OPERATIONS AND CO-OPERATIONS IN MORAVA E-THEORY
MARK HOVEY
Abstract.Let E = En denote the Morava E-theory spectrum, and let be
the Morava stabilizer group of ring spectrum isomorphisms of E. We revis*
*it the
isomorphism ss*LK(n)(E ^ E) ~=C( , E*) of graded formal Hopf algebroids,
and its dual isomorphism E*E ~=E*[[ ]].
Contents
Introduction 1
1. The map 3
2. Pro-free modules 6
3. Ind-'etale algebras 8
3.1. Ind-'etale algebras 9
3.2. Identification of K0E 11
3.3. K0E as an ind-'etale Fpn-algebra 12
4. Geometric points 14
5. Cohomology operations 18
5.1. Duality for E*E 19
5.2. Duality for C(G, R) 21
5.3. The isomorphism 23
6. Hopf algebroid structure 24
6.1. Graded formal Hopf algebroids 24
6.2. E_*E as a graded formal Hopf algebroid 25
6.3. C( , E*) as a graded formal Hopf algebroid 26
6.4. is a map of graded formal Hopf algebroids 29
Appendix A. The topology on 30
References 32
Introduction
Let p denote an integer prime, n denote a nonnegative integer, and let E deno*
*te
the Morava E-theory spectrum Ep,n. This is the Landweber exact ring spectrum
with
E* ~=W Fpn[[u1, . .,.un-1]][u, u-1],
____________
Date: April 22, 2004.
1991 Mathematics Subject Classification. 55N22, 55P42, 57T05.
Key words and phrases. Morava E-theory, Hopf algebroid, Morava stabilizer gr*
*oup, cohomol-
ogy operations, twisted completed group ring.
1
2 MARK HOVEY
where the degree of ui is 0 for allii and the degree of u is 2.nThere is an alg*
*ebra
map BP* -!E* that takes vi to uiup -1for i < n, takes vn to up -1, and takes vi
to 0 for i > n. Morava E-theory is closely related to the Johnson-Wilson theory
E(n); in fact E is a finite free module over the localization LK(n)E(n) of E(n)*
* with
respect to Morava K-theory K(n).
The Morava E-theory spectrum E is very important in algebraic topology. It is
local with respect to K(n), and plays a major role in the structure of the K(n)*
*-local
homotopy category [HS99 ]. Let denote the group of ring spectrum automorphisms
(in the stable homotopy category) of E. Then is a version of the Morava stabi*
*lizer
group [Str00, Proposition 4], and the famous result of Hopkins-Miller [Rez98] s*
*ays
that E is in fact an A1 -ring spectrum and that is isomorphic to the group of
components of the space of A1 self-maps of E.
It is then natural to compute the operations E*E and co-operations E*E of E.
Because E is K(n)-local, it is more natural to look at E_*E = ss*LK(n)(E^E) rat*
*her
than the actual co-operations. This ring turns out to be the completion of E*E *
*at
the maximal ideal m. The answer has then been known to the experts for quite
some time; E_*E is isomorphic to C( , E*), the ring of continuous functions from
the profinite group to E* with its m-adic topology, where m = (p, u1, . .,.un*
*-1) is
the unique homogeneous maximal ideal. Also, E*E is isomorphic to the completed
twisted group ring E*[[ ]].
The first statement of this result seems to have been in an unpublished prepr*
*int of
Hopkins and Ravenel [HR89 ], building on a similar statement for K(n) in Morava*
*'s
seminal paper [Mor85 ]. The first published proof of the co-operation result is*
* due
to Baker in [Bak95 ]; see also [Bak89 ]. A very short treatment is given in [DH*
*04 ,
Proposition 2.2], and a spectrum version of the isomorphism E*E ~=E*[[ ]] appea*
*rs
in Proposition 2.4 of [GHMR03 ]. To this author's eye, at least, these proofs*
* are
condensed and hard to follow. The general assumption is that Morava proved the
result for K(n) in [Mor85 ], and it is just a matter of getting efficiently fro*
*m what
Morava states to the actual result, by lifting using Hensel's lemma. Strickland
approaches the result from a new perspective in [Str00, Theorem 12], as does K.
Johnson in [Joh00]. But the calculation of E_*E is really a side issue in both
papers, so also gets relatively short shrift. There is also an unpublished prep*
*rint of
Daniel Davis [Dav04 ], devoted to the theory of discrete -spectra, that addres*
*ses
the computation of E_*E.
In this paper, the author has tried to present the calculation of E_*E in a f*
*airly
self-contained and conceptual way. Essentially, he has written a proof that he
himself can understand, in the hope that this will also be useful to others. Th*
*ere are,
after all, a great many details to be worked out. The basic idea, following Mor*
*ava,
is that E_*E and C( , E*) should represent the same functor on the category of
rings. However, it is not at all obvious that this is the case. The first reduc*
*tion is to
divide by the maximal ideal m, and to eliminate the grading, reducing us to sho*
*wing
that K0E and C( , Fpn) represent the same functor. Even after identifying as
a profinite topological group with an appropriate version of the Morava stabili*
*zer
group, which itself requires some nontrivial theory of Landweber exact spectra *
*and
profinite groups, this is still_not clear. What is fairly_simple to check is th*
*at K0E
and C( , Fpn) have the same Fp-valued points, where Fp is the algebraic closure*
* of
Fp. The we use the fact that both algebras are ind-'etale over Fpn to complete *
*the
proof.
OPERATIONS AND CO-OPERATIONS IN MORAVA E-THEORY 3
Here is an outline of the paper. Following Strickland [Str00], we first defin*
*e the
map
: E_*E -!C( , E*)
in Section 1. We also show that is profinite and acts continuously on E*. We
then prove, again following Strickland's proof, that both E_*E and C( , E*) are
pro-free E*-modules in Section 2. It follows that to prove is an isomorphism,*
* it
suffices to prove that the map
: K0E -!C( , Fpn)
is an isomorphism, where K = E=m is a variant on Morava K-theory. At this
point, our proof diverges from Strickland's, as we now introduce the notion of *
*ind-
'etale algebras in Section 3. We prove that both K0E and C( , Fpn) are ind-'eta*
*le_
Fpn-algebras. It follows_that is an isomorphism if and only if AlgFpn( , Fp) *
*is
an isomorphism,_where Fp denotes the algebraic closure of Fp. We then prove that
AlgFpn( , Fp) is an isomorphism in Section 4, completing the proof that is an
isomorphism. This requires identification of the group , also done by Strickla*
*nd
in [Str00] but with many details left out.
Having completed the proof that is an isomorphism in Theorem 4.11, we turn
to cohomology operations in Section 5. We define the twisted completed group ri*
*ng
E*[[ ]] and show that the natural map
E*[[ ]] -!E*E
induced by the inclusion of in E0E is an isomorphism. We turn to Hopf algebro*
*id
structure in Section 6, defining the notion of a graded formal Hopf algebroid, *
*show-
ing that both E_*E and C( , E*) carry such structure, and proving that preser*
*ves
it. We close the paper with an appendix where we show that the topology on is
entirely determined by its group structure. This is also proved in [Str00], but*
* again
with details left out.
The author would like to thank Neil Strickland, whose paper [Str00] was the
inspiration for this one. It was also Strickland who first suggested the word "*
*'etale"
to the author, which turned out to be the key idea to making the proof more
conceptual. He would also like to thank Daniel Davis for his careful reading of*
* the
paper and his encouragement.
1. The map
In this section we define the map
: E_*E -!C( , E*),
and we show that is a first countable profinite group. This will require use*
* of
the natural topology, studied in [HS99 , Section 11], on morphisms in the K(n)-
local category. This natural topology is not very complicated, however, and we
will summarize the properties of it that we need. Also, we will frequently use *
*the
notation X^eY to mean LK(n)(X ^ Y ). With this notation, E_*X = ss*(Ee^X).
Our construction of will follow Strickland [Str00, Theorem 12]. As mentioned
above, there is a natural topology on [X, Y ], for X, Y K(n)-local spectra [HS9*
*9 ,
Section 11] (or objects in any algebraic stable homotopy category [HPS97 , Sec-
tion 4.4]). Given any F that is small in the K(n)-local category (this means [F*
*, -]
commutes with coproducts in the K(n)-local category), and any map h: F -! X,
4 MARK HOVEY
we define Uh to be the set of all f 2 [X, Y ] with fh = 0. The Uf define a basis
of neighborhoods of 0 in [X, Y ], and then V [X, Y ] is open if and only if f*
*or all
v 2 V , there is an h such that v + Uh V . Equivalently, V is open if and onl*
*y if
for all v in V there is an h such that fh = vh implies f 2 V .
This natural topology has a number of good properties. First of all, it is st*
*raight-
forward to check that the composition map
[X, Y ] x [Y, Z] -![X, Z]
is continuous (see [HPS97 , Proposition 4.4.1(a)]). The smash product map
[X, Y ] x [Z, W ] -![X^eZ, Y e^W ]
is also continuous [HS99 , Proposition 11.3], but it is slightly harder to chec*
*k this. In
good cases, the natural topology is the obvious one. For example, the natural t*
*opol-
ogy on E*X and E_*X is the m-adic topology when K(n)*X is finite-dimensional
(that is, when X is strongly dualizable in the K(n)-local category). This is pr*
*oved
in [HS99 , Proposition 11.9] for a different version of E, but the proof is the*
* same
for our E.
There is a map
oe : x E_*E -!E*
defined by letting oe(fl, a) be the composite
e^fl ~
Sm -a!Ee^E 1--!Ee^E -! E.
Now E_*E, E*, and [E, E] are sets of maps in the K(n)-local category, so th*
*ey
have a natural topology as described above. The map oe is built from composition
maps and the map
[E, E] -![Ee^E, Ee^E]
that takes fl to 1e^fl. By the remarks above on the natural topology, we see th*
*at oe
is continuous. Its adjoint is therefore a map
: E_*E -!C( , E*),
where continuity is determined by the natural topologies. It is a straightforwa*
*rd
diagram chase to verify that is a map of E*-modules. Using the fact that
consists of ring spectrum automorphisms, one can also check by a diagram chase
that is a map of E*-algebras.
We will of course need to determine the topological group . For the moment,
however, we content ourselves with some basic structural results.
Proposition 1.1. The set [E, E] is a first countable compact Hausdorff topologi*
*cal
ring in the natural topology, and the topology is defined by a set of open left*
* ideals.
Furthermore, the action of [E, E] on [X, E] is continuous for all K(n)-local sp*
*ectra
X.
Proof.The group [X, Y ] is always a topological group in the natural topology. *
*Since
the multiplication in [E, E] is given by composition, it is continuous, and so *
*[E, E]
is a topological ring. The same argument shows that the action of [E, E] on [X,*
* E]
is continuous. Now suppose F is small in the K(n)-local category. The prototypi*
*cal
F would be LnV (n - 1), if it existed. This spectrum would have
E*(LnV (n - 1)) ~=E*=m,
OPERATIONS AND CO-OPERATIONS IN MORAVA E-THEORY 5
which is obviously finite in each degree. One of the main themes of [HS99 ] is *
*that
every small object in the K(n)-local category behaves as if it were built from *
*finitely
many copies of the putative spectrum LnV (n - 1). In particular, the argument
of [HS99 , Theorem 8.6] shows that E*F is finite in each degree. Then [HS99 ,
Proposition 11.5] implies that [E, E] is profinite, and hence compact Hausdorff.
To see that [E, E] is first countable, note that there are only countably many
(isomorphism classes of) small objects in the K(n)-local category by the results
of [HS99 , Section 8]. Given a small object F , we have just seen that [F, E] i*
*s finite.
Hence there are only countably many basic neighborhoods Uh of 0, so [E, E] is f*
*irst
countable. It is clear that the Uh are left ideals.
Corollary 1.2. The group is a first countable Hausdorff topological group who*
*se
topology is defined by a set of open subgroups. Furthermore, the action of on*
* E*
is continuous when E* is given the m-adic topology.
Proof.In general, if A is a topological ring defined by a set of open left idea*
*ls, then
Ax is a topological group defined by a set of open subgroups. Indeed, if U is an
open left ideal in A, then one can check that V = (e + U) \ Ax is an open subgr*
*oup
in Ax . If {Ui} defines the topology on A, then the corresponding set {Vi} defi*
*nes
the subspace topology on Ax . It is clear that the multiplication map is contin*
*uous;
to see that the inverse map is continuous, note that (x(x-1V x))-1 = x-1V . Our
group is a subgroup of [E, E]x , so is also a first countable Hausdorff topol*
*ogical
group whose topology is defined by a set of open subgroups.
Proposition 1.1 shows that the action of on E* is continuous when E* is giv*
*en
the natural topology, and we have seen already that the natural topology on E* *
*is
the m-adic topology.
We want to conclude that is profinite. For this, we need to know that
is compact, according to [DdSMS99 , Definition 1.1]. Since [E, E] is compact,*
* it
suffices to show that is closed in [E, E].
Lemma 1.3. Suppose A is a compact Hausdorff topological ring. Then Ax is closed
in A.
Proof.Note that Ax = B \ C, where B is the set of all elements with a right
inverse and C is the set of all elements with a left inverse. We will show that*
* B is
closed; a similar proof shows that C is closed as well. Let ~: AxA -!A denote t*
*he
multiplication map, and let D denote ~-1(1), which is closed since A is Hausdor*
*ff.
Then B = ss1D, but ss1 is a map from the compact space A x A to the Hausdorff
space A, so is a closed mapping.
Theorem 1.4. The group is a first countable profinite group.
Proof.In view of Corollary 1.2 and Lemma 1.3, it suffices to show that the set *
*S of
ring spectrum maps in [E, E] is closed. Note that S = S1 \ S2, where S1 is the *
*set
of maps compatible with the multiplication ~ and S2 is the set of maps compatib*
*le
with the unit j. Certainly S2 is closed, since it is the inverse image of j und*
*er the
continuous composition
[E, E] -![S, E] x [E, E] O-![S, E].
6 MARK HOVEY
Here the first map sends f to (j, f), and the second map is composition. On the
other hand, S1 is the equalizer of the maps
*
[E, E] ~-![E ^ E, E]
and
[E, E] -! [E, E] x [E, E] ^-![E ^ E, E ^ E] ~*-![E ^ E, E].
The maps ~* and ~* are continuous since they are just composition, and the map
^ is continuous as well ([HS99 , Proposition 11.3]). Proposition 11.5 of [HS99 *
*] tells
us that [E ^ E, E] is Hausdorff, and the equalizer of two continuous maps into a
Hausdorff space is always closed.
2.Pro-free modules
In this section, we show that both E_*E and C( , E*) are pro-free E*-modules.
This allows us to conclude that is an isomorphism if and only if the induced *
*map
: K0E -!C( , Fpn)
is an isomorphism. This section will use the results on pro-free modules from [*
*HS99 ,
Appendix A]. That appendix is completely algebraic and does not depend on the
body of [HS99 ].
Recall that, if R is a complete local ring, an R-module is called pro-free if*
* it is
the completion of a free module. Our particular ring E* is graded, so when we t*
*ake
the completion of a graded module we must do so in the graded sense. However, a*
*ll
our E*-modules will be evenly graded, and since E* has a unit in degree 2, ther*
*e is
an equivalence of categories between evenly graded E*-modules and E0-modules.
This equivalence takes an evenly graded E*-module M to M0, and an E0-module
N to N[u, u-1], with N in degree 0.
We need to show that the E*-modules we deal with are pro-free. For this, we
recall the spectrum K = E=m. This spectrum can be made into a ring spectrum
by using, for example, the theory of bordism with singularities or its modern r*
*e-
placement [EKMM97 , Chapter V]. It is in fact a field spectrum, and is additi*
*vely
isomorphic to a wedge of suspensions of K(n). Note that K* ~=Fpn[u, u-1].
The following lemma is very useful.
Lemma 2.1. If K*X is concentrated in even degrees, then E*X and E_*X are pro-
free E*-modules. Furthermore, in this case K*X ~=(E_*X)=m and K*X ~=E*X=m.
Proof.This lemma is proven, for a slightly different spectrum E, in Proposition*
*s 2.5
and 8.4(f) of [HS99 ]. Let G(k) denote the theory E=(p, u1, . .,.uk-1). There i*
*s an
obvious exact sequence relating G(k) to G(k + 1). Using these exact sequences, *
*one
can work back from K = G(n) to E = G(0). In so doing, one sees that G(k)_*X is
evenly graded and that G(k + 1)_*(X) ~=G(k)_*(X)=uk. Hence K*X ~=(E_*X)=m.
It also follows that (p, u1, . .,.un-1) is a regular sequence on E_*X. Theorem *
*A.9
of [HS99 ] then guarantees that E_*X is pro-free. The same method works for E*X
as well.
Proposition 2.2. The E*-module E_*E is pro-free and concentrated in even di-
mensions.
OPERATIONS AND CO-OPERATIONS IN MORAVA E-THEORY 7
Proof.By Lemma 2.1, it suffices to check that K*E is concentrated in even degre*
*es.
Landweber exactness of E implies that K*E ~=K*BP BP* E*, but the Atiyah-
Hirzebruch spectral sequence implies that K*BP ~=K*[t1, t2, . .].with |ti| = 2(*
*pi-
1) as in [Rav86 , Lemma 4.1.7].
To show that C( , E*) is pro-free will require more work. We begin by analyzi*
*ng
C(G, R) when G is profinite and R is discrete.
Lemma 2.3. Suppose G = limG=U is a profinite group and R is a discrete com-
mutative ring. Then the natural map of rings ae: colimUF (G=U, R) -!C(G, R) is
an isomorphism, where U runs through the open normal subgroups of G,
This lemma works when R is graded as well.
Proof.The map ae sends a map G=U -! R to the composite G -! G=U -! R.
Since the reduction maps are surjective, ae is injective. Now suppose f :G -! R
is continuous. For each x 2 G, choose an open normal subgroup Nx such that
f(xNx) = f(x). The xNx form an open cover of G, so there is a finite subcover
{x1Nx1, . .,.xkNxk}. Let N = Nx1\ . .\.Nxk. We claim that f is induced by a
function G=N -! Z. Indeed, suppose x and y are congruent modulo N. Now x
must lie in some xiNxi, so x is congruent to xi modulo Nxi. It follows that y is
also congruent to xi modulo Nxi, and so f(x) = f(xi) = f(y), as required.
In fact, the dependence of C(G, R) on R is very simple.
Proposition 2.4. Suppose G is a profinite group and R is a discrete commutative
ring. Then the natural map oe :R C(G, Z) -! C(G, R) defined by oe(r, f)(g) =
f(g)r is an isomorphism.
Again, this proposition holds if R is graded as well, where C(G, Z) is though*
*t of
as a graded ring concentrated in degree 0.
Proof.When G is finite, the proposition is clear since both sides are free R-mo*
*dules
of rank |G|, and oe takes the basis {1 g*} to the basis {g*}, where g* is the f*
*unction
that takes g to 1 and everything else to 0. When G = lim(G=U) is profinite, we
take the direct limits of the isomorphisms for G=U and use Lemma 2.3.
We need a version of Proposition 2.4 when R is complete in the a-adic topolog*
*y,
for some ideal a. In this case, if S is a discrete ring such as C(G, Z), we def*
*ine Rb S
to be the completion of R S with respect to the image of a S in R S.
Proposition 2.5. Suppose G is a profinite group and R is a commutative ring
that is complete in the a-adic topology for some ideal a. Then there is a natu*
*ral
isomorphism
~=
Rb C(G, Z) -! C(G, R).
As usual, this isomorphism will work in the graded case as well, as long as a*
* is
homogeneous.
Proof.Take the inverse limit of the isomorphisms
oe :R=ai C(G, Z) -!C(G, R=ai),
of Proposition 2.4.
We can now prove that C( , E*) is pro-free.
8 MARK HOVEY
Theorem 2.6. Suppose G is a first countable profinite group and R is a commu-
tative ring that is complete in the a-adic topology for some ideal a. Then C(G,*
* R)
is pro-free as an R-module.
This theorem holds in either the graded or ungraded case.
Proof.Proposition 2.5 tells us that C(G, R) ~=Rb C(G, Z). Hence it suffices to
show that C(G, Z) is a free abelian group. But Lemma 2.3 tells us that
C(G, Z) ~=colimUF (G=U, Z).
Since G is first countable, we can make this colimit run over a chain
. . .Uk . . .U0.
Each group F (G=Uk, Z) is a finitely generated free abelian group, and each map
F (G=Uk, Z) -! F (G=Uk+1, Z) is a split monomorphism. Hence C(G, Z) is a free
abelian group.
We then have the following general lemma about maps of pro-free modules.
Lemma 2.7. Suppose R is a Noetherian regular complete local ring with maximal
ideal m. A map f :M -! N of pro-free modules is an isomorphism if and only
R=m R f is an isomorphism.
We are of course interested in this lemma when R is graded, but the proof is *
*the
same in any case.
Proof.It is proved in [HS99 , Proposition A.13] that if R=m R f is a monomorphi*
*sm,
then f is a split monomorphism. The cokernel C of f is then a summand in the
complete module N, but C=mC = 0. It follows that C = 0.
We then have the following theorem.
Theorem 2.8. The map : E_*E -! C( , E*) is an isomorphism if and only if
the map : K0E -!C( , Fpn) induced by is an isomorphism.
Proof.Since E_*E and C( , E*) are pro-free by Proposition 2.2 and Theorem 2.6,
Lemma 2.7 implies that is an isomorphism if and only =m is an isomorphism.
Lemma 2.1 then tells us that E_*E=m ~=K*E. Proposition 2.5 and Proposition 2.4
imply that
C( , E*)=m ~=(E*b C( , Z))=m
~=(E* C( , Z))=m ~=(E*=m) C( , Z) ~=C( , K*).
Now K*E is concentrated in even dimensions by Proposition 2.2, as is C( , K*).
Since both contain the unit u in degree 2, =m is an isomorphism if and only if*
* the
degree 0 part of =m is an isomorphism.
3.Ind-'etale algebras
The object of this section is to show that
: K0E -!C( , Fpn)
___
is_an isomorphism if and only if AlgFpn( , Fp) is an isomorphism of sets, where
Fp is the algebraic closure of Fp, and AlgFpn is the category of Fpn-algebras. *
*We
will accomplish this by showing that both K0E and C( , Fpn) are ind-'etale Fpn-
algebras, and proving a general result about maps between such algebras.
OPERATIONS AND CO-OPERATIONS IN MORAVA E-THEORY 9
3.1. Ind-'etale algebras. We first develop the algebraic theory of ind-'etale a*
*lge-
bras. For a general commutative ring k, a finitely presented k-algebra is call*
*ed
'etale if, whenever N is a square 0 ideal in a k-algebra C, the natural map
Algk(A, C) -!Alg k(A, C=N)
is an isomorphism. A is called smooth if this map is always surjective, and unr*
*am-
ified if it is always injective. These definitions are taken from [DG70 , Secti*
*on I.4.3],
where of course one works with arbitrary schemes.
If k is a field, the category of 'etale k-schemes is equivalent to the catego*
*ry of G-
sets, where G is the Galois group of the separable closure ks over k [DG70 , I.*
*4.6.4].
Note that G is profinite, and so a G-set means a set with a continuous action o*
*f G.
The equivalence takes X to the set X(ks) of its ks-valued points. The equivalen*
*ce
works because if X is an 'etale k-scheme, then X k ks is constant [DG70 , I.4.*
*6.2].
Now, if A is an 'etale algebra, then SpecA kks is both constant and affine, so *
*must
be a finite constant scheme. Thus we have the following well-known proposition.
See Proposition 18.3 and Theorem 18.4 of [KMRT98 ].
Proposition 3.1. Suppose k is a field. A k-algebra A is 'etale if and only if A*
* kks
is isomorphic to a finite product of copies of ks. Moreover, the category of '*
*etale
k-algebras is anti-equivalent to the category of finite G-sets, where G denotes*
* the
Galois group of ks=k.
Now, the algebras we have to deal with are definitely not finitely presented,*
* so
we make the following definition.
Definition 3.2. If k is a commutative ring, we define a k-algebra A to be ind-'*
*etale
if A is a filtered colimit of 'etale k-algebras.
The main reason we introduce this definition is the following lemma.
Lemma 3.3. If k is a field and G is a profinite group, then C(G, k) is an ind-'*
*etale
k-algebra.
Proof.Lemma 2.3 tells us that C(G, k) ~=colimF (G=U, k), where U runs through
the open normal subgroups of G. Each F (G=U, k) is obviously 'etale, since it i*
*s a
product of copies of k even before tensoring with ks.
The terminology "ind-'etale" is justified by the following proposition.
Proposition 3.4. For a commutative ring k, the category of ind-'etale k-algebras
is equivalent to the category of ind-objects in the category of 'etale k-algebr*
*as.
Here, given a category C, the category of ind-objects in C is the category of*
* all
functors F :I -!C, where I can be any filtered small category. The morphisms in
this category from F to G are defined to be limffcolimfiC(F (ff), G(fi)). See [*
*SGA4 ,
Section I.8.2].
Proof.The colimit is an obvious functor from ind-objects to ind-'etale k-algebr*
*as.
It is essentially surjective by definition. To see that it is fully faithful, w*
*e compute
Alg k(colimAff, colimBfi) ~=limffAlgk(Aff, colimBfi)
~=limffcolimfiAlgk(Aff, Bfi),
using the fact that Affis finitely presented as a k-algebra.
10 MARK HOVEY
It follows from Proposition 3.1 that, if k is a field, the category of ind-'e*
*tale k-
algebras is anti-equivalent to the category of pro-objects in the category of f*
*inite
G-sets.
Proposition 3.5. Let G be a profinite group. The limit functor from the category
of pro-objects in finite G-sets to profinite G-sets is an equivalence of catego*
*ries.
The morphisms in the category of profinite G-sets are continuous equivariant
maps from X to Y , denoted CG (X, Y ).
Proof.The limit functor is essentially surjective by definition. To determine w*
*hether
it is fully faithful, we compute
CG (limXff, limYfi) ~=limfiCG (limXff, Yfi).
There is a canonical map
oe :colimffCG (Xff, Yfi) -!CG (limXff, Yfi).
We claim that this map is an isomorphism. To see that oe is surjective, note th*
*at
a continuous G-map f : limXff-! Yfiis determined by a partition of limXffinto
open and closed sets Uy = f-1 (y) for y 2 Yfi, with gUy Ugy. A basis for the
topology on limXffis given by the sets ss-1fl(Vfl), where ssfl:limXff-! Xflis t*
*he
evident map and Vflis an arbitrary subset of Xfl. Because our indexing category*
* is
filtered, this basis is closed under finite intersections and finite unions. Si*
*nce limXff
is compact, this means that a set that is both open and closed must be of the f*
*orm
ss-1fl(Vfl). Again using the fact that the indexing category is filtered, we co*
*nclude
that there is a fl and subsets Vy in Xflsuch that Uy = ss-1fl(Vy) for all y 2 Y*
*fi. It
follows that f is in the image of CG (Xfl, Yfi), and hence that oe is surjectiv*
*e. This
implies that the inverse limit functor is full.
To see that oe is injective, we use an argument we learned from Neil Strickla*
*nd.
Suppose we have two maps f :Xff-!Yfiand f0: Xff0-!Yfiwith f O ssff= f0O ssff0.
Since the diagram {Xfl} is filtered, we can assume ff = ff0. Choose an x 2 Xffw*
*ith
f(x) 6= f0(x). Let I denote the indexing category of the pro-object X, and let C
denote the category of all pairs (fl, i) where i: fl -!ff is a morphism in I. T*
*hen C is
itself a filtered category, and we can define a functor from C to finite sets b*
*y taking
(fl, i) to (X(i))-1(x). The inverse limit of this functor is ss-1ff(x), which m*
*ust be
empty since f Ossff= f0Ossff. By [DdSMS99 , Proposition 1.1.4], it follows tha*
*t there
exists an i: fl -! ff such that (X(i))-1(x) = ;. Since there are only finitely *
*many
elements x and I is filtered, it follows that there exists a morphism j :ffi -!*
*ff such
that (X(j))-1(x) is empty for all x with f(x) 6= f0(x). Hence f O X(j) = f0O X(*
*j),
and so f and f0 represent the same element of colimffCG (Xff, Yfi).
We reach the following conclusion.
Theorem 3.6. Suppose k is a field with separable closure ks. Let G be the Galois
group of ks over k. The functor that takes a k-algebra A to Algk(A, ks) defines*
* an
anti-equivalence of categories between ind-'etale k-algebras and profinite G-se*
*ts. In
particular, if f :A -!B is a map of ind-'etale k-algebras, then f is an isomorp*
*hism
if and only if the induced map
Algk(B, ks) -!Alg k(A, ks)
is an isomorphism.
OPERATIONS AND CO-OPERATIONS IN MORAVA E-THEORY 11
3.2. Identification of K0E. We have seen already in Lemma 3.3 that C( , Fpn) is
an ind-'etale Fpn-algebra. To show that K0E is an ind-'etale Fpn-algebra will r*
*equire
considerably more work. The goal of this section is to show that
K0E ~=Fpn V V T [t0, t-10] V Fpn
where V is BP* thought of as an ungraded ring, and V T is BP*BP thought of as
an ungraded ring. Thus (V, V T [t0, t-10]) is the Hopf algebroid that represent*
*s the
groupoid of p-typical formal group laws and arbitrary isomorphisms between them
(see [Rav86 , p. 365]).
We begin by defining R* = BP*[u, u-1], where u has degree 2. Then R* is a
free BP*-algebra, so is obviously Landweber exact. Hence we get a commutative
ring spectrum R, and an isomorphism R*X ~=R* BP* BP*X. This is easy to see,
but see [HS99 , Section 2] for details on Landweber exact spectra. In particul*
*ar,
R*R ~=R* BP* BP*BP BP* R*.
Lemma 3.7. There is an isomorphism of Hopf algebroids
o :(V [u, u-1], V T [t0, t-10][u, u-1]) ~=(R*, R*R).
Proof.A map of graded rings from R* to S is a pair (F (x, y), u) where F (x, y)*
* is
a homogeneous p-typicalPformal group law over S and u is a unit in S2. To say
that F (x, y) = i,jaijxiyj is homogeneous means that each aijis homogeneous of
degree 2(i + j - 1). On the other hand, a map of graded rings from V [u, u-1] to
S is a pair (F0(x, y), y) where F0 is p-typical formal group law over S0 and u *
*is a
unit in S2. Given (F (x, y), u) we define
X
F0(x, y) = aiju1-i-jxiyj = uF (u-1x, u-1y).
i,j
One can readily see from [Rav86 , Lemma A2.1.26] that F0 is p-typical. Con-
versely, given (F0(x, y), u) we define F (x, y) = u-1F0(ux, uy), which is homog*
*e-
neous and p-typical. This one-to-one correspondence gives us the desired isomor-
phism V [u, u-1] ~=R*.
Similarly, a map of graded rings from R*R to S is a quintuple
(F (x, y), F 0(x, y), u, v, OE),
where F (x, y) and F 0(x, y) are homogeneous p-typical formal group laws over S,
u, v are units in S2, and OEPis a homogeneous strict isomorphism from F to F 0,
in the sense that OE(x) = bixi with bi homogeneous of degree 2i - 2. On the
other hand, a map of graded rings from V T [t0, t-10][u, u-1] to S is a quadrup*
*le
(F0(x, y), F00(x, y), u, _), where F0 and F00are formal group laws over S0, u i*
*s a unit
in degree 2, and _ is an arbitrary isomorphism from F0 to F00. Given (F, F 0, u*
*, v, OE),
we define F0(x, y) = uF (u-1x, u-1y) and F00(x, y) = vF 0(v-1x, v-1y) as above.*
* We
define _(x) = vOE(u-1x). The reader can check that _ is defined over S0, and has
leading term vu-1x. We have
F00(_x, _y) = vF 0(v-1_x, v-1_y) = vF 0(OE(u-1x), OE(u-1y))
= vOEF (u-1x, u-1y) = vOE(u-1F0(x, y)) = _F0(x, y).
Hence _ is an isomorphism from F0 to F00. Conversely, given (F0, F00, u, _). we*
* let
w be the leading coefficient (d_(x)=dx)(0) of _ and define
v = uw, F (x, y) = u-1F (ux, uy), F 0= v-1F00(vx, vy), and OE(x) = v-1_(ux).
12 MARK HOVEY
We leave to the reader the check that OE(x) is a homogeneous strict isomorphism,
and that these constructions are inverse to one another.
We then have the following proposition.
Proposition 3.8. E0E ~=E0 V V T [t0, t-10] V E0.
Note that E0 is a V -algebra via the map that takes vi to ui for 0 i < n, t*
*akes
vn to 1, and takes vi to 0 for i > n.
Proof.The map BP* -!E* used to build E*(-) factors through R*. Using Landwe-
ber exactness of E, we conclude that
E*E ~=E* R* R*R R* E*.
Using Lemma 3.7, we conclude that
E0E[u, u-1] ~=E*E ~=E0[u, u-1] V [u,u-1]V T [t0, t-10][u, u-1] V [u,u-1]E0[u,*
* u-1]
~=(E0 V V T [t0, t-10] V E0)[u, u-1].
The proposition follows.
Corollary 3.9. K0E ~=Fpn V V T [t0, t-10] V Fpn.
In this corollary, the map V -! Fpn sends all vi to 0 except vn, which goes t*
*o 1.
Proof.Because E is Landweber exact, E*E is a flat E*-module. In particular, by
tensoring the sequences
0 -!E*=(p, u1, . .,.ui-1) ui-!E*=(p, u1, . .,.ui-1) -!E*=(p, u1, . .,.ui) -!0
with E*E, we see that (p, u1, . .,.un-1) is a regular sequence on E*E. Hence
K*E ~=E*E=m, and so K0E ~=E0E=m. Since In = (p, v1, . .,.vn-1) is an invariant
ideal in V T , the corollary follows from Proposition 3.8.
3.3. K0E as an ind-'etale Fpn-algebra. Having identified K0E, we can begin the
process of proving that K0E is an ind-'etale Fpn-algebra. We begin by identifyi*
*ng the
representing scheme for automorphisms of the Honda formal group law Fn. Recall
that this is the p-typical formal group law over an Fp-algebra S whose classify*
*ingn
map takes vi to 0 for all i 6= n and takes vn to 1. Its p-series is [p]Fn(x) = *
*xp .
Proposition 3.10. Let S be an Fp-algebra. A power series _(x) 2 S[[x]] is an
X1 F
n pi
automorphism of the Honda formal group law Fn if and only if _-1(x) = tix
n i=0
where tpi = ti for all i and t0 is a unit.
The only if half of this proposition is proved in the proof of [Str00, Theore*
*m 12],
and the if half is stated there.
Proof.Suppose first that _(x) is an automorphism of Fn. Since Fn is p-typical,
X1 F
n pi -1
Lemma A2.1.26 of [Rav86 ] implies that _-1(x) = tix . Since _ (x) is also
i=0
an automorphism of Fn, it commutes with the p-series. We conclude that
X1 F
n pn pi -1 pn
ti(x ) = (_ (x)) ,
i=0
OPERATIONS AND CO-OPERATIONS IN MORAVA E-THEORY 13
so 1 1
X Fn i+n X Fn pn i+n
tixp = ti xp ,
n i=0 i=0
Hence tpi = ti for all i.
X1 F
n pi pn
Conversely, suppose _-1(x) = tix , ti = ti for all i, and t0 is a unit.
i=0
Define F (x, y) = _Fn(_-1x, _-1y). Then F is a formal group law, and _ is an is*
*o-
morphism from Fn to F . Furthermore, F is p-typical by [Rav86 , Lemma A2.1.26].
X1 F
j
Thus [p]F (x) = sjxp for some elements sj of S. Note that sj is the image of
j=1
vj under the classifying map of F , so it suffices to show that sj = 0 for all *
*j except
n, and that sn = 1.
We do this by comparing the p-series. On the one hand, we have
1XF 1 1 1
j X Fn -1 pj X Fn X Fn pi pi+j
_-1[p]F (x) = _-1( sjxp ) = _ (sjx ) = tisj x .
j=1 j=1 i=0 j=1
On the other hand, we have
X1 F
n pi+n
_-1[p]F (x) = [p]Fn(_-1x) = tix ,
i=0
n
using the fact that tpi = ti. Looking at the smallest power of x that occurs, we
conclude that t0s1 = 0 (if n > 1), so s1 = 0 since t0 is a unit. Continuing in *
*this
fashion we see that sj = 0 for j < n and sn = 1. Hence
X1 F 1 1 1
n pi+n X FnX Fn pi pi+j X Fn pi+n
tix +Fn tisj x = tix .
i=0 i=0 j=n+1 i=0
Cancelling and again recursively looking at the smallest power of x that occurs*
*, we
find that sj = 0 for all j > n as well.
Corollary 3.11. The representing ringnfor automorphismsnof the Honda formal
group law Fn is Fp[t0, t1, . .].=(tp0-1- 1, tp1 - t1, . .)..
We then get the following description of K0E, also obtained by Strickland in *
*the
proof of [Str00, Theorem 12].
n-1 pn
Proposition 3.12. K0E ~=Fpn[t0, t1, . .].=(tp0 - 1, t1 - t1, . .). Fpn.
Proof.By Corollary 3.9, we have
K0E ~=Fpn V V T [t0, t-10] V Fpn
where the map V -! Fpn takes ui to 0 for all i 6= n and takes un to 1. This ring
represents the functor that assigns to a ring R the set of triples (r, s, _), w*
*here
r, s: Fpn -!R are ring homomorphisms and _ is an isomorphism of formal groups
from r*Fn to s*Fn, where Fn is the Honda formal group law classified by the map
V -!Fpn. In fact, Fn is actually defined over Fp, so r*Fn = s*Fn = Fn. We
conclude from Corollary 3.11 that
n-1 pn
K0E ~=Fpn Fp[t0, t1, . .].=(tp0 - 1, t1 - t1, . .). Fpn,
14 MARK HOVEY
which proves the proposition.
Theorem 3.13. K0E is an ind-'etale Fpn-algebra.
Proof.Define
n-1 pn pn
R0k= Fpn[t0, t1, . .,.tk]=(tp0 - 1, t1 - t1, . .,.tk - tk)
and define Rk = R0k Fpn = R0k Fpn(Fpn Fpn). It is clear from Proposition 3.12
that K0E ~=colimRk as Fpn-algebras. Thus it suffices to show that Rk is an 'eta*
*le
Fpn-algebra. Since
Alg Fpn(Rk, -) ~=AlgFpn(R0k, -) x AlgFpn(Fpn Fpn, -),
it suffices to show that both R0kand Fpn Fpn are 'etale. It is easy to check *
*that
Fpn Fpn is an 'etale Fpn-algebra. Indeed, an Fpn-algebra map from Fpn Fpn
to A is the same thing as a ring map Fpn -!A. Now suppose N is a square-zeron
ideal in A.n If f, gn:Fpn -! A become equal in A=N,nthen (f(x) - g(x))p = 0,
and so f(xp ) =_g(xp_) for all x 2 Fpn. Since xp = x, we conclude that f = g.
Conversely,_if f: Fpn -!A=N is a ring map, we define f(x)nfor x 2 Fpn as follow*
*s.
Write f(x) = z + N for some z 2 A, and define f(x) = zp , One can readily verif*
*y __
that this is independent of the choice of z, and is therefore a ring map that l*
*ifts f.
The proof that R0kis 'etale is very similar. If f, g :R0k-! A become equalnin
A=N, then again f(xpn) = g(xpn) for all x 2 R0k. In particular,_since tpi = ti,*
* we
see that f(ti) =_g(ti)_for all i, and so f = g. Also, given f: R0k-!nA=N, choos*
*e_
yi2 A such that f(ti)n= yi+ N. Then definenf :R0k-!Anby f(ti) = ypi.nSince f
is a ring map, ypi - yi2 N. Hence (ypi)pn = ypi. Similarly, since yp0-1 - 1 2 N,
n+1-p __
we see that yp0 = 1, and so y0 is a unit. Hence f is a ring map lifting f.
By combining Theorem 2.8 with Theorem 3.6, Lemma 3.3, and Theorem 3.13,
we get the following theorem.
Theorem 3.14._The map : E_*E -! C( , E*) is an isomorphism if and only if
AlgFpn( , Fp) is an isomorphism, where : K0E -!C( , Fpn) is induced by .
4.Geometric points
In this section, we prove that
: E_*E -!C( , E*)
is an isomorphism by calculating the effect of
: K0E -!C( , Fpn)
___
on geometric points; that is, on Fpn-algebra homomorphisms to Fp.
___
Proposition 4.1. Alg Fpn(K0E, Fp) is isomorphic to the set of pairs (ff, fi), w*
*here
ff is an automorphism of the Honda formal group law Fn defined over Fpn, and fi
is an element of the Galois group of Fpn over Fp.
Proof.By_Proposition_3.12 and Corollary 3.11, an Fpn-algebra homomorphism_from
K0E to Fp is equivalent to_an_automorphism of Fn defined_over_Fp together with
a ring homomorphism Fpn -!Fp. A ring homomorphism Fpn -!Fp is equivalent to
an_element of the Galois group of Fpn over Fp. Also, any endomorphism of Fn over
Fp is in fact defined over Fpn by [Rav86 , Theorem A2.2.17].
OPERATIONS AND CO-OPERATIONS IN MORAVA E-THEORY 15
___
Proposition 4.2. Alg Fpn(C( , Fpn), Fp) is isomorphic to . The isomorphism
takes x 2 to the map that evaluates at x.
Proof.In view of Lemma 2.3, we have
___ ___
Alg Fpn(C( , Fpn), Fp) ~=limUAlg Fpn(F ( =U, Fpn), Fp),
as U runs over the open normal subgroups of . But F ( =U, Fpn) is generated as
an Fpn-algebra by the orthogonal idempotents x* for x 2 =U, wherePx* is the
function that takes x to 1 and_everything else to 0. Note that x2 =Ux* = 1.
There are no idempotents in Fp except 0 and 1. It follows easily that
___
AlgFpn(F ( =U, Fpn), Fp) ~= =U
where the isomorphism takes x 2 =U to the map that evaluates at x. Taking
inverse limits gives the desired result.
Thus to proceed any further, we must identify the group . Strickland does th*
*is
in [Str00, Proposition 4], but we will fill in many missing details.
We first prove some preliminary results about Landweber exact spectra. Some
of this will require some of the results of [HS99 , Section 2.1], where the aut*
*hors
study Landweber exact spectra whose homotopy is evenly graded. Recall that, if
(A, ) is a Hopf algebroid and M and N are -comodules, then M ^ N denotes the
comodule which is isomorphic to the tensor product of the left A-modules M and
N, with the diagonal coaction of .
Lemma 4.3. Suppose F and G are evenly graded Landweber exact spectra. Then
the natural map MU*F ^ MU*G -! MU*(F ^ G) of MU*MU-comodules is an
isomorphism.
Proof.Any evenly graded Landweber exact spectrum is a minimal weak colimit of
finite spectra that have only even-dimensional cells, by [HS99 , Proposition 2.*
*12].
Such a spectrum is called evenly generated in [HS99 , Section 2.1]. The lemma at
hand is true for any evenly generated spectra F and G. Indeed, the lemma is cle*
*ar
for finite spectra with only even-dimensional cells, and then one takes a suita*
*ble
colimit.
Lemma 4.4. Suppose F and G are Landweber exact spectra. Then F ^ G is also
Landweber exact.
Proof.Landweber exactness implies that
(F ^ G)* ~=F* MU* MU*MU MU* G*
as MU*-modules. Now let (A, ) be any Hopf algebroid. An A-module B is called
Landweber exact if B A (-) takes exact sequences of -comodules to exact se-
quences of A-modules. This is equivalent to B A being flat as a right A-modu*
*le,
by [HS03 , Lemma 2.2]. Now, if B and C are Landweber exact, then B A A C
is obviously Landweber exact since
B A A C A ~=(B A ) A (C A )
is flat, as the tensor product of two flat modules.
16 MARK HOVEY
Proposition 4.5. Suppose F and G are Landweber exact, evenly graded, commu-
tative MU-algebra spectra. Let OEF :MU*F -! F* denote the map induced by the
action of MU on F , and let jF :F* -!MU*F denote the map induced by the unit
of MU.
(1) The map [F, G] -!Hom MU* (MU*F, G*) that takes f to OEG O MU*f is an
isomorphism.
(2) The map f :F -! G is an isomorphism if and only if OEG O MU*f O jF is
an isomorphism.
(3) The map f :F -! G is a map of ring spectra if and only if OEG O MU*f is
a map of MU*-algebras.
Proof.Part 1 is proved in [HS99 , Corollary 2.17]. For part 2, we note that
OEG O MU*f O jF = OEG O jG O f* = f*.
For part 3, if f is a map of ring spectra, then MU*f is a map of MU*-algebras, *
*so
OEG O MU*f is also a map of MU*-algebras. Conversely, suppose OEG O MU*f is a
map of MU*-algebras. Then it must take 1 to 1, from which it follows easily that
f is compatible with the unit. To see that f is compatible with the multiplicat*
*ion,
we first prove that MU*f is a map of MU*-algebras. Indeed, consider the diagram
below.
MU*F MU* MU*F -MU*f-MU*f-------!MU*G MU* MU*GOEG-OEG----!G* MU* G*
? ? ?
~?y ~?y ?y~
MU*F ----!MU MU*G ----! G*
*f OEG
The right-hand square of this diagram is commutative. We want to show that
the left-hand square is commutative. Because MU*G ~=MU*MU MU* G* is an
extended MU*MU-comodule, it suffices to check that the two composites become
equal upon applying OEG , This is equivalent to checking whether the outer boun*
*dary
of the diagram is commutative. But this is true since OE O MU*f is a map of MU*-
algebras.
We have now shown that MU*f is a map of MU*-algebras. Using the isomor-
phism of Lemma 4.3, we see that the diagram below is commutative.
MU*(F ^ F ) -MU*(f^f)------!MU*(G ^ G)
? ?
MU*~?y ?yMU*~
MU*F ----!MU MU*G
*f
But F ^ F and G ^ G are also Landweber exact and evenly graded by Lemma 4.4.
Hence we can apply part 1 in the form
[F ^ F, G] ~=Hom MU*MU (MU*(F ^ F ), MU*G)
to conclude that F -! G is a map of ring spectra.
Corollary 4.6. Suppose F is a Landweber exact, evenly graded, commutative MU-
algebra spectra. Then the set of ring spectrum automorphisms of F is isomorphic
to the set of F*-algebra homomorphisms f :F*F -! F* such that fjR :F* -!F* is
an isomorphism.
OPERATIONS AND CO-OPERATIONS IN MORAVA E-THEORY 17
Proof.By Proposition 4.5, the set of ring spectrum automorphisms of F is isomor-
phic to the set of maps of MU*-algebras h: MU*F -! F* such that h O jE is an
isomorphism. This is equivalent to the set of F*-algebra homomorphisms
f :F*F ~=F* MU* MU*F -! F*
such that fjR is an isomorphism.
We can now apply this to our spectrum E. Recall that there is a p-typical for*
*mal
group law F over E0, whose classifying map takes vi to ui for i < n, takes vn t*
*o 1,
and takes all other vi to 0.
Proposition 4.7. The group is isomorphic to the set of all pairs (o, OE) wher*
*e o
is an automorphism of the ring E0 and OE is an isomorphism from the formal group
F to o*F .
Proof.In view of Corollary 4.6 and the fact that E* has a unit u in degree 2, we
see that is isomorphic to the set of all E0-algebra maps
f :E0E -!E0
such that fjR is an isomorphism. Proposition 3.8 tells us that
E0E ~=E0 V V T [t0, t-10] V E0,
Since the tensor product is the coproduct in the category of commutative rings,
and V T [t0, t-10] is the representing ring for isomorphisms of formal group la*
*ws, we
get the desired result.
Recall that the correspondence between elements of and E0-algebra maps
f :E0E -! E0 for which fjR is an isomorphism takes fl to (ss0~) O E0(fl). The
following corollary is then immediate.
Corollary 4.8. If fl 2 corresponds to (o, OE) as in Proposition 4.7, then the
induced map
(ss0~) O E0(fl): E0E ~=E0 V V T [t0, t-10] V E0 -!E0
takes a b c to asOE(b)o(c), where sOEdenotes the classifying map of OE.
We can now identify , following [Str00, Proposition 4].
Theorem 4.9. Let 0 denote the automorphism group of the Honda formal group
law Fn as a formal group law over Fpn, and let C denote the Galois group of Fpn
over Fp. Then ~= 0o C. In particular, is isomorphic to the set of pairs (ff*
*, fi),
where ff is an automorphism of Fn defined over Fpn, and fi is an element of the
Galois group of Fpn over Fp.
Proof.Proposition 4.7 and Lubin-Tate deformation theory [LT66 ] show that is
isomorphic to the set of all pairs (ff, fi) where fi is an automorphism of Fpn *
*~=E0=m
and ff is an isomorphism of the reduction Fn of F modulo m with fi*Fn. But Fn
is defined over Fp, so fi*Fn = Fn. Hence ff is an automorphism of Fn defined ov*
*er
Fpn, and fi is an element of C.
There is an obvious action ofPC on 0; ifPoe 2 C, then oe induces an isomorph*
*ism
Fpn[[x]] -!Fpn[[x]] that takes icixi to icoeixi. This isomorphism preserves*
* 0.
We leave to the reader the check that the multiplication of corresponds to the
multiplication on 0 o C.
18 MARK HOVEY
Combining Theorem 4.9 with Proposition 4.1, we see that the two sets
___ ___
AlgFpn(K0E, Fp) and AlgFpn(C( , Fpn), Fp)
___
are abstractly isomorphic. To see that the map AlgFpn( , Fp) is an isomorphism,
though, we need to understand the map better. The main point is the following
corollary.
Corollary 4.10. If fl 2 corresponds to (ff, fi) as in Theorem 4.9, then the m*
*ap
: Fpn V V T [t0, t-10] V Fpn ~=K0E -!C( , Fpn)
has (a b c)(fl) = asff(b)fi(c), where sffis the classifying map of ff.
Proof.Recall that is induced by by dividing by m and taking the induced map
in degree 0. Using the fact that K*E = E_*E=m (see Lemma 2.1), we see that
(a)(fl) is the composite
S0 a-!K ^ E 1^fl--!K ^ E ~-!K.
Since the Lubin-Tate correspondence proceeds by reducing a pair (o, OE) as in P*
*ropo-
sition 4.7 modulo m to obtain (ff, fi), this corollary follows from Corollary 4*
*.8.
We can now complete the proof that is an isomorphism.
Theorem 4.11. The map
: E_*E -!C( , E*)
of Section 1 is an isomorphism.
___
Proof.In view of Theorem 3.14, it suffices to show that AlgFpn( , Fp)_is_an iso-
morphism. By Proposition 4.2, a typical element of AlgFpn(C( , Fpn), Fp) is the
evaluation at fl map Evflfor fl 2 . If fl corresponds to the pair (ff, fi) as*
* in
Theorem 4.9, then Corollary 4.10 tells us that EvflO is the Fpn-algebra map
___
EvflO : Fpn V V T [t0, t-10] V Fpn -!Fp
that takes a b c to asff(b)fi(c), where sffis the classifying map of ff. Under_*
*the_iso-
morphism of Proposition 4.1, this map corresponds to (ff, fi). Thus AlgFpn( , F*
*p)
is an isomorphism.
5. Cohomology operations
In this section, we prove that E*E is isomorphic to the twisted completed gro*
*up
ring E*[[ ]]. The strategy is to construct the commutative square below,
E*[[ ]] ----! E*E
?? ?
y ?y
Hom E*(C( , E*), E*)----! Hom E*(E_*E, E*)
and to show that the vertical maps are isomorphisms.
We begin with the right-hand vertical map in Section 5.1, and we discuss the
left-hand vertical map in Section 5.2. We then complete the proof in Section 5.*
*3.
OPERATIONS AND CO-OPERATIONS IN MORAVA E-THEORY 19
5.1. Duality for E*E. The object of this section is to prove the following theo*
*rem.
Theorem 5.1. Suppose X is a spectrum such that K*X is evenly graded. Then
the natural map
E*X ae-!HomE*(E_*X, E*)
is an isomorphism.
The natural map in question takes an element x 2 Em X to the homomorphism
that takes y 2 E_kX to the composite
-m y 1e^x ~
Sk-m ----! Ee^ -m X --! Ee^E -! E.
One can define an analogous map
ae: F *X -!Hom E*(E_*X, F*)
for any E-module spectrum F .
Note that Theorem 5.1 would be automatic if we had a suitable universal coef-
ficient spectral sequence for E_*(-). Indeed, since K*X is evenly graded, E_*X *
*is
pro-free and so projective in the category of L-complete E*-modules [HS99 , The*
*o-
rem A.9]. We do have a universal coefficient spectral sequence for E*(-), follo*
*wing
Adams' approach [Ada74 , Section III.13]. But our hypotheses do not guarantee t*
*hat
E*X is projective over E*. A universal coefficient spectral sequence for E_*(-)*
* has
in fact recently been constructed [Hov04 ].
We will prove Theorem 5.1 by working with (E=J)*(X), where
J = (pi0, ui11, . .,.uin-1n-1)
is a regular ideal in E*, and E=J is obtained by taking successive cofibers as
in [EKMM97 , Chapter V]. The ring E*=J is a local ring with nilpotent maximal
ideal m. We have the following lemma, which is surely well-known, about modules
over such rings.
Lemma 5.2. Suppose f :M -!N is a map of R-modules, where R is a local ring
with nilpotent maximal ideal m.
(1) f is surjective if and only if R=m R f is surjective.
(2) If M and N are flat and R=m R f is an isomorphism, then f is an iso-
morphism.
It follows easily from this lemma that flat R-modules are in fact free, thoug*
*h we
do not need this fact.
Proof.Suppose that R=m R f is surjective. Given y in N, we show by induction
on k that there is an xk in M such that f(xk) - y 2 mkN. Since m is nilpotent,
taking k large enough shows that f is surjective. The case k = 1 is clear since
R=m R f is surjective. Now suppose that
f(xk) - y = r1z1 + . .+.rjzj
where ri2 mk. For each i, we can find an wi2 M such that zi= f(wi) + ti, where
ti2 mN, by the base case of the induction. It follows that
f(xk - r1w1 - . .-.rjwj) - y 2 mk+1N,
proving the induction step.
20 MARK HOVEY
Now suppose M and N are flat. Let K denote the kernel of f. Since N is flat,
the sequence
0 -!K -!M f-!N -!0
pure, and so K is flat [Lam99 , Corollary 4.86] and K=mK = 0. Since mk=mk+1 is a
free module over the field R=m, we see that K R mk=mk+1 = 0 for all k. Tensori*
*ng
the short exact sequence
0 -!mk=mk+1 -!R=mk+1 -!R=mk -!0
with K, we conclude by induction that K=mkK = 0 for all k. Since m is nilpotent,
we see that K = 0.
With this lemma in hand, we can now analyse (E=J)*(X).
Proposition 5.3. Suppose X is a spectrum such that K*X is evenly graded, and
J = (pi0, ui11, . .,.uin-1n-1) is a regular ideal in E*. Then the natural map
ae: (E=J)*(X) -!Hom E*(E_*X, E*=J)
is an isomorphism.
Proof.Note that
Hom E*(E_*X, E*=J) ~=Hom E*=J((E_*X)=J, E*=J).
and E*=J is a commutative ring with nilpotent maximal ideal m = (p, u1, . .,.un*
*-1).
The plan is thus to use Lemma 5.2.
Since K*X is evenly graded, so is its dual K*X. Thus E*X is pro-free by
Lemma 2.1. In particular, the sequence (p, u1, . .,.un-1) is a regular sequence*
* on
E*X. It follows from [Mat89 , Theorem 16.1] that
(pi0, ui11, . .,.uin-1n-1)
is also a regular sequence on E*X, and therefore that (E=J)*X ~=(E*X)=J, Since
E*X is the completion of a free E*-module, we conclude that (E=J)*X is a free
E*=J-module, and that ((E=J)*X)=m ~=K*X.
Similarly, because K*X is evenly graded, E_*X is evenly graded and pro-free by
Lemma 2.1. The same argument as in the preceding paragraph then shows that
(E=J)*X ~=(E_*X)=J
and is a free E*=J-module. It follows that Hom E*=J((E_*X)=J, E*=J) is a product
of free modules, and so is flat (since E*=J is Noetherian). It also follows th*
*at
((E=J)*X)=m ~=K*X.
Furthermore, there is a natural map
Hom E*=J((E_*X)=J, E*=J) -!Hom E*=J((E_*X)=J, K*) ~=Hom K*(K*X, K*).
This map is surjective, since (E_*X)=J is free. Hence we get an induced surject*
*ion
(Hom E*=J((E_*X)=J, E*=J))=m -!Hom K*(K*X, K*).
We claim this map is an isomorphism. To see this suppose f :(E_*X)=J -!E*=J is
a homomorphism that goes to zero as a map to K*. This means that f(x) 2 m for
all x. In particular, if we let {ei} be a generating set for the free module (E*
*_*X)=J,
we can write
f(ei) = pxi0+ u1xi1+ . .u.n-1xi,n-1
OPERATIONS AND CO-OPERATIONS IN MORAVA E-THEORY 21
for some elements xijin E*=J. But then we can define homomorphisms
gj: (E_*X)=J -!E*=J
for j = 0, 1, . .,.n - 1 by gj(ei) = xij. This gives
f = pg0 + u1g1 + . .u.n-1gn-1,
and so f 2 m Hom E*=J((E_*X)=J, E*=J), as required.
It now follows from an easy diagram chase that if we tensor the map
(E=J)*X -!Hom E*(E_*X, E*=J)
with E*=m, we get the isomorphism
K*X ~=Hom K*(K*X, K*).
Lemma 5.2 completes the proof.
We now prove Theorem 5.1.
Proof of Theorem 5.1.Choose a sequence of ideals
. . .Jk Jk-1 . . .J1
such that each Jk is of the form (pi0, ui11, . .,.uin-1n-1) and the Jk converge*
* to 0 in the
m-adic topology. We have a commutative square
E*X ---ae-! Hom E*(E_*X, E*)
?? ?
y ?y
limk(E=Jk)*(X)----!limaelimkHomE*(E_*X, E*=Jk).
k
The right-hand vertical map is an isomorphism because E* is complete. To under-
stand the left-hand vertical map, we apply the Milnor exact sequence
0 -!lim1(E=Jk)*+1X -!E*X -!lim(E=Jk)*X -!0.
Since K*X is evenly graded, (E=Jk)*X ~=(E_*X)=Jk as we have seen in the proof
of Proposition 5.3. Thus the lim1-terms vanish, and so the left-hand vertical m*
*ap
is an isomorphism. Proposition 5.3 implies that the bottom horizontal map is an
isomorphism as well, completing the proof.
5.2. Duality for C(G, R). We now turn our attention to the twisted completed
group ring E*[[ ]].
In general, whenever G is a profinite group and a is an ideal in a ring R, we*
* can
define the R-module
R[[G]] = limklimU(R=ak)[G=U].
We then have the following theorem.
Theorem 5.4. Let G be a profinite group and a an ideal in a commutative ring R
such that R is complete in the a-adic topology. Then the natural map
R[[G]] -!Hom R (C(G, R), R)
is an isomorphism of R-modules.
22 MARK HOVEY
Proof.There is a natural map
ff: R[G] -!Hom R (C(G, R), R),
where ff(r[g])(f) = rf(g). It is clear that ff is an isomorphism when G is fini*
*te.
Hence we get an induced natural isomorphism
R[[G]] = limk,U(R=ak)[G=U] -!limk,UHomR=ak(C(G=U, R=ak), R=ak).
We must show that the right-hand side of this isomorphism is naturally isomorph*
*ic
to Hom R(C(G, R), R). To see this, we first use Proposition 2.4 and Proposition*
* 2.5
to conclude that C(G, R)=ak ~=C(G, R=ak) This fact together with Lemma 2.3
gives us the following chain of isomorphisms
Hom R(C(G, R), R) ~=limkHom R(C(G, R), R=ak)
~=limkHom R=ak(C(G, R=ak), R=ak)
~=limklimUHom R=ak(C(G=U, R=ak), R=ak),
completing the proof.
Now, when G acts continuously on R, Hom R(C(G, R), R) is the dual of the Hopf
algebroid C(G, R), so is an R-algebra. In view of Theorem 5.4, then, R[[G]] sho*
*uld
also be an algebra, in analogy with the group ring, but in a way that takes into
account the action of G on R.
To see how this works, assume first that an arbitrary group G acts on a com-
mutative ring R by ring isomorphisms. Then the twisted group ring R[G] is
the free R-module generated by the elements of G, with multiplication defined by
(a[g])(b[h]) = abg[gh]. We would like to realize R[[G]] as a completion of the *
*twisted
group ring R[G] with respect to a suitable family of ideals. For this, we need *
*to
assume G is a profinite group, R is a local ring with maximal ideal m that is c*
*om-
plete in the m-adic topology, and G acts continuously on R, and, even better, a*
*cts
through a finite quotient on R=mk for all k. That is, we need to assume that, f*
*or all
k, there is an open normal subgroup Uk of G such that Uk acts trivially on R=mk.
Note that this is automatic from continuity if R=mk is finite, or, in the grade*
*d case,
finite in each degree, as is true for R = E*. Indeed, in this case Aut(R=mk) is*
* finite,
so the homomorphism G -!Aut (R=mk) must factor through a finite quotient.
Assuming that G does act through a finite quotient on each R=mk, we define
ideals I(k, U) of R[G] for each integer k and each open normal subgroup U of G
such that U acts trivially on R=mk. The ideal I(k, U) is the kernel of the surj*
*ection
R[G] -!(R=mk)[G=U] to the twistedPgroup ring. In more concrete terms, I(k, U)
is thePset of all elements ag[g] in R[G] such that for every coset C of U in *
*G, we
have g2Cag 2 mk. We then the twisted completed group ring R[[G]] to be
the completion of R[G] with respect to the ideals I(k, U), so that
R[[G]] = limk,U(R=ak)[G=U].
This is the same R-module as we defined above, since we can take this inverse l*
*imit
over arbitraryTpairs (k, U). Note that the natural map R[G] -!R[[G]] is injecti*
*ve
since kmk = 0. All the statements above work when R is a graded ring, as long
as m is homogeneous and the action of G preserves the grading.
With these definitions, the isomorphism of Theorem 5.4 is in fact an isomorph*
*ism
of R-algebras, using the mulitplication on Hom R(C(G, R), R) dual to the Hopf
OPERATIONS AND CO-OPERATIONS IN MORAVA E-THEORY 23
algebroid structure on C(G, R). We do not need this result, so we leave the pro*
*of
to the interested reader.
5.3. The isomorphism. We can now compute E*E.
Theorem 5.5. The inclusion -!E0E induces an E*-algebra isomorphism
E*[[ ]] -!E*E.
Proof.There is a map E* -!E*E that takes r 2 Ek to er2 EkE, where eris the
composite
k~
E = S0 ^ E r^1--! kE ^ E ---! kE.
The inclusion of then certainly induces a map ff: E*[ ] -!E*E, defined by let*
*ting
ff(r[fl]) be the composite erO fl. To see that ff is an E*-algebra homomorphism*
*, use
the commutative diagram below.
~= r^1 k~
E ----! S0 ^ E ----! kE ^ E ----! kE
? ? ? ?
fl?y 1^fl?y kfl^fl?y ?y kfl
E ----!~=S0 ^ E ----!rfl^1 kE ^-E---! kE
k~
This diagram commutes because fl is a map of ring spectra, and it shows that
ff([fl]r) = (ff[fl])(ffr), so that ff is an E*-algebra homomorphism.
Now, E*E is complete with respect to the m-adic topology, so to show that ff
extends to an E*-algebra homomorphism
fi :E*[[ ]] -!E*E,
it suffices to show that ff is continuous. That is, given k, we must find an m *
*and U
such that ffI(m, U) mkE*E. But the action of on E*E is continuous, so there
is a U such that U preserves mkE*E. This means that the composite
E*[ ] ff-!E*E -!(E*E)=mk
factors through (E*=mk)[ =U], and so ffI(k, U) mkE*E, as required.
Now consider the diagram below.
E*[[ ]] ---fi-! E*E
?? ?
y ?yae
Hom E*(C( , E*), E*)----! Hom E*(E_*E, E*)
The left-hand vertical map is the isomorphism of Theorem 5.4, the right-hand
vertical map is the isomorphism of Theorem 5.1, and the bottom horizontal map is
dual to the isomorphism of Theorem 4.11. One can easily check that the diagram *
*is
commutative. Indeed, it suffices to check that the element [fl] 2 E*[ ] goes to*
* the
same place under both composites. In fact, it goes to the map that takes x 2 E_*
*kE
to the composite
k(1e^fl) k~
S0 x-! kEe^E -----! kEe^E ---! kE.
Hence fi is an isomorphism.
24 MARK HOVEY
6. Hopf algebroid structure
The object of this section is to show that (E*, E_*E) and (E*, C( , E*)) are
both graded formal Hopf algebroids, and that the isomorphism of Theorem 4.11
preserves the Hopf algebroid structure. We begin by defining precisely what we
mean by a graded formal Hopf algebroid. We then show that (E*, E_*E) is a
graded formal Hopf algebroid in Section 6.2, then discuss C( , E*) in Section 6*
*.3.
We show that preserves the Hopf algebroid structure in Section 6.4.
6.1. Graded formal Hopf algebroids. A graded formal Hopf algebroid should
be a cogroupoid object in the category of graded formal rings. Recall from [Str*
*99,
Section 4] that a formal ring A is a topological commutative ring A such that
cosets of open ideals form a basis for the topology on A, and such that A ~=lim*
*A=I
as I runs through the open ideals. Morphisms in the category of formal rings are
continuous homomorphisms. The category of formal rings has all finite colimits.
The pushout S bRT of two continuous homomorphisms R -!S and R -!T is the
completion of the tensor product S R T with respect to the topology defined by
the ideals I R T + S R J, where I and J run through open ideals in S and
T . Note that this definition of b does not conflict with our previous use of *
*the
notation, in Proposition 2.5. We can thus define a formal Hopf algebroid to be
a cogroupoid object in the category of formal rings. The major difference betwe*
*en
a formal Hopf algebroid (A, ) and a Hopf algebroid is that the diagonal has the
form : -! b , and so all diagrams involving must be similarly changed.
Some subtleties arise in the graded case. We define a graded formal ring to be
a graded ring A equipped with a family {Ij} of homogeneous ideals such that for*
* all
j, j0there is a j00such that Ij00 Ij\ Ij0and such that A is the inverse limit,*
* in the
category Ringsgr of graded rings, of the A=Ij. This means that Ak ~=limj(A=Ij)k
for all k. The family of ideals {Ij} will sometimes be referred to as the topol*
*ogy
on A. Maps of graded formal rings R -!S are maps of graded rings such that each
Rk -!Sk is continuous (in the inverse limit topology).
Note that the forgetful functor from Rings grto Rings does NOT preserve
inverse limits. In particular, if k is a field and x has degree 2, the ring k[*
*x] is a
graded formal ring in the x-adic topology, but it is not a formal ring in the x*
*-adic
topology.
Nevertheless, the category of graded formal rings has all finite colimits, wh*
*ere
the pushout of R -! S and R -! T is S bRT , the completion in the graded sense
of the graded ring S R T with respect to the ideals I T + S J, where I and
J are the homogeneous ideals that define the formal structure on R and S. Hence
we can define a graded formal Hopf algebroid to be a cogroupoid object in the
category of graded formal rings.
From an algebro-geometric perspective, it would be much more natural to con-
sider formal graded rings, rather than graded formal rings. Here a formal graded
ring would be a formal ring R equipped with a coaction of the Hopf algebra
Z[u, u-1]. That is, the "grading" would actually be a continuous ring homomor-
phism R -!Rb Z[u, u-1] that is coassociative and counital. The drawback of this
approach for us is that this is just not how we think of completions in algebra*
*ic
topology.P For example, when we form (v-1nBP*)In, we do not allow the element
1 k
k=1v1. It is possible that it would be better to allow such elements, giving *
*us
OPERATIONS AND CO-OPERATIONS IN MORAVA E-THEORY 25
homotopy groups that are only graded in this weak sense. We will stick with gra*
*ded
formal rings though.
6.2. E_*E as a graded formal Hopf algebroid. We now show that (E*, E_*E)
is a graded formal Hopf algebroid. The topology is defined by the powers of the
maximal ideal m of E*. We have seen that E_*E is pro-free in Proposition 2.2,
and so E_*E is a graded formal ring. The usual proof that (R*, R*R) is a Hopf
algebroid [Rav86 , Proposition 2.2.8] depends on knowing that the natural map
R*R R* R*R -!R*(R ^ R)
is an isomorphism. If we are to apply the same argument in this case, we need to
know that
E_*E bE*E_*E -!E_*(Ee^E)
is an isomorphism.
Proposition 6.1. Suppose X is a spectrum such that K*X is concentrated in even
dimensions. Then the natural map
oe :(E_*E E* E_*X)^m-! E_*(Ee^X)
is an isomorphism.
Proof.Since K is a field spectrum, the natural map
ae: K*E K* K*X -!K*(E ^ X) = K*(Ee^X)
is an isomorphism. In particular, K*(Ee^X) is concentrated in even dimensions.
Lemma 2.1 then guarantees that E_*X and E_*(Ee^X) are pro-free, as is E_*E. If
M ~= Fm^and N ~= (F 0)^mare pro-free modules, then (M N)^mis also pro-free,
since it is isomorphic to the completion of F E* F 0. Hence (E_*E E* E_*X)^mis
pro-free as well. In view of Lemma 2.7, to prove the proposition it suffices to*
* check
that oe=m is an isomorphism. Using Lemma 2.1 and the fact that m is an invariant
ideal in E*E, we conclude that oe=m ~=ae, and so oe is an isomorphism.
Corollary 6.2. Suppose X is a ring spectrum such that K*X is concentrated in
even dimensions. Then the natural map
E_*X bE*E_*X -!E_*(Ee^X)
is an isomorphism.
Proof.By Lemma 2.1, E_*X is pro-free, and so in particular is a formal ring with
respect to the ideals mn. The corollary then follows from Proposition 6.1 becau*
*se
m is an invariant ideal in E*.
Recall that if R is a flat commutative ring spectrum, then (R*, R*R) is a gra*
*ded
Hopf algebroid [Rav86 , Proposition 2.2.8]. The only use of the flatness of R i*
*n this
argument is in the isomorphisms
R*X R* R*Y -! R*(X ^ Y )
in case X = Y = R, needed to define the diagonal, and in case X = R, Y = R ^ R
and X = R ^ R, Y = R, needed to prove coassociativity. We can repeat this
argument using the isomorphisms of Corollary 6.2 to deduce the following theore*
*m,
Theorem 6.3. (E*, E_*E) is a graded formal Hopf algebroid.
26 MARK HOVEY
6.3. C( , E*) as a graded formal Hopf algebroid. The object of this section is
to show that (E*, C( , E*)) is a graded formal Hopf algebroid. We will work more
generally with a profinite group G acting continuously on a graded ring R that *
*is
complete in the a-adic topology, where a is a homogeneous ideal.
We begin with a very simple case. Given a group G, the constant group
scheme G is the functor on commutative rings defined by G(S) = C(SpecS, G).
If S has no nontrivial idempotents, then SpecS is connected, and so G(S) = G,
consisting only of the constant maps. The constant group scheme G is in fact the
coproduct in the category of schemes of |G|-many copies of SpecZ. In particular,
it is not affine when G is infinite, because it is not quasi-compact.
However, when G is finite, we have the following well-known proposition.
Proposition 6.4. Suppose G is a finite group. Then the constant group scheme G
is affine and represented by the Hopf algebra F (G, Z) of functions from G to Z.
This proposition follows from the fact that F (G, Z) is the product of |G|-ma*
*ny
copies of Z. The structure maps of the Hopf algebra F (G, Z) are defined as fo*
*l-
lows. The counit ffl: F (G, Z) -! Z is evaluation at the identity element e of*
* G,
the conjugation O is induced by the inverse map in G, and the diagonal is the
composite
o-1G
F (G, Z) -!F (G x G, Z) --! F (G, Z) F (G, Z)
where the first map is induced by the multiplication map of G and oG is the iso-
morphism defined by oG (f f0)(g, h) = f(g)f0(h).
We can generalize Proposition 6.4 to profinite groups as well.
Proposition 6.5. Suppose G is a profinite group. Then the group-valued functor *
*on
commutative rings defined by G(S) = C(SpecS, G), where G is given the profinite
topology, is an affine group scheme represented by the Hopf algebra C(G, Z) of
continuous functions from G to Z.
Note that if S has no nontrivial idempotents, then G(S) = G, just as for the
constant group scheme. There is a natural map from the constant group scheme
to this profinite group scheme G, because every map that is continuous to the
discrete topology on G is also continuous to the profinite topology, and this m*
*ap
is an isomorphism on every S with no nontrivial idempotents. But it is not an
isomorphism on all S unless G is finite, since the constant group scheme is not*
* affine
when G is infinite but the profinite group scheme is, according to Proposition *
*6.5.
Proof.We simply compute
G(S) = C(SpecS, G) ~=limUC(SpecS, G=U)
~=limURings (F (G=U, Z), S) ~=Rings(C(G, Z), S).
This shows that G is affine and represented by C(G, Z). Since G is visibly a gr*
*oup-
valued functor, C(G, Z) is a Hopf algebra.
Note that C(G, Z) is the colimit of the Hopf algebras F (G=U, Z). One can
describe the structure maps in a similar fashion. The counit is again evaluatio*
*n at
e, the conjugation is induced by the (continuous) inverse mapping on G, and the
diagonal is the composite
o-1G
C(G, Z) -!C(G x G, Z) --! C(G, Z) C(G, Z),
OPERATIONS AND CO-OPERATIONS IN MORAVA E-THEORY 27
where oG is again defined by oG (f f0)(g, h) = f(g)f0(h). The map oG is the
colimit of the isomorphisms oG=U , so is an isomorphism.
We now show how to build the Hopf algebroid (R, C(G, R)) from the Hopf algebra
C(G, Z). Recall that a Hopf algebroid is a cogroupoid object in the category of
commutative rings [Rav86 , Appendix 1]. In general, a cogroupoid object in a
category C is a pair of objects (A, ) such that (C(A, X), C( , X)) is a groupo*
*id
that is natural in X for all objects X of C. Here C(A, X) is the object set of
the groupoid and C( , X) is the morphism set. When C has finite colimits, (A, )
is a cogroupoid object if and only if there are structure maps jL, jR :A -! ,
ffl: -! A, O: -! , and : -! qA analogous to the structure maps
of a Hopf algebroid [Rav86 , Appendix 1], where qA denotes the pushout of
-jR-A -jL-! . These structure maps are required to satisfy certain axioms
analogous to those of [Rav86 , Definition A1.1.1].
One of these axioms deserves special attention, because Ravenel's description*
* of
it is not optimal. Define qA e to be the pushout where A acts on the right fa*
*ctor
of though jR instead of jL; simmilarly, define e qA to be the pushout where
A acts on the left factor through jL instead of jR . Then axiom (f) of Ravenel's
definition [Rav86 , A1.1.1], which implies that the composition of a map and its
inverse is the appropriate identity, should be rephrased to say that the follow*
*ing
diagram, and a similar diagram involving jL and 1 q O, are commutative.
----! qA --Oq1--!eqA
flfl ?
fl ?yr
----!ffl A ----!j
R
Here r denotes the fold map, which is the identity on each factor of .
Here is a general result on constructing cogroupoid objects from actions of
cogroups.
Proposition 6.6. Suppose C is a category with finite colimits and C is a cogroup
object of C that coacts on the right on an object A of C. Then (A, A q G) is a
cogroupoid object of C.
A cogroupoid object of this form is sometimes called a split cogroupoid. In
the case of Hopf algebroids, this recovers the definition of a split Hopf algeb*
*roid
given in [Rav86 , Definition A1.1.22].
Proof.Let 0 denote the initial object of C. Since C is a cogroup, it comes with*
* maps
: C -!C q C, ffl: C -!0, and O: C -!C playing the role of the diagonal, counit,
and conjugation. There is also a coassociative and counital map _ :A -! A q C
giving the coaction of C. We then define the left unit jL :A -! A q C to be the
structure map i1 of the coproduct, the right unit jR :A -!AqC to be the coaction
_, and the counit fflA :AqC -!A to be 1qffl. It is then obvious that fflA jL = *
*1A , and
fflA jR = 1A because _ is counital. We define the conjugation OA :A q C -!A q C
to be _ on A and i2O on C. Then OA jL = jR by definition, and OA jR = jL since
O2 = 1. Also, O2A= 1AqC for the same reasons. Finally, the diagonal A is the
composite
A q C 1q---!A q C q C ~=(A q C) qC (A q C),
28 MARK HOVEY
The fact that is coassociative, counital, and compatible with O implies the s*
*ame
facts for A .
We can now apply this to (R, C(G, R)),
Theorem 6.7. Suppose G is a profinite group acting continuously on a ring R that
is complete in the a-adic topology for some ideal a. Then (R, C(G, R)) is a sp*
*lit
formal Hopf algebroid.
This theorem is also true in the graded case, where the action of G must pres*
*erve
the grading, the ideal a must be homogeneous, and R need only be complete in the
graded sense.
Proof.Think of C(G, Z) as a cogroup object in the category of formal rings, whe*
*re
the topology is trivial. Recall from Proposition 2.5 that there is an isomorphi*
*sm
oe :Rb C(G, Z) -!C(G, R).
Define a coaction _ of the cogroup C(G, Z) on R as the composite
-1
R jR--!C(G, R) oe--!Rb C(G, =Z),
where jR (r)(g) = rg. It is easy to check that oe is counital. To see that _ *
*is
coassociative, we use the following commutative diagram.
R --jR--! C(G, R) --oe-- Rb C(G, Z)
? ? ?
jR?y ~*?y ?y1b~*
C(G, R) ---fi-! C(G x G, R) --oe-- Rb C(G x G, Z)
x x x
oe?? ff?? ??1bo
Rb C(G, Z)----! C(G, R)b C(G, Z)---- Rb C(G, Z)b C(G, Z)
jR b1 oeb 1
In this diagram, ~: G x G -! G denotes the multiplication map, fi :C(G, R) -!
C(G x G, R) is defined by (fif)(g, h) = f(h)g, and ff: C(G, R)b C(G, Z) -!C(G x
G, R) is defined by ff(f f0)(g, h) = f0(h)f(g). The reader can check that th*
*is
diagram is in fact commutative. All the maps that go either left or up are isom*
*or-
phisms, using Proposition 2.4 and the fact that o is an isomorphism. Hence we c*
*an
reverse those arrows, and then the equality of the outer composites shows that *
*jR
is coassociative. Hence Proposition 6.6 completes the proof.
We now describe the structure maps of (R, C(G, R)). The left unit is the incl*
*u-
sion of the constant functions, and the right unit jR is defined by jR (r)(g) =*
* rg.
The counit is evaluation at e, and the conjugation O is defined by (Of)(g) =
f(g-1)g. We have the commutative diagram below,
b~* 1bo
Rb C(G, Z) -1---!Rb C(G x G, Z) ---- Rb C(G, Z) C(G, Z)
? ? ?
oe?y oe?y ?yoeb oe
C(G, R) ----!~* C(G x G, R) ----o C(G, R)b RC(G, R)
OPERATIONS AND CO-OPERATIONS IN MORAVA E-THEORY 29
when the o on the bottom line is defined to be the completion of the map defined
by o(f f0)(g, h) = f(g)f0(h)g. Hence o is an isomorphism, and the diagonal is
the composite of the bottom line after reversing o.
6.4. is a map of graded formal Hopf algebroids.
Theorem 6.8. The map
(1, ): (E*, E_*E) -!(E*, C( , E*))
is a map of graded formal Hopf algebroids.
Proof.we begin by showing that jL = jL. Indeed, if a 2 Em and fl 2 , then we
have
( jL(a))(fl) = ~ O (1e^fl) O (1e^j) O a = ~ O (1e^j) O a = a = (jL(a)(fl),
as required. Here we have used the fact that fl O j = j, since fl is a map of r*
*ing
spectra, Similarly, we have
( jR (a))(fl) = ~(1e^fl)(je^1)a = ~(je^1)fl O a = fl O a = afl= (jR (a))(fl*
*),
To see that commutes with ffl, we have
ffl (a) = ( a)(e) = ~ O (1e^e) O a = ~ O a = ffl(a).
And to see that commutes with O we compute:
[O (a)](fl) = [ (a)(fl-1)]fl= fl O ~ O (1e^fl-1) O a
= ~ O (fle^fl) O (1e^fl-1) O a = ~ O (fle^1) O a
= ~ O (1e^fl) O T O a = [ (Oa)](fl).
We are left with proving that is compatible with in the sense that (
)^mO = O . We do this by constructing the commutative diagram below.
E_*E - ---! C( , E*)
? ?
ae?y ?ym*
0
(6.9) E_*(Ee^E) - ---! C( x , E*)
x x
o??~= ~=??o
(E_*E E* E_*E)^m-----! (C( , E*) E* C( , E*))^m
( )^m
Here ae is induced by 1e^je^1, and of course the vertical composites (after inv*
*erting
o) are the diagonal maps. The map 0 will be defined analogously to . Indeed,
we have a map
oe0: x x E_*(Ee^E) -!E*
that takes the triple (fl, fl0, a), where a 2 E_m(Ee^E) to the composite
e^fle^fl0 1e^1e^fl 1e^~ ~
Sm -a!Ee^Ee^E 1-----!Ee^Ee^E ----! Ee^Ee^E --! Ee^E -! E.
Since oe0 is a composite of composition maps and smash product maps, it will be
continuous when every set of maps is given its natural topology (see the discus*
*sion
at the beginning of Section 1). Thus the adjoint of oe0 is our desired map
0: E_*(E ^ E) -!C( x , E*).
30 MARK HOVEY
Now we need to check that the two squares in our diagram (6.9)commute. For
the top square, we have
( 0ae(a))(fl,=fl0)~ O (1e^~) O (1e^1e^fl) O (1e^fle^fl0) O (1e^je^1) O a
= ~ O (1e^~) O (1e^1e^fl) O (1e^je^1) O (1e^fl0) O a
= ~ O (1e^~) O (1e^je^1) O (1e^fl) O (1e^fl0) O a
= ~ O (1e^fl) O (1e^fl0) O a
= [m* (a)](fl, fl0).
It suffices to check that the bottom square commutes before we complete the bot*
*tom
row. We then have
[o( )(a, b)](fl,=fl0)~ O (1e^fl) O (~e^~) O (1e^fle^1e^fl0) O (ae^b)
= ~ O (1e^~) O (1e^fle^fl) O (~e^1) O (1e^fle^1e^fl0) O (*
*ae^b)
= ~ O (1e^~) O (~e^1e^1) O (1e^fle^fle^flfl0) O (ae^b)
= ~ O (1e^~) O (1e^~e^1) O (1e^fle^fle^flfl0) O (ae^b)
= ~ O (1e^~) O (1e^fle^flfl0) O (1e^~e^1) O (ae^b)
= [ 0o(a, b)](fl, fl0),
completing the proof.
Appendix A. The topology on
Recall that we identified the group in Theorem 4.9 as the semi-direct produ*
*ct
0 o C, where 0 is the automorphism group of the Honda formal group law over
Fpn and C is the Galois group of Fpn over Fp, This is an identification of abst*
*ract
groups; for the isomorphism : E_*E -! C( , E*) to be useful, we also need to
understand the topology on . At the moment, we know only that this topology is
profinite. The topology on was described in [Str00], but many details are mis*
*sing
that we fill in here.
As explained in [Rav86 , Lemma A2.2.16], the group 0 is the group of units in
the endomorphism ring D of Fn, which is a noncommutative ring obtained from the
Witt ring W Fpn by adjoining an indeterminate S subject to the relations Sn = p
and Sw = woeS, where oe denotes the generator of C. The group C acts on W Fpn
according to [Rav86 , Lemma A2.2.15]. The subgroup of 0 consisting of the stri*
*ct
isomorphisms is called the Morava stabilizer group Sn in [Rav86 , Section 6.2].
We begin with some well-known facts about the stabilizer group Sn, whose proo*
*fs
can be hard to find in the literature.
Lemma A.1. The Morava stabilizer group Sn is a topologically finitely generated,
pro-p group, and an open subgroup of finite index in 0 o C.
Here a topological group is topologically finitely generated if it has a dense
finitely generated subgroup. Note that it follows from Lemma A.1 that 0 o C, as
a finite extension of Sn, is also topologically finitely generated.
Proof.First note that D itself is a profinite ring. Indeed, by Lemma A2.2.16
of [Rav86 ], D is a free Z(p)-module of rank n2, and since p is central in D, we
conclude that D = limD=(pk) as rings. It follows that 0, as the group of units*
* in
OPERATIONS AND CO-OPERATIONS IN MORAVA E-THEORY 31
D, is a profinite group (see the proof of Corollary 1.2). Hence 0o C is a prof*
*inite
group as well in the product topology.
The stabilizer group Sn is the preimage of 1 under the map of rings D -! Fpn
that takes an endomorphism of Fn to the coefficient of x in it. This map is the
canonical reduction mod p on W and takes S to 0. Its kernel is the 2-sided ideal
(S) generated by S, which is open since Sn = p. Hence this reduction map is
continuous, so Sn is closed. Since Fpn is finite, Sn has finite index in Dx and*
* so
also in 0 o C, and so must also be open.
Since Sn = p, D is complete in the S-adic topology. Hence Sn is the inverse
limit of the groups Gk = ker(D=Sk)x -! (D=S)x . Since S is nilpotent in D=Sk,
this kernel is 1 + Hk, where Hk = ker(D=Sk -!D=S). In particular, Gk is a finite
p-group, so Sn is a pro-p group.
We now prove that Sn is topologically finitely generated. RecallPfrom [Rav86 ,
Lemma A2.2.16] that every element of Sn can be written as 1 + 1i=1eiSi, with
ei 2 gFpn, where gFpnis the set of all e 2 W Fpn such that epn = e. The reducti*
*on
map W Fpn -!Fpn sends gFpnto Fpn by a multiplicative bijection; gFpnis known as
the set of Teichmuller lifts. Now, let
T = {1 + eSk|e 2 gFpn, 1 k _np__p}-.1
We claim that the subgroup generated by T is dense. To see this, we first show *
*that
for all e 2 gFpnand for all k, T contains an element xe,ksuch that xe,k~=1 + eSk
(mod Sk+1). Indeed, this is obvious for all k _np_p-1. For k > _np_p-1, we l*
*et xe,k=
xpe,k-n. To see that xpe,k-nhas the required form, we use the fact that
(1 + eSk-n)p
` '
i-1)=(p-1)i(k-n) (pn-1)=(p-1)p(k-n)
= 1 + peSk-n + . .+. pie(p S + . .+.e S
~=1 + eSk + e(pn-1)=(p-1)Sp(k-n) (mod Sk+1),
since Sn = p, and Se = epS for e 2 gFpn. Now, since k > _np_p-1, we see that
p(k - n) > k, so xpe,k-nis indeed a good choice for xe,k.
Now, in order to see that T is dense, it suffices to showPthat for all k and
e1, . .,.ek 2 gFpn, T contains an element congruent to 1 + ki=1eiSi modulo Sk*
*+1.
We prove this by induction on k, the base case being obvious. For the induction
step, the induction hypothesis guarantees we can find an element
Xk
y ~=1 + eiSi+ aSk+1 (mod Sk+2)
i=1
in T . We can also find an element b 2 gFpnsuch that a + b ~=ek+1 (mod p). Then
Xk
yxb,k+1~=1 + eiSi+ (a + b)Sk+1 (mod Sk+2)
i=1
Xk
~=1 + eiSi+ ek+1Sk+1 (mod Sk+2),
i=1
as required, using the fact that Sn = p.
32 MARK HOVEY
Theorem A.2. This isomorphism ~= 0 o C of Theorem 4.9 is a continuous
isomorphism of profinite groups. Furthermore, a subgroup of is open if and on*
*ly
if it has finite index.
This theorem is saying that the topology on is completely determined by the
group structure. This is believed to be true for a general profinite group, but
remains an open question [CR02 ].
Proof.By [DdSMS99 , Theorem 1.17], the open subgroups of a topologically finit*
*ely
generated pro-p group such as Sn are precisely the subgroups of finite index.
By [And76 , Proposition 2], this remains true for any finite extension, such as*
* 0oC,
of such a group. It follows that the isomorphism 0 o C -! of Theorem 4.9 is
continuous. It is therefore a homeomorphism as well, since it is a map from a
compact space to a Hausdorff space.
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Department of Mathematics, Wesleyan University, Middletown, CT 06459
E-mail address: hovey@member.ams.org