Recognizing Hopf algebroids defined by a group action
Ethan S. Devinatz*
Department of Mathematics, University of Washington,
Box 354350, Seattle, Washington 98195, USA
phone: 12066854777
fax: 12065430397
email: 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 Ralgebra w*
*ith maximal
ideal m and that S is a profinite group with identity e acting continuously on *
*A (with the
madictopology) via Ralgebra homomorphisms. Then define = Map c(S, A), the a*
*lgebra
of continuous functions from S to A, and give the madicfiltration obtained b*
*y regarding
as an Aalgebra 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 madic
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) = s1a
ffl(f)= f(e)
c(f)(s) = s1(f(s1)).
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) = s12(f1(s1)) . f2(s2)
__
is an isomorphism. Then define = oe1 . , 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 madictopology. With some addi*
*tional
assumptions on the map R ! A, we give necessary and sufficient conditions on *
*as a right
Aalgebra 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, . .,.un1]][u, u1],*
* 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 Ktheory 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 padicintegers Zp *
*is isomorphic
to the complete Hopf algebroid determined by this action. Our main result repr*
*oves this
without using the LubinTate 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 Gmodule.) 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 Ralgebra 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 madictopology. 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 Kalgebra. (For definiteness, if (B, ) is a Hopf algebr*
*oid, we use jR
to provide with a Balgebra structure.)
In such a case, __
S Hom Aalg( , A) ! Hom Kalg( , K)
as profinite groups (see Remark 2).
*
* Q t
Remark 1. If L is any field, a finite Lalgebra B is separable if and only if B*
* = i=1Li as
Lalgebras, 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 madictopology, then Hom Aalg( , A) has a canonical monoid *
*structure.
Indeed, recall that Hom cRalg(A, A), the set of continuous Ralgebra endomorph*
*isms of A, is
the set of objects of a groupoid with morphisms Hom cRalg( , A). Hom Aalg( , *
*A) is therefore
the set of morphisms whose source is the identity map in Hom cRalg(A, A). If f*
* and g are in
Hom Aalg( , 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 Aalg( , A) to be the composition of the morphisms f0 and g in the above*
* groupoid.
More explicitly, the product f * g in Hom Aalg( , 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 Aalg( , A) need not be a group. It is a gro*
*up, however,
if __
Hom Aalg( , A) ! Hom Kalg( , K)
*
* __
is a bijection and K is an algebraic extension of k. To see this, observe that *
*if f : ! K
is a Kalgebra 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 Kalg( , 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 Kalgebras_then Hom Kalg( , K) i*
*s profinite as a
set. Under the conditions of the theorem, Hom Kalg( , 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))(s1), 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 Aalgebra structure.
Proof.We begin with the following recollections. If M is an madically complet*
*e finitely
generated Amodule and N is a submodule of M, then N is complete with the madic
topology, and, by the ArtinRees lemma (see [9; Theorem 15]), this topology agr*
*ees with the
subspace topology. The ArtinRees lemma also implies that M=N is madically com*
*plete_
this in turn implies that any finitely generated Amodule is madically complet*
*e. Next
observe that Map c(S, ?) is exact on the category of finitely generated Amodul*
*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, . .,.xi1) ([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 madically complete Aalgebra 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 Kalgebras. The reduction map
__ __
Hom Aalg( , C) ! Hom Kalg( , C)
*
* __
is then a bijection whenever C is an madically complete Aalgebra, 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
Bbimodule_that is, a module over the Lalgebra Be B L Bop. Note that if B *
*is
commutative, any Bmodule may be regarded as a Bbimodule 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) Lalgebra which is the direct limit *
*of finite
separable Lalgebras. Then HHi(B, M) = 0 for all i > 0 and Bmodules M.
Proof.Write B = limBff, where each Bffis a finite separable Lalgebra. It is w*
*ellknown
!
(andQoriginally proved in [7]) that HHi(Bff, M) = 0 for all i > 0 and Bffbimod*
*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 Amodule( , C) ! Hom Kmodule( , C) is surjective.
Proof of Step 1. It suffices to show that
Hom Amodule( , C=mi+1C) ! Hom Amodule( , 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 Aalg( , C) ! Hom Kalg( , C) is onetoone.
Proof of Step 2. We prove that
Hom Aalg( , C=mi+1C) ! Hom Aalg( , C=miC)
is onetoone for each i 1. __ __
Suppose that h 2 Hom_Aalg(_, 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 Amodule derivations *
*from to
miC=mi+1C. But
__ i i+1 i i+1
DerK( , m C=m C) = Hom Kmodule( _=K, m C=m C) = 0,
so f = g. __ __
Step 3. Hom Aalg( , C) ! Hom Kalg( , C) is surjective.
Proof of Step 3. Again we prove that
Hom Aalg( , C=mi+1C) ! Hom Aalg( , C=miC)
is surjective for each i 1.
Let g : ! C=miC be a map of Aalgebras. By Step 1, there exists an Amodule*
* map
f : ! C=mi+1C lifting g. Then define an Amodule map cf : A ! miC=mi+1C_*
*by_
cf(s t) = f(s)f(t)  f(st). We may_and_will_regard cf as a Kmodule_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 *
*Amodule 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
Amodule lift f of g such that c = cf. Since cf = 0 if and only if f is an Aa*
*lgebra lift,
we have that dg_= 0 if and only if g lifts to an Aalgebra 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 Aalgebra structure. Then the map
S h!Hom Aalg( , A)
given by h(s)(f) = f(s) is a bijection. If S is a profinite group acting contin*
*uously on A via
Ralgebra homomorphisms and Hom Aalg( , 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!HomAalg(Map (Sff, A), A) = Hom Aalg( 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 Aalg( 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 Aalg(Map c(S, A), A) ! limHom Aalg(Map (Sff, A), A)
ff
9
is a bijection. By Lemma 1,
Hom Aalg(Map c(S, A), A)=limHom Aalg(Map c(S, A), A=mj)
j
= limHom Aalg(Map c(S, A)=mjMap c(S, A), A=mj)
j
= limHom Aalg(Map c(S, A=mj), A=mj).
j
But Map c(S, A=mj) = limMap (Sff, A=mj), so
!ff
limHom Aalg(Map c(S, A=mi), A=mj)= limlimHom Aalg(Map (Sff, A=mj), A=mj)
j ffj
= limlimHom Aalg(Map (Sff, A), A=mj)
ffj
= limHom Aalg(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 Ralgebra 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 Kalgebra. The isomorphisms
__
S Hom Aalg( , A) ! Hom Kalg( , K)
follow from Proposition 2 and Lemma 4. *
* __
Conversely, suppose (A, ) satisfies the conditions of i and_ii. Let Sff= Hom*
* Kalg(_ff, K),
and let S be the profinite group (see Remark 2) limHom Kalg( ff, K) = Hom Kal*
*g( , K).

By Proposition 2, S = Hom Aalg( , A).
Now define
f : ! Map (Hom Aalg( , A), A)
by f(t)(h) = h(t). I claim that f is an isomorphism onto Map c(Hom Aalg( , A),*
* A). Assum
ing this claim, define a continuous action of S on A by sa = f(jL(a))(s1). 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 madically complete*
* and therefore,
by Lemma 2, f is the unique algebra homomorphism lifting
f~: __! Map c(Hom Kalg(__, K), K) Map (S, K).
But Map c(S, A) is also madically 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 Kalg( 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
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