ={a 9 ord 0a>/ c} for c 9 where C, denotes the ring of convergent power series in n variables with complex coefficients. Rank Theorem. -- Let g(x,y) 9 0,,+~ q be a vector of convergent power series in two sets of variables x and y satisfying g(x, O) = O. There is a c o c N such that for any c >t Co for which the induced map f:E c ~F:a~g(x,a(x)) has constant rank at 0 there are local analytic isomorphisms u of Ec at 0 and v ofF at 0 linearizing f, i.e. such that vfu -1= Tof.. The number c o is related to the highest order occurring in a standard basis of the image of the tangent map Tof Constant rank is equivalent to saying that the kernels of the tangent maps T~f form a flat family of modules. The linearizing automorphisms will in general not be given by substitution. The above Rank Theorem allows to apply methods of differential calculus in the infinite-dimensional context. For instance, we shall prove that for the infinite-dimensional Lie group G = Aut(C", 0) of local analytic automorphisms of (s ", 0) the stabilizer group Gh of a given power series h under the natural action of G on 0, is a submanifold and thus a Lie subgroup (Theorem 4.1). ~ts Lie algebra consists of the vector fields which annihilate h. Similar results hold for the contact group K and show that hyper- surfaces X C ((l", 0) are determined up to isomorphism by their Lie group of embedded automorphisms (Theorem 4.2). Example. -- For h(x,y) = x.y one has to investigate the equationy.a + x.b + a.b = 0 with unknown series a and b. The corresponding map f: (x,y).C{ x,y } 2 -+C{x,y}: (a, b) ~--~ y.a + x.b + a.b is of constant rank at 0. The tangent maps off are given by r,a.b,f(v , w) -- v.(y + b) + w.(x + a). Their kernels form a flat family. The Rank Theorem applies and f can be linearized at 0 into (a, b) ~y.a + x.b. Let us briefly indicate how the Rank Theorem above is proven. As in all theorems of this type o~e estimates the size of the terms of order 1> 2 in comparison to the tangent map at 0. In the present situation, the spaces are filtered by Banach spaces. The mapf is shown to respect these filtrations and thus induces by restriction Banach analytic maps. Each of those can be linearized locally by the Rank Theorem for Banach spaces. A RANK THEOREM FOR ANALYTIC MAPS BETWEEN POWER SERIES SPACES 97 The point is to show that the linearizing automorphisms of each level glue together to well-defined analytic automorphisms u and v of E, and F at 0. It is here that the restriction off to E, C E is used. Moreover, in the construction of u and v, scissions of O,-linear maps E ~ F are needed and have to be estimated. This relies on the Division Theorem for power series in the version of Grauert and Hironaka with norm estimates. The first three sections present the notion of constant rank and the main results. This is applied in section 4 to show the smoothness of stabiliTer groups. Sections 5 to 7 are auxiliary and collect technical tools needed in the proofs. The results of the present paper were announced in [HM1]. The first named author thanks the members of the Max-Planck-Institut in Bonn for their hospitality during part of the work on this article. 1. Constant r~nk and flatness For our purposes a definition of constant rank weaker than the one given in Bourbaki [B] is convenient. For an analytic f: U ~ F with U C E open we do not assume that there is some closed J such that Im T,f~J = F holds pointwise for all a near ao. We only require that this equality holds analytically in a, namely that any analytic curve in F decomposes uniquely as a sum of curves in Im Tof and J. Let us give this sentence a precise meaning. For a topological vector space F (Hausdorff, locally convex and sequentially complete), consider the tangent bundle TF ----- F � F and subbundles Jx* = F � J where J c F is a closed subspace, together with the tangent bundle map Tf: TE It~ -+ TF : (a, b) ~-* (f(a), T,,f(b)). For the germ of an analytic curve y : (C, 0) ~ E let I'v(TE) denote the vector space of germs of analytic sections of TE over y, i.e. germs of analytic maps (C, 0) ~ TE : t ~-* (y(t), b(t)), with induced map Tf: r~(rE) -+ r,~(TF) : (V, b) ~. (fit, (Tvf) b). Define rtv(Jx, ) in an obvious way as space of sections over fit with values in J. Thenfis said to have constant rank at a 0 ~ U if the image of T~e f has a topological direct complement J Im T,~f~J ----- F such that for all analytic germs y:(C, 0) ~ E with y(0) = a 0 Tf(Pr(TE)) | I~(J~) -- I'sv (TF) lS 98 HERWIG HAUSER AND GERD MOLLER (as an algebraic direct sum). This means that any analytic section o of TF over fy can be written uniquely as a sum of a section z of JF over f~ and a composition (Tf) p where 0 is a section of TE over y o= (Tf) p We sometimes say thatfhas constant rank at a 0 w.r.t.J. On the other hand, f is called flat at a0 if for all analytic germs y : (C, 0) ~ E with y(0) = a 0 the evaluation map Ker(Tf: P,(TE) ~ rtv(TF)) -+ Ker(T,of: E ~ F) is surjective. This means that for any b ~ E with T~0f(b ) = 0 there is an analytic germ t ~ b t in E with b o = b and Tvc,f(b,) = 0 for all t (" firing of relations "). In the special case where the tangent map T~o f off at a o is injective, f is automatically flat at a 0. Proposition 1.1. ~ Let f: U ~ F, U r E open, be analytic. a) If f has constant rank at a o ~ U it #flat at a o. b) Assume that f is flat at ao ~ U and that the image of T~o f admits a topological direct complement J in F. Then for every analytic germ y : (C,, O) -+ E with y(0) = ao one has Tf(r.r(TE)) n = 0. Proof. -- a) Let 7 : (C, 0) -+ E with y(0) = a o be analytic, and fix b E Ker T~o f. The analytic germ t ~ Tv,~f(b ) can be written Tv(.f(b ) = t.'b(t) with an analytic germ ~: (G, 0) ~ F using power series expansion and T~of(b ) = O. Then t~ (f(y(t)), b'(t)) is contained in Ptv(TF). Suppose that f has constant rank at a o w.r.t.J. The existence part of the constant rank condition allows to write = + d, with analytic germs t ~ c~ in E and t ~-, d, in J. Then the analytic germ t w-, b t = b -- t. c t in E satisfies b o = b and Tv,,j;(b,) = t.d~ ~J for all t. The uniqueness part of the constant rank condition implies Tvl,f(b~) = 0 for all t. b) Let b t be an analytic curve in E with c t = Tvc,f(b, ) ~J for all t. In particular, Co = T~f(bo) = 0. By assumption there is an analytic curve ~, in E with ~o = bo and Tv(,,f(~) = 0 for all t. This yields Tv(,,f(b , -- ~,) = c,. Since b 0 -- o b -= 0 one can factor out t on both sides and then prove by induction on the order that c t = 0 for all t. This proves the Proposition. In the cases we shall be concerned with the existence part of the constant rank condition will be automatic (and thus, roughly speaking, constant rank will be equivalent to flatness). A RANK THEOREM FOR ANALYTIC MAPS BETWEEN POWER SERIES SPACES 99 Let I:E--> F be a continuous linear map. A scission of l is a continuous linear map a:F-~E with lal=l. Then id--a/:E-~E and la:F-,F are continuous projectors onto Ker al = Ker l and lm la := Im l inducing decomposi- tions E = Ker l | Im(id -- al) and F =Im l | Ker la. If E and F are Banach scales (cf. sec. 6) l admits a scission ff and only ff Im l and Ker l admit topological direct complements in F, respectively E. If Im l@J = F and Ker l | L = E then l]L : L ---> Im I is bijective with continuous inverse (the Closed Graph Theorem holds in Banach scales). If n : F -~ Im l is the projector with kernel J then a = (ILL)-1 7t is a scission of l. Proposition 1.2. -- Let E and F be Banach scales, U C E be open and f: U --> F be an analytic map which is compatible at a o e U. Assume that l = T~o f admits a compatible scission : F ~ E with corresponding decomposition F=I| I=Iml, J=Kerla. a) For every analytic curve y : (C, 0) ~ E with y(0) = a 0 one has I'tv(TF ) = Tf(Fv(TE)) + I',v(JF). b) If f is flat at a o (e.g. if f has constant rank at a o w.r.t, some other complement J" of I) f has constant rank at a o w.r.t.J. Proof. -- As b) is immediate from a) by Proposition 1.1 b) we have only to prove part a). Let b(t) be an analytic curve in F. By [P2, Proposition 2] the germs y : (C, 0) ~ E and b : (C, 0) ~ F have values in E,, respectively F, for sufficiently small s and are analytic as E~ respectively F,-valued map germs. We are thus reduced to the case where E and F are Banach spaces. Let L(E, F) be the Banach space of continuous linear maps E -+ F. By [U, 1.8] the map U -+ L(E, F) : b v--, T~f is analytic. Moreover, the invertible elements of L(F, F) form an open subset, and the inversion mapping ~-* ~?- ~ is analytic, see [U, 2.7]. For a E U consider ~a = Tof~ln + idy -- la E L(F, F). As ~o ---- id we may assume after shrinking U that all ~o are invertible. We see that a ~-~ q);1 is analytic. Then c(t) = ~la*?~ b(t) and d(t) = (id~ -- l~) q)v~l b(t) define analytic curves in E, respectively J satisfying b(t) = + d(t). This proves the Proposition. In section 5, remark b), we deduce: Corollary. -- Let f: U § ~, be an analytic map given in a neighborhood U s OP, of 0 by substitution in a power series g(x,y). Then f has constant rank at 0 if and only if f is flat at O. Examples. -- a) f: Oa + Ox : a ~-* xa + a ~ has G~teaux-differentials T~f : 01 ~ O x : b ~ (x + 2a) b. I00 HER.WIG HAUSER AND GERD M~LLER Since Tof is injective, f is flat at 0, hence of constant rank. But Im T,f = 0x for a(0) + 0 and Im To f-= (x). Thus for a close to 0 the images of the tangent maps T,f do not have a simultaneous direct complement. b) f: 01 ~ 0~ : a ~ a q- (l/x) (a 2 -- a"(0)) has G~teaux-differential To f= id. Againfhas constant rank at 0 and for a close to 0 the kernels of the tangent maps T,f do not have a simultaneous direct complement, see the example of section 6. 2. The Rank Theorem in Banach scales For the notion of Banach scales and compatible maps see section 6. Rank Theorem 2.1. -- Let E = U, E,, F = U, F, be Banack scales and f: U -+ F be an analytic map in a neighborhood U C E of 0 with f(O) = O. Set l = T0f, I = Im l, f = l + h. Assume that l admits a scission ~ : F -+ E, lol ~ l, with the following properties: (i) f has constant rank at 0 w.r.t, to the topological direct complement J = Ker lo of I. (ii) ah:U -~ E is compatible at 0 and its restrictions satisfy I (~h), 1,<~ c.r for some constants 0 < c < l[e(e + 1) and 0 < r, and all small s > O. Then there are local analytic isomorphisms u of E at 0 and v of F at 0 linearizingf, i.e. such that vfu-' =l. Iff and n are compatible with the filtrations, assumption (i) can be replaced by the condition thatfhas constant rank at 0 (w.r.t. some other complement J'), see Pro- position 1.2. If the image of l admits a topological direct complement J then constant rank is necessary forf to be linearizable near 0. In fact, l has constant rank at 0 w.r.t. J since T, l = l for all a ~ E. Observe that (i) plus (ii) imply in particular Bourbaki's rank condition. Proof. - Set u = id a + oh:U ~E. Then u is analytic and compatible at 0 with u(0)-= 0 and To u = id a . By the norm estimate in (ii) the Inverse Mapping Theorem 6.2 applies: u is a local analytic automorphism of E at 0. Note that l~f =- l~l + l~h = l + l~h = lu, hence l~fu-1 = l near 0. Replacing f by fu-~ we may assume that l~f----- l. The map v = id F -- (ida, -- l~)f~la is a local analytic automor- phism of F at 0 with inverse id F + (id F --l~)f~l~. Assume for the moment that f =f~l near 0. Then vf = f -- (idF -- 1o) f~rlaf = f -- (id, -- &r) fal = f -- (idF -- I~) f =l f= t A RANK THEOREM FOR ANALYTIC MAPS BETWEEN POWER SERIES SPACES 10l near 0 as required. The equality f=firl follows from the constant rank condition: a being a scission of l, ~l is a linear projection onto a topological direct complement of K=Kerl in E. It suffices to show that T~f[K =0 for all a~E close to 0. But laf= l from above implies f= l + (id F -- l~)f, hence T~f= l + (idF -- l~) Tof. Fix a point a in an absolutely convex open neighborhood of 0 in U, and consider the analytic curve ~- : (C, 0) ~ E, u = t.a. For b ~ K = Ker I we get Tv,,f(b ) = (idp -- l,) Tv,,f(b), so that (fv, Tvf(b)) ~ Tf(r~(TE)) n p,,(J,). By assumption (i) Tv,~f(b ) must be 0 for t close to 0. Then T,f(b) = 0 by analytic continuation. This proves T,f]~ = 0 and the Theorem. Corollary. -- Assume that f is compatible at 0 and that l admits a compatible sdssion a : F ~ E such that I, for some constaats 0 < c < 1]e(e + 1) and 0 < r, and all small s > O. a) If I is injective f admits locally at 0 an analyt# left inverse g. In particular, f is locally injective. b) Ill is surjective f admits locally at 0 an analytic right inverse g. In particular, f is open at O. Proof. -- a) Since I ks injectivefks flat at 0 and thus has constant rank at 0. There- fore f can be linearized locally. From the equality &rl = l we obtain an analytic map g with fgf = f. Moreover, injectivity of l implies that f ks locally injective. Thus gf = idr. b) In case l is surjective one has l~r = ida. The beghming of the proof of the Rank Theorem shows that lafu-t = l, hence fu-~ ~ = l~ = id r. 3. The Rank Theorem in power series spaces Fix a weight vector Z = ()'0,)") ~ R1 +" with Z-linearly independent components ),, t> 1. It induces a Banach scale structure on O, = [.J O,(s) where O,(s) are the sub- spaces on which [bl,=Z~[b~ls x'~ for b=Z~b,x~O, is finite. Moreover, d~q, becomes a Banach scale by I b[, = Z,s xo''-t' Ib, l~ for b = (bx, ...,b,) ~O~. For subsets M C 0q, we set M, = M c~ (0q,)0. For a submodule I of Oe, denote by [~(I) the maximal weighted order of a minimal standard basis of I, cf. section 5. For 0t ~N" set I ~] = ~1 + .-. + ~,. Let m, denote the maximal ideal of 0,. 102 HERWIG HAUSER AND GERD MI~I,LER e p q Rank Theorem 3.1. -- Let E = d~, ~, E~=m,.~, and F = 0,. Let g(x,y) ~+p be a vector of convergent power series in two sets of variables x and y with g(x, O) -- O. Let I = (0 v g(x, 0)) C g~, be the submodule generated by the partial derivatives w.r.t, y, and write g(x,y) = ~m,(x)y ~. For any c ~N with [3(m*. I) < ord(m~) + c. I a I for all ] ~ ] >>. 2 and suck that f: E, ~ F : a ~ g(x, a(x)) has constant rank at 0 there are local analytic isomorphisms u and v of Eo and F at 0 linearizing f vfu- ' = ToY Remark. -- There always exists a c o with [~(m[ I) ~< ord(m~) + c. [ ~ [ for c >t Co, see Remark c), section 5. Proof. -- We reduce to the Rank Theorem for Banach scales. Equip E~ = m[. O,P with [[a[I. =s-~Z, la,[, for a= (ax, ...,a,) EEl. The pseudonorm is contractive, [[ a [1, ~< 1[ a l I,, for a e E= and s < s', because X~ >I 1. Set f= l + h with l = Tof and h : E~ ---> F of order >/ 2. Let h, = ~h,, be the power series expansion of h, in continuous homogeneous polynomials and set ] h, [, = Y~ [ h0k [ r k for r > 0, cf. [U]. We prove: a) l : E~ ~ F admits a compatible scission a : F ~ E, satisfying [ a, [ ~< c x.s- ~ for some constant c I > 0 and all small s ~> 0 where ~ = ~(m~ I). b) For any constant c2 > 0 there is an r > 0 such that, for all small s > 0, l h, [,< c~.s~.r. c) There are constants 0 < c3< l[e(e + 1) and r> 0 such that, for all small s> O, I (ah), 1,~ < c3.r. Let us show a): Let N be the number of monic monomials of degree c in the r 19 variables xx,..., x,. Let p:O~ 'v ~E, = m,.O, be the canonical surjective O,-linear map. Provide 0,~'~ with the Banach scale structure given by lal,=Y~,la, l,, fora=(a,...,a~.v) eO, ~'v. Then p is compatible and satisfies [ P~ [ ~< 1 for all small s > 0, since ~. i> 1 and thus Ix ~ 1, ~< s ~ for monic monomials x ~ of degree c. By Theorem 5.2 the O,-linear map lp : 0~'~ -+ F = ~. admits a compatible scission x : d~, -+ 0~, "~ with I "~, I < q.s-~ A RANK THEOREM FOR ANALYTIC MAPS BETWEEN POWER SERIES SPACES 103 for some constant c t > 0 and all small s > 0. Then ~r := Oz : F ~ E, is a compatible scission of l w~th i o, i~ q.s-~, proving a). To see b), note first that one has, by definition of [h0 ],, Ih, l,=Ek~>z sup [El,t= ~m.a sl~ k IlaI[,~<l ~< ~k~>z sup El,l= klm~l,[a s[,.r~< Zl,l~> zlm,],.s'l'l.r I~1. Now set d~(s) ---Im= I~ -~. It follows from X,~> 1 and the assumption on ~ that d,(s) decreases as s goes to 0. Moreover Y~l,l>~z d,(so).r I'1 ~< sff ~.Zl,l~> 2 [ m, [,o.r j=l converges for small s o and r since g(x,y) converges. Therefore for any c2 > 0 there is an r > 0 such that [ h, 1, ~ s ~. Zl=l~z d~(so).rill ~< sf~.c~.r for all s <~ s o. This is b), and c) follows from a) and b). Thus 2.1 applies and the Theorem is proven. Examples. -- a) Consider g(x,y) = xy -k y 9" where I, = (x) '+t. As f: (~)" ~ ~ : a ~ xa + a ~ has constant rank at 0 it can be linearized locally near 0 for c/> I. b) The map f: Oz -+ 0~ : a F--~ xya + (x ~ + y~) a ~ has constant rank at 0. Further- more, I = (xy), and ~(I) > 2. Hence Theorem 3. I cannot be applied to linearize f on 0 z. But one can use the Rank Theorem for Banach scales directly. Provide d~z with the Banach scale structure associated to Z = (Zl, Z2) = (1, 1). These weights are not permitted in 3.1. The natural projection 02 ~ I and division by xy induce a compatible scission o. One calculates I%l~< s-2 and [h,[,~< 2s er z for the quadratic part h(a) = (x ~ -t-.y ~) a ~. Thus the norm estimate (ii) of Theorem 2.1 is satisfied and f: 9 z ~ d~ z can be linearized locally at 0. 4. Application: stabilizer groups We apply the preceding results to prove the smoothness of stabilizer groups. Let G = Aut(C", 0) be the group of germs of analytic automorphisms of (C", 0) and p be a natural number. The contact group K is the semidirect product K = Aut(C ", 0) ~< GL~(0,) 104 HERWIG HAUSER AND GERD MI~LLER defined through the right action of G on GLp(0,). It is an open subset of k = m, ~," | 0. ~2. By Proposition 6.1, one can prove that K and G are infinite-dimensional Lie groups in the sense of [Mi, sec. 5]. The tangent space k of K has a natural Lie algebra structure [v, w] = ad(v) w where k ~k : v ~ ad(v) w is the tangent map of K -+k :g ~ Ad(g) w and Ad(g) :k ~ k is the tangent map of conjugation with g. Identifying elements of O~ with derivations of 0, the Lie algebra of G is g = ( D 9 Der 0,, D(m,) C m, } with the usual bracket [D, E] = DE -- ED. The bracket on k equals [(D, v), (E, w)] ---- ([D, E], [v, w] -k Dw -- Ev). The contact group acts analytically on the space 0~, of rows from the right by h(% a) = (h o ~).a. Theorem 4.1. -- For h 9 O~ the stabilizer group Ka=((%a) 9 is a submanifold of K, hence a Lie subgroup. Its Lie algebra is k n={(D,v) 9 Dh+hv----0}. Proof. -- a) For fixed c E N consider the finite-dimensional algebra V c ---- O,/m~ and the finite-dimensional algebraic group Ac = Aut V, ~< GLp(V,). There is a natural surjective homomorphism of Lie groups ~ : K -+ A c. Let Kc ----- Ker ~,, H = K n and H, ---- H n K c. We apply the Rank Theorem to show that for c sufficiently large, H, is a submanifold of K~. Writing ~(x) = x + +(x), a(x) = 1 + b(x) we identify K~ as a manifold with the space Ec = m~. 0~, + ~2. Then He equals the zero-fiber of the analytic map f: Eo ~ r b) = h(:, + +).(1 + b) -- h. Its G~teaux-differential is given by T,+,b,f(D, v) = Dh(x + +).(1 + b) + h(x + +).v. If T0f(D , v) = 0 then r(+,b,f(D(x + +), v(x + +).(1 + b)) = 0. Hence f is flat at 0. The Rank Theorem implies that f can be linearized locally. b) Similarly as in [Mti 1, see. 2] it is shown using Artin's Approximation Theorem [A] that the image B, = :r,(H) is an algebraic subgroup of A0. Below we shall see that ~, : K ~ A, admits locally at 1 9 Ao an analytic section , : U ~ K which restricts to a section U n Bc ~ H of the restriction H ~ B,. Then %-x(U) --* U x K, :g ~ (n~(g), (a~,(g))-' g) is an analytic isomorphism with inverse U � K c -+ ~-~(U) : (g,,g) ~(gc) g. A RANK THEOREM FOR ANALYTIC MAPS BETWEEN POWER SERIES SPACES 105 It restricts to a bijective map rc~-'(U) c~ H ~ (U c~ Be) � He. Therefore H is a submanifold of K. r To prove the existence of a let a~ be the Lie algebra of A~ and ~c :k ~ a~ the tangent map of r~. Moreover let h = k h and bc = ~(la). Using again the Approxi- mation Theorem it is checked that b e is the Lie algebra of B~. By solving an appropriate initial value problem one can show that the Lie group K admits an exponential map exp: k ~ K, i.e., an analytic map such that for all (D, v) e k the map C ~ K : t ~ exp t(D, v) is a one-parameter subgroup with initial velocity vector (D, v), see also [P1, sec. 1]. Since the exponential map of the finite-dimensional Lie group A, is a local analytic isomorphism near 0 ~ a e and since the exponential map of K maps la into H, see step d) below, the existence of the section ~ follows from the corresponding infinitesimal assertion: rk e :k ~ a e admits a continuous linear section .r:a~--*k which restricts to a section b e ~ It of the restriction h - ~ b c. For this take the canonical section % : ac -+ k which maps (De, re) c a. onto the unique polynomial of degree ~< c -- 1 in the fiber of (D e, v~). Then adjust "% by defining = (id k-p~)oT 0:a~k where r :k ~ 0~ is given by ~(D, v) -- Dh q- hv and p : 0~ ~k e is a scission of r res- tricted to k c - Ker 9 e. This -r is a continuous linear section of ~,. To show that it maps b e into h, take (D~, v~) ~ bc and let (D, v) ----- "r0(D~, v~) q- (D', ~') be an ele- ment of h mapped onto (D,, v~). Then (D', v') ek~ and e~0(D~, v,) : -- r v'). This implies r v,) = 0 and ~(D,, re) e h. d) It remains to show that the Lie algebra of K h is k h and that exp maps k~ into K~. Clearly { (D, v) ~ k, exp t(D, v) e K h for all t } C 7": K h C k~. Conversely, if (D, v) e kh, calculation shows that (hoqh).a,=0 for (qh,a,) =expt(D,v). Therefore exp t(D, v) ~ K h. This proves the Theorem. Let now X C (C", 0) be a reduced hypersurface defined as zero-fiber X = h-X(0) of some h e 0,. Let to, = x) be the group of embedded automorphisms of X. Thus G x ={~ ~G, ho ~ e 0~.h}. 14 106 HERWIGHAUSER AND GERD M~LLER For q~ ~ G x the factor a E 0, such that h o q~ = h. a is a unit which is uniquely determined by q0. Therefore the projection K -+ G : (9, a) ~-~ 9 induces an isomorphism of groups K h -% G x. Thus we may view G x as a Lie group. If X is isomorphic to another hypersurface Y C (C", 0), say defined by g ~ 0,, then h and g are in the same K-orbit: g = (h o 9). a for some (% a) ~ K. Conjugation with (q~, a) induces an isomorphism K h -% K o of Lie groups. In particular, the Lie group structure on G x is independent of the chosen equation h, and isomorphic hypersurfaces X, Y have isomorphic Lie groups Gx, Gy. The converse is also true: Theorem 4.9.. -- Two reduced hypersurfaces X, YC (C", 0), n/> 3, are isomorphic if and only if the# Lie groups Gx, Gy of embedded automorphisms are isomorphic. Proof. -- Let Dx, 0 = { D ~ g, Dh e h. O. }. The projection k = g | O, ~ g induces a continuous isomorphism of Lie algebras k h --% Dx, 0. Both k h and Dx, 0 are r modules of k, respectively g, hence closed subspaces. By the Closed Graph Theorem [Gr, chapter 4.1.5, Theorem 2] we conclude that k h -% Dx. 0 is even an isomorphism of topological vector spaces. Therefore, if f: G x ~ Gy is an isomorphism of Lie groups, the tangent map off at the unit induces an isomorphism Dx. 0 -% Dy, 0 of topological Lie algebras. By [HM2, part II, Theorem and Comments d), e) in section 1] the topo- logical Lie algebra Dx, o determines X up to analytic isomorphism. Therefore X and Y are isomorphic. 5. Scissions of O.-linear maps In order to construct and control scissions of 0,-linear maps we need a version of the Grauert-Hironaka Division Theorem with special attention paid to norm estimates [Ga], [Hau]. For fixed X = (X0, K) E R a+" + with Z-linearly independent components X~, ele- ments ofN a +" are ordered by ix < j~ ifk(ix) < X(j~), where X(ix) = X0(i -- 1) + 2~ ),~ x k. Define a total order on the set of monic monomial vectors (0, ..., O, x ~, O, ..., O) of dF, by setting (0, ..., 0, x ~, 0, ..., 0) < (0, ..., 0, x ~, 0, ..., 0) if j~ < ix where i and j denote the position of x ~ and x ~. Denote by in(a) and in(I) the initial monomial vector and initial module of elements and submodules of O~,. Set A(I) =(b no monomial of b belongs to in(I)}. For a standard basis ml, ..., m~ of I with initial terms ~tx, ..., ~%, partition the support of in(I) as a disjoint union U~= 1 M~ with M~ C supp(O,, bq)- Then set V(I) = { a ~ O. ~, supp(a~. ~q) C M~ for all i }. Any weight vector X as above induces on 0, a Banach scale structure [hi,= Z=Ib=I s~'= forb=~,b,x'sO,,. A RANK THEOREM FOR ANALYTIC MAPS BETWEEN POWER SERIES SPACES 107 In the sequel when dealing with O,-linear maps 1: 0, ~ ~ d7~ we consider on the domain space O, ~ the Banach scale structure defined by Ibl0 = ~,1 b~[~ and on the target space ~. the one given by I b l, = Z, s~,,-,I b, [~ Division Theorem 5.1. -- Let l : O~, -+ Oq. be an O,-linear map, say l( a) = a. m = Y~, a~ m, . Assume that the raps form a standard basis of I = Im l. Set K = Ker l and let A(I) and V(I) be defined as above. a) I~ A(I) ----- d~,, K@V(I) = dT~, and these decompositions are compatible. b) There is a constant c > 0 such that for all small s the following holds: For any e e (O~,)~ the unique elements a ~ V(I), and b e A(I)~ with e = Y~ a i m~ + b satisfy (min, I m, I,).l a I~ + Ibl~ Z [ a, I, 1 m,t. + Ibl, < c[Za, m, + bl~ Proof. -- We may assume that all m~ are 4: 0. For s > 0 with ml, ..., m~ the continuous linear map u, : V(I),| A(I). -+ (d~,), : (a, b) ~-. a.m + b will be shown to be bijective for small s. To this end supply the Banach space V(I)~ | A(I), with the norm II(a,b)l]~ = Z, I a,l.]~,l, + I b I~ By definition of V(I) the map v, : 7(I), @ A(I), ~ (O~,)~ : (a, b) ~ a. Vt + b is bijective, bicontinuous of norm 1, and its inverse v: ~ has norm 1 as well. Decompose u, into u,=v,+w~ where w~ =a.m' with m; =m~--~q. There are constants c>0 and r such that I~', I, ~< ]m~ l, <c [~., [, and I m, h < s'l~, ], for all i and all sufficiently small s. This yields tw, l<~s* and ] wsv~-I ]~ st<~ 1 for small s, say s ,< s 0. Using the geometric series one sees that u, v~ -1 = id + w, v~ -~ is invertible with I(u, v~-1)-1] ~ < -- =: c'. l-s; 108 HERWIG HAUSER AND GERD MOLLER Consequently u, is invertible and l u~ l l <~ c ' fors~<s 0. This implies part b) of the Theorem. Since trivially I, n A(I), = 0 we obtain moreover I, | A(I), = (0~,), and K, | V(I), = (0~),. This gives a). Example. -- Consider l: 0~-+0 2 given by m i = x +y, m,. = x--y. Assume X i<x*. Then 1.mr-- 1.m,=2y, but there is no constant c>0 such that for all small s I 1 I, I re, l, +1- l l, I re, l, = 2( sx' + sx') is bounded by c12y I, = The reason is that mr, m 2 are not a standard basis. We now use the Division Theorem to estimate the norms of projections onto sub- modules I C 0~ and of scissions of 0,-linear maps l : ~,~ -+ 0~,. Consider the Banach scale structures on 0,~ and 0~, as defined above. Let ~i = (0, ..., 0, x% 0, ..., 0) with entry in the jcth place the unique minimal monomial generator system of in(I). The number ~(I) = max~ X(j i ~,) is called the weighted order of I w.r.t, the weight X. It equals the maximal weighted order of the elements of a minimal standard basis of I. Theorem 5.2. -- Let l : 0, ~ -~ O", be an O,-linear map with image I =Im l and [5 = [5(I). a) The projection ~i : Oq, ~ I induced by the decomposition 0q, = I 9 A(I) is compatible and satisfies for some constant c > 0 and all small s > 0 I (~,), I~ c. b) l admits a compatible continuous linear scission ~ : t~, ~ 0~, satisfying for some constant c > 0 and all small s > 0 Proof. -- By the choice of the norms, a) follows from part b) of the Division Theorem. For b) choose a minimal standard basis mi,..., m~, ~ I with ~ = in m~. Consider the 0,-linear map l' : 0, ~' ~ 0~, : a ~ ]~ a~ m~. The Division Theorem gives a compatible decomposition Ker l' | V'(I) = 0~. '. Then 7' = (l' Iv,) -1 : I -+ O. r is a compatible continuous linear section of l'. Moreover there is a constant c' > 0 such that c'l A RANK THEOREM FOR ANALYTIC MAPS BETWEEN POWER SERIES SPACES 109 for a e V'(I) and all small s. By definition of ~ we have s ~ = min~ [ ~, 1, and thus Choose an 0,-linear map p : 0~, ' -+ 0~, with commuting diagram Observe that p is compatible with I P0 ] ~< c" for some constant c" > 0 and small s. Then a = p-r' ~I is a compatible continuous linear scission of l satisfying a norm estimate as claimed. Remarks. -- a) For an 0,-linear map l : 0, ~ ~ 0a, with image I one trivially has Im l~163 I,. The existence of the compatible scission ~ implies that Im l, = I, for small s. This is false if instead of the Banach scale structure defined by Z-linearly inde- pendent weights X0, ),~, ..., X, as above we would have taken different Banach scale structures. For example define [ a [, = Y~ ] a,, I s'+' for a = Y_, a~, x~y ' 9 C{ x,y } = 0,. Let l:02--->02:a~a.(x-y) with I=Iml= (x--y). For fixed s>0 set I k--1 a = Z ~oX'~ -1-/ tt-o ~ I We then have la.(x_y)[ = ~ 2 ,_o~ <~176 lal.= =~176 Thus a.(x --y) ~ I, but r l,. b) We can now prove the corollary of section 1: Let f:U---0~ be an analytic map in a neighborhood U C 0~ of 0 given by substitution in some g(x,y). Thenf has constant rank at 0 if and only if f is fiat at 0. In this casefis of constant rank at 0 w.r.t. A(I), where I = Im I. Indeed, let a be the scission constructed above. Then Ker/a = Kern I = A(I). Now our claim follows from the results of section 1. c) For any submodule I of 0~ the weighted order ~(m~ I) increases at most linearly with coefficient 1 in c. To see this, consider the inclusions m~.in(I) (2 in(m~, I) C in(I). Let ~ be an element of the minimal monomial generator system of in(m~ I). Then = x'v with u an element of the minimal monomial generator system of in(I) and I10 HERWIG HAUSER AND GERD M1DLLER x" E ~,. If I ~ ] > c, 7 belongs to m~.in(I) but is not a minimal monomial vector of this module w.r.t, the component-wise order on N". The first inclusion implies that it neither is minimal in in(m~I) which is a contradiction. Therefore [a[~ c and I) .< + c. 6. B*o*ch scales For details and the basic notions of this and the next section we refer the reader to [BS, He, Gr, U]. A Banach scale is a topological vector space E, Hausdorff, locally convex, and sequentially complete together with pseudonorms I I,, s c R and s> 0, such that (i) if s < s' then I1,-< I I,', (ii) E, := { a ~ E, I a [, < oo } is a Banach space with norm ] I,, (iii) E = [J, E, as topological vector space with the final topology. We often say that E is a Banach scale. But the reader is reminded that the Banach spaces E, with their fixed norms ] ], are part of the structure. Remarks. -- a) For r eR, r > 0, let B,(0, r) = { a a E,, ] a ], < r} be the open ball of radius r. Then B(0, r) := [3, B,(0, r) is open in E. Indeed, for s < s' and a E E,, C E, we have I a l, ~< I a I,', hence B,.(0, r) C B,(0, r). Therefore B(0, r) n E,. = B,.(0, r) u [.J,<,,(B,(0, r) n E,,) is open in E,. as E,, C E, is continuous for s < s'. b) Let F be a closed linear subspace of a Banach scale E = U s E,. Then F = U, F, with F, = E, n F is a Banach scale. In fact, it is immediately seen that a subset A C F is relatively closed if and only if A n F, is closed in F, for all s. Thus the final topology of F with respect to the inclusions F, C F coincides with the relative topology, which proves (iii). c) Let F be a Banach scale. Every decomposition F = I | J as an algebraic direct sum of closed linear subspaces is in fact topological. To show this it is enough to prove that the projection ~ : F ~ I with kernel J is continuous. By the Closed Graph Theorem [Gr, chapter 4.1.5, Theorem 2] it suffices to show that graph n is closed. But this is obvious from graph ~ ={(a, b) eF| a -- b e J}. A continuous linear map I:E ~ F between Banach scales is called compatible if, for small s, l restricts to a map l0 : E, -+ F,. Then l, is necessarily a continuous linear map. Since the relative topology on F, induced from F is a Hausdorff topology coarser than its Banach space topology this follows from the Closed Graph Theorem, see [Gr, chapter 1.14, Corollary to Theorem 10]. A decomposition F = I | as a direct sum of closed linear subspaces is called compatible if F, = Io | with I0 = F, n I, J, -- F, n J for small s. Then clearly the projections onto I and J are compatible. A RANK THEOREM FOR ANALYTIC MAPS BETWEEN POWER SERIES SPACES 111 An analytic (i.e. continuous and G~teaux-analytic) map f: U ~F, U CE open is called compatible at a e U if, for small s, f restricts to an analytic map f,: U,-+F, where U, is a suitable neighborhood of a contained in U n E,. In that case the chain rule gives that Tar , = 7"ofl, for the G~tteaux-differentials Taf:E~F and Taf,:E,~F ,. Proposition 6.1. -- Let E and F be Banach scales, U C E an open set and f: U ~ F a map. Then f is analytic if and only if, for all analytic curves "~ : T --~ E with T C C open and ~,(T) C U, the composition f o "( : T -+ F # analytic. Proof. -- [He, Theorem 3.2.7 d)]. Inverse Mapping Theorem 6.2. -- Let E = U, E,, F -= [.J, Fo be Banach scales, and let f: U ~ F be an analytic map in a neighborhood U C E of 0 with f(O) = O. Assume: (i) The Gdteaux-differential l -~ To f: E -+ F is an isomorphism of topological vector spaces. (ii) The composition l -~ of is compatible at 0 with restrictions (l -~ of), : U, -~ F,. (iJi) There are constants 0 ( c ( 1/e(e § 1) and r > 0 such that for all small s > 0 ](1-1 of), -- idE, I,~< c.r. Then f is a local analytic isomorphism at O. Proof. -- We proceed exactly as in the usual proof of the Inverse Mapping Theorem in Banach spaces. Upon replacingfby l -~ of we may assume F = E and I = To f= ids. Write f = ids § h and define rccursively g0=0, g" +l _id s_hog.. As ordh >/ 2 it is shown by induction (using Proposition 6.3 below) that ord(g "+a -- g") > n. Let ~ be the unique formal power series satisfying ord(g -- g") ~ n for all n. Then also ord(fo ~ --fog") > n. Since ord(fo g" -- ids) = ord(g" -- g" + ~) > n we conclude thatfo ~ = ida., i.e. ~ is a formal right inverse off We shall show that converges on the open set B(0, r') = [J, B,(0, r') for r' -~ ((l/e) -- c).r. Note that r' > 0 since c < 1/e. Let us first prove by induction that for small s. Indeed, using assumption (iii) we have I g~ ],' = ](idE -- h o g"-~), I,' r' + ]h, ],.10"-'J,, r'§ I, <~ r' § e 112 HERWIG HAUSER AND GERD MI~ILLER But then, since ord(~- g")> n, we also have ILl,,<. -. This implies that ~ induces analytic maps g, : B,(0, r') ~ E, with [g, I,' <` r/e as well as an analytic map g:B(0, r') -+ E. Thus the following holds: fog = idB(0.,,~, g(0) = 0, Tog : ida. ce [g, -- idE, 1,' <` c'. r' with c'-- 1 -- r ~ Using c < I/e(e + 1) we see that c'< 1/e. Hence the whole argument above can be applied to g with r and c replaced by r' and c'. Setting r" = ((l/e) - c').r' we find an analytic map ft." B(0, r")~ E with g off= idB~0.,,, ~. Uniqueness of the inverse implies f=f],,o,,",. Thus f: B(0, r") ~f(B(0, r")) is an analytic isomorphism. Example. -- We indicate what can happen if the norm estimate (iii) in the Inverse Mapping Theorem is violated. View E = 01 = G { x } as a Banach scale as in section 5. Then ~ : E -~ E : a ~ a -- a(0) is a compatible continuous linear map. Define f: E -~ E by - a (0) f(a) =a+ --a=--- x x It is analytic and compatible at O. Its G~tteaux-differential is given by ~(2ab) raf(b ) = b + -- In particular, To f=-id z. Nevertheless f is not a local analytic isomorphism at 0. In fact, if a E E is invertible, a(0)4: 0, we have for b := a(O).(x q-2a) -1 that Taf(b ) :- (l/x) 7~((x + 2a) b) ----- 0. Hence there are points a s E arbitrarily close to 0 such that T,,f is not injective. This phenomenon can be explained by comparing the size of the terms of order/> 2 at 0 with Tof = ida. For this let us calculate If, -- ida, ],. We have f(a)= a + P(a) where P(a)= (1/x).n(a 2) is a continuous homogeneous polynomial of degree 2. For fixed s we have ] 7z(a 2) I, --=- [ a2 -- a2(0) 18 <` ] a2 Is <` [ a I~. If a = x.s -~ then [ a l, = 1 and I n(a2) I, = 1. This impfies I P, [ = s-~ and If. -- ida, l, = I P, l, = re's-" But we cannot find a constant c such that r2.s -~ <~ c.r for all small s. Note that T.fis injective for a ~ (x), say of order /> 1. Moreover, Taf: 0 x ~ 01 is surjective for all a close to 0. Indeed, x + 2a is not in (x) * for a close to 0. tIence given c ~ 9 1 we can solve cx = (x + 2a) b for b. Then Taf(b) = c. A RANK "tHEOREM FOR ANALYTIC MAPS BETWEEN POWER SERIES SPACES 113 Proposition 6.3. -- Let f: U---> F, g:V ~ G be ana~tic with U C E, V C F open and f(U) C V. Then g of: U ~ G is analytic and for a ~ U T,(g of) -~ T1,~,g o T,f. Moreover, if 0 ~ U, f(O) ~ 0 ~ V and g(O) -~ O, the expansion of g of at 0 is given by the composition formula for formal power series applied to the expansions of g and f at O. Proof. -- The first part is standard. For the second, one can assume by the Hahn- Banach-Theorem that E = G = C. Let g ---- 2~ g~ be the expansion of g in homogeneous polynomials and gi be the corresponding/-linear symmetric maps. Let then i be fixed. As ~ is continuous there are a constant c > 0 and a continuous seminorm p on F such that [~(ba .... , b~) I ~< c.p(b~) ... p(b,) for all b~, ..., b, ~ F. Fix a near 0. By [BS, Proposition 4.1] there are constants M > 0 and 0 ~ ]0, 1 [ such that p(f~(a)) <. M.0 ~ for all k. Consequently ~>, ~k~+ ... *~.=k I g~(fk~(a), " .,f~.(a)) I ~< ~ ~ cM ~ 0 ~ 0~( oo. As ~, is /-linear and continuous we obtain gdf(a)) -= ~,(f(a), ...,f(a)) = + ... g,(L,(a), This is the power series expansion of g, of. For sufficiently small r > 0 we have g(f(a)) == X;,g,(f(a)) for I ~ I < r, the series converging uniformly on [a[< r, see [BS, proof of Theorem 6.4]. By Weier- strass' Theorem on (locally) uniform sequences of analytic functions = g,(L,(a), is the power series expansion on [ a[ < r. 7. Analytic maps between power series spaces We now specialize to power series spaces. Equip O, p with pseudonorms defined through weight vectors as in section 5. Then O](s) ---- { a ~ d~, ~, I a 1, < oo } is a Banaeh space and d~ = O, 0,~(s) with the final topology a Banach scale [GR]. Partial differen- tiation and integration O, ~ O, as well as algebra homomorphisms 6, ~ 6,, and On-module homomorphisms are continuous linear maps, hence analytic. The next result characterizes analytic maps with finite-dimensional domain [Mfi2, sec. 6]: 15 114 HERWIG HAUSER AND GERD MI[II.LER Proposition 7.1. -- a) For a map f: T-+ d~,, T CC ~ open, the following conditions are equivalent: (i) f is analytic. (ii) f is continuous and for all jet maps ~ :~W, ~ (O,/m~,+') ' the composition =,of is analytic. (iii) For every t o 9 T there are open neighborhoods T'C T oft0, VC C" of 0 and an ana- lyt# map g : V � T' ~ C ~ such that for t 9 T' :fit) (x) = g(x, t 1. (iv) For every t o 9 T there are an open neighborhood T' C T of t o and an s ~ 0 such that f(T') C d~,(s) and f: T' ~ O~,(s) is analytic. b) Assume that (i) to (iv) hold. Let g(x, t o + t)= Z,,~g~ x ~ t ~ be the power series expansion of g near (0, to). For t 9 C ~ let f~(t) = Yq~r ~ ~g~ x~ t~ 9 0~. 9 Then f(t o + t) = Y~kfk(t) is the power series expansion off near t o. c) For t e T' as in (ill) above, the Gdteaux-differential Ttf : C ~ --> Oq. is given by T,f(ba, .... b~) = 2~, Oq g(x, t).b,. The main examples of analytic maps between power series spaces are given by substitution: Proposition 7.2. -- Let g:V � W-+C" be analytic, where V x W C C ~ x C ~ is an open neighborhood of O. Let U = { a 9 ~,, a(O) 9 W }, and define f: U ~ ~ by substitution in g : f(a) (x) : g(x, a). a) f is analytic. For a c U its Gdteaux-differential To f: 0~ -~- O~ is given by Toy(b,, ..., b,) = ]E, Ovi g(x , a).b,. b) Let g(x,y) = ~,~g~ x~ y ~ be the expansion of g. For a 9 0~, let fk(a) = Z I ,! =~ Y~ g~ x ~ a ~ ~ 0~. Then f(a) = Zkf~(a) # the power series expansion off near O. c) f is compatible at O. Proof. - We may assume q = 1. a) Propositions 6.1 and 7.1 imply thatfis ana- lyric. For ~(:U and b 9 ~, let h(t)=f(a+tb), t 9 close to 0. Then h(t) (x) ---- g(x, a + tb) =: h'(x, t). By Proposition 7.1 c) we have Tof(b) = ~ h(t) I,- o = 0, ~(x, t)l, =o = }-], Ou, g(x, a).b,. b) For small s the fk are continuous homogeneous polynomials O.~(s) ---> O,(s) of degree k. For a e B,(0, r), r sufficiently small, the series ]~kf~(a) converges to f(a) in O,,(s). Hence f restricts to an analytic map B,(0, r) ~ O.(s). This proves b) and c). A RANK THEOREM FOR ANALYTIC MAPS BETWEEN POWER SERIES SPACES 115 REFERENCES ARTIN, M., On the solutions of analytic equations, Invent. Math., ,5 (1968), 277-291. C A ] [BS] BOCHNAX, J., SicI~a~, J., Analytic functions in topological vector spaces, Studia Math., 39 (1971 ), 77-112. BOURBAKI, N., Variltts diff~rentielles et analytiques, Hermann, I%7. [BI COLO~mEAU, J.-F., Differentiation et bornologie, Th6se, Universit6 de Bordeaux, 1973. COl GALLmO, A., Th6or6me de division et stabilit6 en g~omdtrie analytique locale, Ann. Inst. Fourier, 29, 2 [Gal (1979), 107-184. [Gr] GROTHBNDmCK, A., Topological vector spaces, Gordon and Breach, 1973. [GR] GRAU~nT, H., REMMBRT, R., Analytisehe Stellenalgebren, Springer, 1971. t-IAMILTON, R. S., The inverse function theorem of Nash and Moser, Bull. Amer. Math. Sor 7 (1982), 65-222. [Ham] I-L~usER, H., La construction de la ddformation semi-universelle d'un germe de vari~tfi analytique complexe, [Haul Ann. Sci. fie. Norm. Sup. Paris (4), 18 (1985), 1-56. t-IAusER, H., MOLLER, G., Automorphism groups in local analytic geometry, infinite dimensional Rank [HMI] Theorem and Lie groups, C. R. Acad. Soi. Paris, 818 (1991), 751-756. HAUSER, H., MOLLER, G., Affine varieties and Lie algebras of vector fields, Manuscr. Math., 80 (1993), [HM2] 309-337. [He] t-~RV~., M., Analytieity in infinite dimensional spaces, Studies in Math., 10, De Gruyter, 1989. LESLIE, J., On the group of real analytic diffeomorphisms of a compact real analytic manifold, Trans. [El Amer. Math. Sot., 274 (1982), 651-669. [Mi] MXLNOR, J., Remarks on infinite-dimensional Lie groups, in Relativitg, groupes et topologie II, Les Houches, Session XL, 1983 (eds : DE Wrr'r et STox~), p. 1007-1057, Elsevier, 1984. MOLLER, G., Reduktive Automorphismengruppen analytischer C-Algebren, J. Reine Angew. Math., 364 [Mall (1986), 26-34. MOLLER, G., Deformations of reductive group actions, Proe. Camb. Philos. Soc., 108 (1989), 77-88. [Mii2l PISANELta, D., An extension of the exponential of a matrix and a counter example to the inversion theorem [Vll ofa holomorphlc mapping in a space H(K), Rend. Mat. Appl. (6), 9 (1976), 465-475. PISANELH, D., The proof of Frobenius Theorem in a Banach scale, in Functional analysis, holomorphy and [P21 approximation theory (ed. G. I. ZAPATA), p. 379-389, Marcel Dekker, 1983. PIS,~ELH, D., The proof of the Inversion Mapping Theorem in a Banach scale, in Complex analysis, func- [P31 tional analysis and approximation theory (ed. J. MujmA), p. 281-285, North-Holland, 1986. UPMEXER, H., Symmetric Banach manifolds and Jordan C*-algebras, North-Holland, 1985. [Ul H~ H~ Institut ft~r Mathematik Universit/it Innsbruck A-6020 Innsbruck Austria G~ M~ Fachbereich Mathematik Universit~it Mainz D-55099 Mainz Germany Manuscrit refu le 7 juillet 1992. Rgvisg le ler april 1994.

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