Recognizing Hopf algebroids defined by a group action Ethan S. Devinatz* Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195, USA phone: 1-206-685-4777 fax: 1-206-543-0397 e-mail: devinatz@math.washington.edu Abstract Let A be a complete noetherian regular local ring, and suppose that S is a pr* *ofinite group acting continuously on A via ring homomorphisms. Let = Map c(S, A), the alge* *bra of continuous functions from S to A. Then (A, ) has a canonical structure of a co* *mplete Hopf algebroid, determined by the action of S on A. We give necessary and sufficient* * conditions for a general complete Hopf algebroid to be of this form. Applications to Morav* *a theory are also discussed. Keywords: Hopf algebroid, Morava theory __________ *Partially supported by a grant from the NSF. 1 2 Suppose that A is a complete noetherian regular local ring and an R-algebra w* *ith maximal ideal m and that S is a profinite group with identity e acting continuously on * *A (with the m-adictopology) via R-algebra homomorphisms. Then define = Map c(S, A), the a* *lgebra of continuous functions from S to A, and give the m-adicfiltration obtained b* *y regarding as an A-algebra via the map jR : A ! Map c(S, A) given by jR(a)(s) = a for all a 2 A, s 2 S. (The regularity of A ensures that t* *he m-adic filtration on agrees with the filtration F i Map c(S, miA); see Lemma 1.) * *There are also filtration preserving maps jL : A ! ffl: ! A : ! b A c : ! defined by jL(a)(s) = s-1a ffl(f)= f(e) c(f)(s) = s-1(f(s-1)). As for , begin by recalling (cf. [4; Lemma 3.14]) that the map oe : Map c(S, A)b AMap c(S, A) ! Map c(S x S, A) defined by oe(f1 f2)(s1, s2) = s-12(f1(s1)) . f2(s2) __ is an isomorphism. Then define = oe-1 . , where __ : Map c(S, A) ! Map c(S x S, A) is induced by the multiplication map S x S ! S. With these maps, (A, ) become* *s a complete Hopf algebroid over R in the sense of [3]. In this note, we consider the inverse problem. That is, suppose given a comp* *lete Hopf algebroid (A, ) over R, where A is a complete noetherian regular local ring wi* *th maximal ideal m, m is an invariant ideal, and has the m-adictopology. With some addi* *tional assumptions on the map R ! A, we give necessary and sufficient conditions on * *as a right A-algebra for (A, ) to be isomorphic to the complete Hopf algebroid arising as* * above from the action of a profinite group S on A. We also explicitly identify the group S. Although this result is purely algebraic, we are motivated by examples in sta* *ble homotopy theory. Fix a prime p and positive integer n, and let En denote the Landweber * *exact spectrum as in [6]. The coefficient ring En* is W Fpn[[u1, . .,.un-1]][u, u-1],* * where |ui| = 0, |u| = -2, and W Fpn denotes the ring of Witt vectors with coefficients in the f* *ield Fpn of pn elements. Let E^n*En = ß*LK(n)(En ^ En), where LK(n)denotes localization with r* *espect to the nthMorava K-theory K(n). The main result of Morava's theory asserts that th* *ere is an action of a certain profinite group Gn_called the extended Morava stabilizer gr* *oup_on En* such that the complete Hopf algebroid (En*, E^n*En) over the p-adicintegers Zp * *is isomorphic to the complete Hopf algebroid determined by this action. Our main result repr* *oves this without using the Lubin-Tate theory of liftings of formal groups. (The reader m* *ay wish to 3 compare this with the approach taken by Hovey [8]). Moreover, if G is any close* *d subgroup of Gn, one can construct a öc ntinuous homotopy G fixed point spectrum" EhGn[6]* *, and there is a strongly convergent spectral sequence H**c(G, En*X) ) EhGn*X for any finite spectrum X. However, formulas for the action of Gn on En* are v* *ery com- plicated (see [5]); this makes the direct calculation of H**c(G, En*X) inaccess* *ible, except in certain very special cases. (See for example [11]; this is essentially the * *only nontrivial situation where En* can be explicitly identified as a G-module.) On the other h* *and, H**c(G, En*X) = Ext**Mapc(G,En*)(En*, En*X), the cohomology of the (complete) Hopf algebroid (En*, Mapc(G, En*)) arising fro* *m the action of G on En*. Now Map c(G, En*) is a quotient of E^n*En = Map c(Gn, En*), and t* *here are good_or at least reasonable_formulas for the structure of E^n*En, making (at le* *ast partial) calculations of ExtE^n*En(En*, ?) sometimes feasible. An explicit determination* * of the quotient Map c(G, En*) might then allow one to make computations of H**c(G, En*X). Our m* *ain result does not produce such a description for an arbitrary group G; it does, however,* * provide a "recognition principle"; that is, given a quotient Hopf algebroid (En*, ), we * *can determine whether it is (En*, Mapc(G, En*)) for a given closed subgroup G of Gn. In practice, one often makes calculations of ExtE^n*En(En*, ?) using a Bockst* *ein spectral sequence. Such a technique can also work for ExtMapc(G,En*)(En*, ?) and will be* * carried out in a special case in forthcoming work. This approach requires an explicit under* *standing of Map c(G, En*=m) as a quotient of E^n*En=mE^n*En, but not one of Map c(G, En*). Here then is our main result. For the rest of this paper, assume that A is a * *complete local ring and R-algebra with maximal ideal m and residue field K. Assume also that R* * is a local ring with residue field k and that the map R ! A is a local ring homomorphism. Theorem Assume in addition that A is noetherian and regular and that K is a se* *parably closed algebraic extension of k. Suppose also that (A, ) is a complete Hopf al* *gebroid over R, with m an invariant ideal and given the m-adictopology. Then (A, ) (A, Map* *c(S, A)) for some profinite group S if and only if i. =mi__is flat over A=mi for all_i 1 * * __ ii.(K, ) (A=m, =m ) = lim(K, ff) as Hopf algebroids over k, where each* * ffis a -!ff finite separable K-algebra. (For definiteness, if (B, ) is a Hopf algebr* *oid, we use jR to provide with a B-algebra structure.) In such a case, __ S Hom A-alg( , A) -! Hom K-alg( , K) as profinite groups (see Remark 2). * * Q t Remark 1. If L is any field, a finite L-algebra B is separable if and only if B* * = i=1Li as L-algebras, where each Li is a separable field extension of L. This condition i* *s satisfied if and only if B=L, the module of Kähler differentials, is trivial (see [10; I, P* *ropositions 3.1, 3.2, 3.5]). 4 Remark 2. If (A, ) is a complete Hopf algebroid over R, where, once again, m i* *s invariant and is given the m-adictopology, then Hom A-alg( , A) has a canonical monoid * *structure. Indeed, recall that Hom cR-alg(A, A), the set of continuous R-algebra endomorph* *isms of A, is the set of objects of a groupoid with morphisms Hom cR-alg( , A). Hom A-alg( , * *A) is therefore the set of morphisms whose source is the identity map in Hom cR-alg(A, A). If f* * and g are in Hom A-alg( , A), let f0 be the composition f ''L g -! A -! -! A. Then the source of f0 is the same as the target of g, so we may define the prod* *uct of f and g in Hom A-alg( , A) to be the composition of the morphisms f0 and g in the above* * groupoid. More explicitly, the product f * g in Hom A-alg( , A) is given by X (f * g)(t) = g(jL(f(t0))g(t00), P where : ! b A sends t to t0 t00. The reader may check that this opera* *tion is associative and that the structure map ffl : ! A is the identity. Without further assumptions, Hom A-alg( , A) need not be a group. It is a gro* *up, however, if __ Hom A-alg( , A) ! Hom K-alg( , K) * * __ is a bijection and K is an algebraic extension of k. To see this, observe that * *if f : ! K is a K-algebra homomorphism, then f O jL is an endomorphism_of_K fixing k. Sinc* *e K is an algebraic extension, it must be an isomorphism. (If (K, ) is a Hopf algebra* *, then f O jL is the identity, but this condition_is not satisfied in the situations we are i* *nterested_in.)_The inverse of f in Hom K-alg( , K) is now the composition (f O jL)-1 O f O c, wher* *e c : ! is the structure map_corresponding to taking the inverse of a morphism. __ Finally, if is the direct limit of finite K-algebras_then Hom K-alg( , K) i* *s profinite as a set. Under the conditions of the theorem, Hom K-alg( , K) is a profinite group. Remark 3. The theorem applies to ungraded complete Hopf algebroids. If (En*, * **) is a graded complete Hopf algebroid and * is concentrated in even dimensions, then * *we may apply the theorem to ((En)0, 0). If the hypotheses of the theorem are satisfie* *d, we obtain 0 Map c(S, E0) and hence (y) * Map c(S, En*), since multiplication by jR(u) is an isomorphism from 2k to 2k+2. Then define * *the action of S on u by su = (jL(u))(s-1), where we use the identification of (y). With th* *is definition, (En*, *) (En*, Mapc(S, En*)) as graded complete Hopf algebroids. Remark 4. The alert reader may have noticed that the theorem cannot possibly ap* *ply to (En)0, 0), since (En)0=m = Fpn is not separably closed. But, if * is a quotie* *nt of E^n*En, n __ we have that xp = x for all x 2 0. This allows the theorem to go through anywa* *y. Let us begin the proof of the main theorem. Our first result is the only plac* *e where we use the regularity of A. 5 Lemma 1. Suppose that A is noetherian and regular, and let S be a profinite set* *. Then mjMap c(S, A) = Map c(S, mj), where Map c(S, A) is given the evident A-algebra structure. Proof.We begin with the following recollections. If M is an m-adically complet* *e finitely generated A-module and N is a submodule of M, then N is complete with the m-adic topology, and, by the Artin-Rees lemma (see [9; Theorem 15]), this topology agr* *ees with the subspace topology. The Artin-Rees lemma also implies that M=N is m-adically com* *plete_ this in turn implies that any finitely generated A-module is m-adically complet* *e. Next observe that Map c(S, ?) is exact on the category of finitely generated A-modul* *es; to prove this, we need only show that Map c(S, M) ! Map c(S, M=N) is an epimorphism. But* * this follows from the exact sequence limMap c(S, M=mjM) ! limMap c(S, M=N + mjM) ! lim1Mapc(S, N=N \ mjM) j- -j -j together with the fact that the inverse system {Map c(S, N=N \ mjM)} is Mittag-* *Leffler, since any continuous map from S into N=N \ mjM factors through a finite quotien* *t of S. Therefore, if I is any ideal of A, IMap c(S, A) -! Map c(S, I) if and only if Map_c(S,_A)_ -! Mapc(S, A=I). IMap c(S, A) Now write m = (x1, . .,.xd), where x1, . .,.xd is a regular sequence; that is* *, for each i with 1 i d, xi is not a zero divisor on A=(x1, . .,.xi-1) ([9; Theorem 36])* *. Write Ij = (xj1, . .,.xjd). Since mjd Ij, A=Ij is discrete. I claim that (1.1) IjMap c(S, A) -! Map c(S, Ij). Assuming this, we have that mjMap_c(S,_A)_ j -! m Map c(S, A=Ij), IjMap c(S, A) and, moreover, since mj=Ij is discrete, mjMap c(S, A=Ij) -! Map c(S, mj=Ij). The desired result now follows from the diagram jMap(S,A) 0____//IjMap c(S, A)__//mjMap c(S, A)____//_m____c_IjMapc(S,A)//_0 | || || || fflffl| fflffl| fflffl| 0_____//Mapc(S, Ij)___//_Mapc(S, mj)___//Mapc(S, mj=Ij)__//0. We will prove (??) by showing that (1.2) (xj1, . .,.xjt)Map c(S, A) -! Map c(S, (xj1, . .,.xjt)) 6 for all t, by induction on t. Indeed, if (??) holds for t = i, then Map c(S, A) j j (1.3) ____________________jj-!Mapc(S, A=(x1, . .,.xi)). (x1, . .,.xi)Map c(S, A) Now (xj1, . .,.xjd) is a regular sequence ([9; Theorem 26]), so multiplication * *by xji+1is a homeomorphism from A=(xj1, . .,.xji) to xji+1(A=(xj1, . .,.xji)). Thus Map c(S, A=(xj1, . .,.xji)) j j (1.4) ________________________jjj-!Mapc(S, A=(x1, . .,.xi+1)), xi+1Mapc(S, A=(x1, . .,.xi)) and therefore, by (??), ___Mapc(S,A)_ _____Map_c(S,_A)_____ (xj1,...,xji)Mapc(S,A) j j = ___________________hi-!Map c(S, A=(x1, . .,.xi+1)). (xj1, . .,.xji+1)Map c(S,xA)ji+1_Mapc(S,A)_ (xj1,...,xji)Mapc(S,A) This completes the inductive step and the proof. The next result is the technical heart of our recognition principle. Its proo* *f will be im- mediate to anyone familiar with the basic theory of formally 'etale algebras; w* *e, however, include a proof for the convenience of the reader. __ Proposition 2. Let be an m-adically complete A-algebra and write_ = =m . Su* *ppose that =mi is flat over A=mifor all i, and suppose further that is the direct* * limit of finite separable K-algebras. The reduction map __ __ Hom A-alg( , C) ! Hom K-alg( , C) * * __ is then a bijection whenever C is an m-adically complete A-algebra, and we writ* *e C C=mC. We first separate off a key fact which will be used in the proof. Suppose that B is a not necessarily commutative algebra over a field L and M * *is a B-bimodule_that is, a module over the L-algebra Be B L Bop. Note that if B * *is commutative, any B-module may be regarded as a B-bimodule in an evident way. In* * any event, there is a cochain complex P *(B, M) with P n(B, M) = Hom L(B(n), M) and differential ffi : P n(B, M) ! P n+1(B, M) given by Xn ffif(b1, . .,.bn+1)=b1f(b2, . .,.bn+1) + (-1)nf(b1, . .,.bibi+1, . .,* *.bn+1) i=1 +(-1)n+1f(b1, . .,.bn)bn+1, where B(n)= B__L_._.-.LBz_____". n times The homology of this complex is the Hochschild cohomology HH*(B, M) = Ext*Bc(B,* * M) (see for example [2; Section 2]). The next result gives the main fact we need. 7 Lemma 3. Suppose that B is a (commutative) L-algebra which is the direct limit * *of finite separable L-algebras. Then HHi(B, M) = 0 for all i > 0 and B-modules M. Proof.Write B = limBff, where each Bffis a finite separable L-algebra. It is w* *ell-known -! (andQoriginally proved in [7]) that HHi(Bff, M) = 0 for all i > 0 and Bff-bimod* *ules M. Now let *P t(Bff, M) denote the (cochain complex associated to the) cosimplicial * *replacement ofQthe inverse system {P t(Bff, M)} ([1; Chapter XI, x5]), and consider the dou* *ble complex *P *(B ff, M). Since ( t Q * t i t P (B, M) i = 0 Hi( P (Bff, M)) = limP (Bff, M) = -ff 0 i > 0 (because Hom L(?, M) is exact and lim iBff= 0 for all i > 0), it follows from t* *he spectral -!ff Q sequence of the double complex that the total cohomology of *P *(Bff, M) is H* *H*(B, M). On the other hand, ( Q s Q s i * Mff i = 0 H (P (Bff, M)) = , 0 i > 0 where Mff= {m 2 M : bm =Qmb 8 b 2 Bff}. But we are assuming Mff= M; therefore the total cohomology of *P *(Bff, M) is M, concentrated in degree 0. This com* *pletes the proof. Proof of Proposition 2 . The proof consists of_3_parts. Step 1. Hom A-module( , C) ! Hom K-module( , C) is surjective. Proof of Step 1. It suffices to show that Hom A-module( , C=mi+1C) ! Hom A-module( , C=miC) is surjective for all i 1; for this we only need Ext1A=mi+1( =mi+1 , miC=mi+1C) = 0. But, since =mi+1 is flat over A=mi+1, __ i i+1 Ext*A=mi+1( =mi+1 , miC=mi+1C) = Ext*K( , m C=m C) = 0. __ __ Step 2. Hom A-alg( , C) ! Hom K-alg( , C) is one-to-one. Proof of Step 2. We prove that Hom A-alg( , C=mi+1C) ! Hom A-alg( , C=miC) is one-to-one for each i 1. __ __ Suppose that h 2 Hom_A-alg(_, C=miC), and let ~h: ! C be its mod m reduc* *tion. Regard miC=mi+1C as a (resp. )-module by pulling back along ~h(resp. ~hcompo* *sed with the reduction). If f and g are algebra homomorphisms from to C=mi+1C which re* *duce to h, define d : ! miC=mi+1C by d(t) = f(t) - g(t). Then __ i i+1 d 2 DerA( , miC=mi+1C) = DerK( , m C=m C), 8 where, for example, DerA( , miC=mi+1C) denotes the set of A-module derivations * *from to miC=mi+1C. But __ i i+1 i i+1 DerK( , m C=m C) = Hom K-module( _=K, m C=m C) = 0, so f = g. __ __ Step 3. Hom A-alg( , C) ! Hom K-alg( , C) is surjective. Proof of Step 3. Again we prove that Hom A-alg( , C=mi+1C) ! Hom A-alg( , C=miC) is surjective for each i 1. Let g : ! C=miC be a map of A-algebras. By Step 1, there exists an A-module* * map f : ! C=mi+1C lifting g. Then define an A-module map cf : A ! miC=mi+1C_* *by_ cf(s t) = f(s)f(t) - f(st). We may_and_will_regard cf as a K-module_map K * * ! miC=mi+1C. Now make miC=mi+1C a -module by pulling_back along ~g: ! C=mC. Then one can check that cf is a cocycle in P *( , miC=mi+1C). If f0 is another * *A-module lift of g, then cf0- cf = ffih, where h(t) = f0(t) - f(t) 2_miC=mi+1C; from this it * *follows that this construction yields a cohomology_class dg 2 HH2( , miC=mi+1C) depending on* *ly on g. It also follows that if c 2 P 2( , miC=mi+1C) is a representative of dg, then t* *here exists an A-module lift f of g such that c = cf. Since cf = 0 if and only if f is an A-a* *lgebra lift, we have that dg_= 0 if and only if g lifts to an A-algebra map f : ! C=mi+1C.* * But by Lemma 3, HH2( , miC=mi+1C) = 0. This completes the proof. The next result gives part of the main theorem. Lemma 4. Suppose in addition that A is noetherian and regular, and let S be a p* *rofinite set. Let = Map c(S, A) with the evident A-algebra structure. Then the map S -h!Hom A-alg( , A) given by h(s)(f) = f(s) is a bijection. If S is a profinite group acting contin* *uously on A via R-algebra homomorphisms and Hom A-alg( , A) is given the monoid structure of Re* *mark 2, then h is a group isomorphism. Q Proof.Write S = limSff, where each Sffis finite. Map c(Sff, A) = S A, and the* * map ff ff Y Sffhff-!HomA-alg(Map (Sff, A), A) = Hom A-alg( A, A) Sff sends an element s 2 Sffto the algebra homomorphismQwhich is projection onto th* *e coordi- nate indexed by s. Let esbe the elementQof SffA withPa 1 in the coordinate in* *dexed by s and with 0's elsewhere. If f 2 Hom A-alg( SffA, A), then s2Sfff(es) = 1 and f(es* *)f(et) = 0 whenever s 6= t. Since A is a domain [9; Theorem 36], this implies that there e* *xists s0 such that f(es) = 1 when s = s0 and is 0 otherwise. Hence hffis a bijection. To complete the proof of the first part, it now suffices to show that the can* *onical map Hom A-alg(Map c(S, A), A) ! limHom A-alg(Map (Sff, A), A) -ff 9 is a bijection. By Lemma 1, Hom A-alg(Map c(S, A), A)=limHom A-alg(Map c(S, A), A=mj) -j = limHom A-alg(Map c(S, A)=mjMap c(S, A), A=mj) -j = limHom A-alg(Map c(S, A=mj), A=mj). -j But Map c(S, A=mj) = limMap (Sff, A=mj), so !ff limHom A-alg(Map c(S, A=mi), A=mj)= limlimHom A-alg(Map (Sff, A=mj), A=mj) -j -ffj- = limlimHom A-alg(Map (Sff, A), A=mj) -ffj- = limHom A-alg(Map (Sff, A), A). -ff Finally, the reader may check that h is a group homomorphism if S is a profin* *ite group acting continuously on A via R-algebra homomorphisms. Proof of Theorem. First suppose (A, ) (A, Mapc(S, A)) for some profinite g* *roup S. By Lemma 1, =mi = Map c(S, A=mi) = limMap (Sff, A=mi) !ff __ __ Q and is therefore flat over A=mi. For part ii, let ff= Map (Sff, K). Then ff * * SffK and so is a separable K-algebra. The isomorphisms __ S Hom A-alg( , A) -! Hom K-alg( , K) follow from Proposition 2 and Lemma 4. * * __ Conversely, suppose (A, ) satisfies the conditions of i and_ii. Let Sff= Hom* * K-alg(_ff, K), and let S be the profinite group (see Remark 2) limHom K-alg( ff, K) = Hom K-al* *g( , K). - By Proposition 2, S = Hom A-alg( , A). Now define f : -! Map (Hom A-alg( , A), A) by f(t)(h) = h(t). I claim that f is an isomorphism onto Map c(Hom A-alg( , A),* * A). Assum- ing this claim, define a continuous action of S on A by sa = f(jL(a))(s-1). The* * reader may then check that f : ! Map c(S, A) is in fact an isomorphism of complete Hopf * *algebroids. To prove the claim, start by observing that Map (S, A) is m-adically complete* * and therefore, by Lemma 2, f is the unique algebra homomorphism lifting f~: __! Map c(Hom K-alg(__, K), K) Map (S, K). But Map c(S, A) is also m-adically complete (see proof of Lemma 1); hence there* * is a unique lift of ~fto an algebra map f0 : ! Map c(S, A). This implies that f = f0. The proof of the claim will now be completed by showing that f0 : =mi! Map c(S, A=mi) 10 * * __ is an isomorphism for all i. If i = 1, this follows from the fact (see Remark 1* *) that ffis a finite product of copies of K and hence that __ __ ff-! Map(Hom K-alg( ff, K), K). In general, there is the following commutative diagram, where the rows are exac* *t: 0_______//mi =mi+1 __________//_ =mi+1___________//_ =mi_______//_0 | | | | | | fflffl| fflffl| fflffl| 0____//Mapc(S, mi=mi+1)___//Mapc(S, A=mi+1)__//Mapc(S, A=mi)___//0 But =mi+1 is flat over A=mi+1, therefore __ i i+1 i i+1 K m =m -! m =m , and the left vertical map may be identified with the isomorphism ~f K mi=mi+1. * *Hence, if =mi ! Map c(S, A=mi) is an isomorphism, so is =mi+1 ! Map c(S, A=mi+1). References [1] A. K. Bousfield, D. M. Kan, Homotopy Limits, Completions and Localizations,* * 2nd corrected edition, in: Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, New York, 1987. [2] R. L. Cohen, Pseudo-isotopies, K-theory, and homotopy theory, in: E. Rees, * *J. D. S. 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