VARIETIES AND LOCAL COHOMOLOGY FOR CHROMATIC GROUP COHOMOLOGY...

VARIETIES AND LOCAL COHOMOLOGY FOR CHROMATIC GROUPCOHOMOLOGY RINGS.

J. P. C. GREENLEES AND N. P. STRICKLAND

Abstract. Following Quillen [20, 21], we use the methods of algebraic geometry to studythe ring E∗(BG) where E is a suitable complete periodic complex oriented theory andG is a finite group: we describe its variety in terms of the formal group associated toE, and the category of Abelian p-subgroups of G. This also gives information about theassociated homology of BG.

For example if E is the complete 2-periodic version of the Johnson-Wilson theory E(n)the irreducible components of the variety of the quotient E∗(BG)/Ik by the invariantprime ideal Ik = (p, v1, . . . , vk−1) correspond to conjugacy classes of Abelian p-subgroupsof rank ≤ n− k. Furthermore, if we invert vk the decomposition of the variety into irre-ducible pieces corresponding to minimal primes becomes a decomposition into connectedcomponents, corresponding to the fact that the ring splits as a product.

Contents

1. Informal introduction 22. Description of results. 43. Schemes and varieties 94. Admissible cohomology theories 145. The proof of Theorem 2.4 176. Multiple level structures 227. The geometric Frobenius map 288. Thickenings 299. Pure strata 3510. The E-homology of BG. 3911. Some examples. 41Appendix A. Proof of the theorem of Hopkins-Kuhn-Ravenel 43Appendix B. The Evens norm map 44Appendix C. Varieties and reduction mod Ik. 45References 47

The authors thank the University of Chicago for its hospitality during Autumn 1994 when this workwas begun, and the Transpennine Topology Triangle for providing the opportunity to meet later. The firstauthor also thanks the Nuffield Foundation for its support.

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2 J. P. C. GREENLEES AND N. P. STRICKLAND

1. Informal introduction

The article investigates the interface between equivariant topology and the chromaticapproach to stable homotopy theory, by giving a geometric description of the ring E∗(BG)for a certain class of complete periodic complex oriented theories E∗(·) and finite groups G.The archetypal examples of such theories are the complete, 2-periodic versions E = En ofthe Johnson-Wilson theories E(n) with coefficients E(n)∗ = Z(p)[v1, v2, . . . , vn−1, vn, v

−1n ]

for some prime p, and we restrict attention to this case in the introduction. To obtaina general description, we follow the example of Quillen [20, 21], and concentrate on thegeometric properties of the ring E∗(BG). We use Quillen’s descent argument to reduceto the study of E∗(BA) for Abelian p-subgroups A. The Abelian case is translated intothe theory of formal groups, and the resulting questions studied by extending the secondauthor’s theory of multiple level structures [23]. The work is also strongly influenced bythat of Hopkins-Kuhn-Ravenel [14], discussed in [13].

The description of the ring E∗(BG) allows us to understand the first author’s localcohomology approach to the homology E∗(BG) [6] from the chromatic point of view, andsets the results of [10, 11] in a wider context. The slogan is that equivariant topology istrivial over pure chromatic strata, and thus the important geometry concerns the way thestrata are attached to each other.

Quillen’s method involves considering the cohomology E∗(EG ×G Z) of the Borel con-struction for a finite G-complex Z, even if one is only concerned with the special caseZ = ∗. Our results apply to all such Z, but it may help the reader if we spend the restof the introduction summarising the highlights when Z = ∗. We explain the geometriclanguage in Section 3 below, but the outline should be clear without precise definitions.The essential distinction is between schemes, which encode all the ring theoretic informa-tion, and varieties, which ignore nilpotents and only capture a crude picture of the ring.Nonetheless many important features, such as the dimension, connected components andirreducible components are visible at the level of varieties. Readers comfortable with theformal framework may wish to skip directly to the next section, where our results arestated precisely and in appropriate generality.

Following Morava’s chromatic philosophy we concentrate on the invariant prime idealsIk = (p, u1, . . . , uk−1) of the complete local ring E0. Indeed, if we let X = spf(E0) denoteits formal scheme, the geometric counterpart of the filtration

0 = I0 ≤ I1 ≤ I2 ≤ · · · ≤ In,

is the filtration

X = X0 ≥ X1 ≥ X2 ≥ · · · ≥ Xn,

where Xk = spf(E0/Ik) is the formal subscheme defined by Ik. Evidently Xk is a formalaffine space of dimension n − k. One of the themes will be that we can understandphenomena over X by restricting to the subschemes Xk, and that they will be especiallysimple over the pure strata X ′k = Xk \Xk−1. The notational convention that a subscript kdenotes restriction to the kth chromatic stratum and that a dash denotes restriction to apure stratum will remain in force throughout the paper.

CHROMATIC GROUP COHOMOLOGY RINGS 3

Now consider the formal scheme X(G) = spf(E0(BG)), which is finite over X; we shallgive a description of the primary features of its underlying variety. Let Xk(G) denote therestriction of X(G) to the part over Xk, and note that this is spf(E0(BG)/Ik). On theother hand, we are often interested in the rings (E/Ik)

0(BG); for example if k = n it issimply the Morava K-theory K(n)0(BG) made 2-periodic. If, as often happens, E0(BG)is free over E0 then E0(BG)/Ik is actually equal to (E/Ik)

0(BG); we shall see that in anycase their varieties agree.

Next we describe the irreducible components of Xk(G), and how they behave over purestrata. In fact there is a decomposition

Xk(G) =⋃(A)

Yk(G,A)

into irreducible components all of which have dimension n − k, where the indexing set isthe set of conjugacy classes of Abelian p-subgroups A of rank ≤ n− k. Furthermore, overthe pure stratum X ′k this decomposition becomes a disjoint union

X ′k(G) =∐

Y ′k(G,A).

Indeed, there is a single formal scheme Levelk(A∗) for each Abelian p-group of rank ≤

n − k so that at the level of varieties, Y ′k(G,A) = Level′k(A∗)/WG(A) where WG(A) =

NG(A)/A is the Weyl group of A in G. At one extreme, spf(E0(BG)) has one irreduciblecomponent for each conjugacy class of Abelian p-subgroups of rank ≤ n whilst at theother, spf(K(n)0(BG)) is itself irreducible. The schemes Levelk(A

∗) classify ‘multiple levelstructures’ (see Section 6) for the formal group G of E; they are formal spectra of completeregular local rings Dk(A) of dimension n− k.

Translating geometric statements back into ring theory, our decompositions state thatthe minimal primes of E0(BG)/Ik are in bijective correspondence with the conjugacyclasses of Abelian p-subgroups of rank ≤ n − k. Moreover, if we invert uk the ring splitsup to F -isomorphism as a product of rings of invariants of well-understood rings:

u−1k E0(BG)/Ik

F−iso−−−→∏(A)

D′k(A)WG(A).

Here WG(A) = NG(A)/A, but the action factors through NG(A)/CG(A), so that in theAbelian case the action is trivial. In the Abelian case G = A, we identify u−1

k E0(BA)/Ikexactly, and it decomposes in the same way:

u−1k E0(BA)/Ik =

∏B

D′k(A,B).

The pieces correspond in the sense that D′k(A,B)red = D′k(B), and D′k(A,B) is |A/B|ktimes larger than D′k(B). In particular D′k(A,A) is reduced and D′k(A,A) = D′k(A).

The reader may also wish to consider applications to E∗(BG) in Sections 10 and 11before confronting the general results.


2. Description of results.

In this section we give precise statements of our main results.Let p be a prime. We shall consider the following class of cohomology theories. In this

paper, E∗(Y ) will always denote the unreduced cohomology of a space Y , and the reducedcohomology of a based space will be indicated by a tilde.

Definition 2.1. Let pCP∞ : CP∞ −→CP∞ be the map classifying the p’th tensor power ofthe canonical line bundle. A p-local commutative ring spectrum E is admissible if

(a) E0 is a complete local Noetherian ring.(b) E1 = 0.(c) E2 contains a unit.

(d) pCP∞ induces a nonzero self-map of E0(CP∞).

These conditions will be discussed further in Section 4. Examples include the cohomology

theory obtained from E(n) by adjoining a (pn−1)’st root of vn, or various spectra obtainedfrom this by tensoring with the Witt ring of Fpn or Fp, or killing some generators (providedthat p > 2). For suitable versions of elliptic cohomology, the spectrum LK(2)Ell will be afinite product of admissible ring spectra. The version of integral Morava K-theory used byIgor Kriz in [16] is also admissible. It can be shown that every admissible ring spectrumE is K(n)-local for some n, called the height of E (this is the same as the height of theformal group law associated to E).

We will specify one example more precisely. Let W be the Witt ring of Fpn, and considerthe following graded ring:

E∗ = W [[u1, . . . , un−1]][u, u−1].

The generators uk have degree 0, and u has degree −2. We take u0 = p and un = 1 anduk = 0 for k > n. There is a map BP ∗ −→E∗ sending vk to up

k−1uk. Using this, we definea functor from spectra to E∗-modules by

E∗(X) = E∗ ⊗BP∗ BP∗(X).

The BP ∗-module E∗ is Landweber exact, so this functor is a homology theory, which weshall call Morava E-theory. It is represented by an admissible ring spectrum.

Convention 2.2. Throughout the whole of this paper the following notation is fixed.

• E is an admissible cohomology theory of height n,• G is a finite group, and• Z is a finite G-complex.

Our main results will give a crude picture of the ring

E∗G(Z) = E∗(EG×G Z).

In particular, when Z is a point this is E∗(BG), as discussed in the introduction. Our ap-proach is influenced heavily by the work of Quillen [20, 21] and Hopkins-Kuhn-Ravenel [13,14].


Because of axiom (c), the ring E∗G has period two. Our methods cannot see E1G(Z), and

anyway it is zero in many cases, so we focus attention on E0G(Z), which is an ungraded

commutative ring. We shall prove in Corollary 5.4 that it is a finitely generated moduleover E0, and thus a complete semilocal Noetherian ring.

It will be convenient to give our results in geometric language. We will set up a suitabletechnical context in Section 3. For the moment, we just define the following affine objects:a scheme is a covariant representable functor from Noetherian rings to sets, and a varietyis a functor from algebraically closed fields to sets that is represented by a Noetherianring. We will define formal schemes in Section 3. We write X(Z,G) for the formal schemespf(E0

G(Z)) represented by E0G(Z), or for its underlying scheme spec(E0

G(Z)), or for itsunderlying variety. We omit Z from the notation when it is a point, so X(G) meansX(∗, G) = spec(E0(BG)).

We also write X for spf(E0), and G for spf(E0(CP∞)), which turns out to be a formalgroup. We will construct a “chromatic” filtration of X by closed formal subschemes:

X = X0 ≥ X1 ≥ . . . ≥ Xn.

In terms more familiar to topologists, we have Xk = spec(E0/Ik), where Ik is the ideal

generated by the coefficients of x, xp, . . . , xpk−1

in the p-series of a suitable formal grouplaw over E0. We also consider

X ′k = Xk \Xk+1 = spec(u−1k E0/Ik) = k’th pure stratum of the filtration,

which are schemes, but not formal schemes. One of the themes will be that we canunderstand phenomena over X by restricting to the subschemes Xk, and that they will beespecially simple over the pure strata X ′k. We adopt the permanent notational conventionthat a subscript k denotes restriction to the k’th chromatic stratum and that a dash denotesrestriction to a pure stratum. Thus

Xk(Z,G) = X(Z,G)×X Xk

(which is a formal scheme) and

X ′k(Z,G) = X(Z,G)×X X ′k(which is an ordinary scheme).

The idea is to describe X(Z,G) by working up from X(A), where A is Abelian. Thiscan be described completely in terms of the formal group G. Suitably interpreted (seeProposition 4.10), the answer is that

X(A) = Hom(A∗,G),

where A∗ is the character group Hom(A, S1). We define

Homk(A∗,G) = Hom(A∗,G)×X Xk = spf(E0(BA)/Ik) = Xk(A),

and similarly for Hom′k(A∗,G).

If G were a group in the category of sets (rather than the category of formal schemes),we could argue as follows: any map φ : A∗ −→G factors uniquely as the projection to aquotient of A∗, followed by a monomorphism. The quotients of A∗ are precisely the groups


B∗ where B ≤ A. We would thus have a decomposition Hom(A∗,G) =∐

B≤A Mon(B∗,G).As the category of formal schemes is more complicated than the category of sets, this doesnot quite work out as expected. The right substitute for the notion of a monomorphismA∗ −→ G turns out to be a level-A∗ structure on G. This concept is essentially due toDrinfel’d [2], and is studied in detail in the present context in [23]. When working over thesubscheme Xk, it is useful to refine this notion and use pk-fold level structure instead; therelevant theory is developed in Section 6, where we prove the following theorem.

Theorem 2.3. There are formal schemes Levelk(A∗,G) = spf(Dk(A)) (for 0 ≤ k < n)

with the following properties.

1. Levelk(A∗,G) is a closed subscheme of Homk(A

∗,G).2. Levelk(A

∗,G) is flat of degree |A|k|Mon(A∗, (Qp/Zp)n−k)| over Xk. In particular, itis empty if rankp(A) > n− k.

3. The evident map

Levelk(B∗,G)� Homk(B

∗,G)� Homk(A∗,G)

is a closed embedding.4. The evident map of formal schemes∐

B≤ALevelk(B

∗,G) −→Homk(A∗,G)

induces an isomorphism of varieties∐B≤A

Level′k(B∗,G) −→Hom′k(A

∗,G).

This will be proved as Theorem 6.4.We now consider a general finite group G. Let A denote the category whose objects are

the Abelian p-subgroups of G, with morphisms

A(A,B) = MapG(G/A,G/B) ' {g ∈ G | Ag ≤ B}/B.We have a functor

Aop ×A −→ Formal Schemes

given by (A,B) 7→ π0(ZA)×Hom(B∗,G), where ZA is the subspace of A-fixed points. We

denote its coend by∫ A

π0(ZA)×Hom(A∗,G) (see [18, Chapter IX] for the theory of endsand coends).

Theorem 2.4. There is a natural map of formal schemes∫ A

π0(ZA)×Hom(A∗,G) −→X(Z,G),

which induces an isomorphism of varieties. This gives a natural map of formal schemes∫ A

π0(ZA)× Homk(A∗,G) −→Xk(Z,G)


and a natural map of ordinary schemes∫ A

π0(ZA)×Hom′k(A∗,G) −→X ′k(Z,G),

both of which induce isomorphisms of varieties.

This will be proved as Theorem 5.10.We also show that the last of the above coends, where we have restricted to a pure

chromatic stratum, can be greatly simplified. Recall that Levelk(A∗,G) is empty unless

rankp(A) ≤ n− k.

Theorem 2.5. There is a natural map of formal schemes(∐A∈A

π0(ZA)× Levelk(A∗,G)

)/G −→Xk(Z,G)

such that the induced map of schemes(∐A∈A

π0(ZA)× Level′k(A∗,G)

)/G −→X ′k(Z,G)

is an isomorphism on the underlying varieties.

This will be proved as Theorem 9.2.In the case k = 0 our result is not the best available. The main theorem of [14] can be

reformulated as follows:

Theorem 2.6 (Hopkins-Kuhn-Ravenel). There is a natural isomorphism of schemes(∐A∈A

X(ZA)×X Level′0(A∗,G)

)/G −→X ′0(Z,G),

where X(ZA) = spf(E0(ZA)). Moreover, it is easy to see that X(ZA) has the same varietyas π0(ZA)×X.

We outline a proof of this in Appendix A; although it does not differ in an essential wayfrom that of [14] we feel the difference of language and inaccessibility of [14] make thisworthwhile.

One can deduce from Theorem 2.5 that the underlying variety of X ′k(Z,G) splits as adisjoint union of pieces indexed by (

∐A π0(ZA))/G where the coproduct runs over Abelian

subgroups of p-rank ≤ n−k, and that the irreducible components of Xk(Z,G) are indexedby the same set. In particular, when Z = ∗ the set of pieces correspond to conjugacy classesof Abelian subgroups of p-rank ≤ n− k. For some applications, we need to know that thescheme itself splits. Moreover, we would like a splitting that is valid after completing at Ik,rather than just reducing modulo Ik. In other words, we would like to study the scheme

X ′k(Z,G) = spec((u−1k E0

G(Z))∧Ik

).


For this we need some further notation. Let A ≤ G be Abelian and z an element of π0(ZA).The pair (A, z) defines a point of the finite G-set

∐A π0(ZA). We write NG(A, z) for its

stabiliser and [A, z] for its orbit, and put WG(A, z) = NG(A, z)/A.We will prove the following theorem.

Theorem 2.7. There is a closed subscheme Y ′k(Z,G,A, z) ≤ X ′k(Z,G) depending only onthe orbit of (A, z), such that

X ′k(Z,G) =∐[A,z]

Y ′k(Z,G,A, z),

where the coproduct is indexed by orbits. Moreover, there is a map of schemes

Level′k(A∗,G)/WG(A, z) −→Xk ×X Y ′k(Z,G,A, z),

which is an isomorphism on the underlying varieties.

This will be proved as Theorem 9.4.In the case Z = ∗ we write Y ′k(G,A) = Y ′k(∗, G,A, ∗), so that X ′k(G) =

∐(A) Y

′k(G,A).

Using the material developed in Section 8, we can make this somewhat more explicit whenG is Abelian. In particular, we have the following theorem.

Theorem 2.8. If A is a finite Abelian p-group and B ≤ A then Y ′k(A,B) is flat over X ′k,with degree |A|k|Mon(B∗, (Qp/Zp)n−k)|.

This will be proved as Theorem 9.5.We believe that we can prove the following result, but we have not yet worked out all

the details.

Conjecture 2.9. In good cases, the map π0LK(k)F (BG+, E) −→π0F (BG+, LK(k)E) is theprojection OX′k(G) � OY ′k(G,1). It is probably enough for E to be Landweber exact and E∗Gto be free over E∗ and concentrated in even degrees.

One application of these results is to the study of EG∗ (EG) = E∗BG, in the spirit

of [10, 11], and equally to EG∗ (EG × Z). For this discussion, we shall assume (as in the

conjecture) that E is Landweber exact and that E∗G is free over E∗ and concentrated in evendegrees. This is known to be true for a large class of groups. In the cases where E∗(BG)is known, it is rather complicated, containing many pieces like E∗/(p

∞, . . . , u∞n−1), often inodd degrees. Usually it can be calculated using local cohomology at the augmentation idealJ = ker(E0

G � E0); in fact, if G is a p-group or E admits an equivariant E∞ structure,there is a spectral sequence

E∗2 = H∗J(E0G) =⇒ EG

∗ (EG) = E∗(BG).

The groups H∗J(M) occuring here are the derived functors of the functor

ΓJM = H0J(M) = {m ∈M | JNm = 0 for N � 0}.

Our description of the variety suggests we should consider the Cousin complex associatedto the chromatic filtration of X(G), which has the form

C•(E0G) = (p−1E0

G −→u−11 E0

G/p∞ −→· · · −→u−1

n−1E0G/I

∞n−1 −→E0

G/I∞n ).


Note that the k’th term is a module over (u−1k E0

G)∧Ik

= OX′k(G). There is a spectral sequence

Es,t1 = HsH t

JC•(E0

G) =⇒ Hs+tJ (E0

G).

We will show using our splitting of X ′k(G) that this is concentrated on the line t = 0, givingthe case Z = ∗ of the following calculation.

Theorem 2.10. Assuming that E is Landweber exact and E∗G(Z) is a free module overE∗ concentrated in even degrees, there is a natural isomorphism

H∗J(E0G(Z)) = H∗ΓJC

•(E0G(Z)).

This will be proved as Theorem 10.1. Geometrically, the point is that

X ′k(G) =∐(A)

Y ′k(G,A),

and only the component indexed by A = 1 meets V (J), and this lies entirely over V (J) inthe sense that

V (J)×X(G) X′k(G) = Y ′k(G, 1),

and the local cohomology of this one factor is concentrated in degree zero.We should point out that the complex ΓJC

•(E0G) is still highly non-trivial. For example

if G is Abelian and of rank r then J may be generated as a radical ideal by r elements,and so the local cohomology must vanish above degree r. If r < n it is far from obviousthat the above complex is exact in degrees greater than r.

We conclude this introduction with a brief outline of the various sections of the paper.Section 3 contains some preliminary material about schemes and varieties, and Section 4develops the basic theory of admissible ring spectra. In Section 5, we prove Theorem 2.4,following Quillen’s proof of the parallel result in ordinary cohomology. In Sections 6 to 8,we develop the theory of multiple level structures. In Section 9, we combine this withTheorem 2.4 to prove Theorem 2.5 and Theorem 2.7. The applications to the calculationof E∗(BG) are developed in Section 10. In Appendix B, we outline an alternative approachto some of our results (under some quite restrictive hypotheses) using a variant of the Evensnorm map.

3. Schemes and varieties

In this section we set up the technical framework for our use of schemes and varieties.All rings will be assumed to be commutative and p-local.

First, recall that the Jacobson radical JA of a ring A is the intersection of its maximalideals, and that a ring is said to be semilocal if it has only finitely many maximal ideals. Asemilocal ring is said to be complete if it is complete with respect to its Jacobson radical.The complete semilocal Noetherian rings are precisely the finite products of complete local


Noetherian rings. We define three categories:

R = { complete Noetherian semilocal rings }R = { Noetherian rings }K = { algebraically closed fields }.

In R, we consider only maps f : A −→B such that f(JA) ≤ JB; equivalently, f must becontinuous with respect to the J-adic topologies on A and B. In the other two cases, weconsider all ring maps. Of course, any ring map between fields is injective.

Definition 3.1. If A ∈ R, we define a functor spf(A) : R −→Sets by

spf(A)(B) = R(A,B).

We define a formal scheme to be a functor X : R −→Sets such that X ' spf(A) for someA.

If X is a formal scheme, we write OX for the set of natural maps from X to the forgetfulfunctor. These form a ring, and an isomorphismX ' spf(A) gives an isomorphism OX ' Aby Yoneda’s lemma. Thus, the functors A 7→ spf(A) and X 7→ OX give an equivalence

between Rop and the category of formal schemes. We prefer to think about the functorcategory because many examples arise in topology where the functor is easier to describethan the representing ring. See [23] for some basic theory of formal schemes in this sense,and extensive discussion of some examples. See [24] for more examples and topologicalmotivation, in a rather more general technical framework. We will make heavy use of thefunctorial viewpoint in Sections 6 to 8 of the present work.

We next define schemes and varieties.

Definition 3.2. If A ∈ R, we define a functor spec(A) : R −→Sets by

spec(A)(B) = R(A,B).

We define a scheme to be a functor X : R −→Sets such that X ' spec(A) for some A. IfX is a scheme, we write OX for the ring of natural maps from X to the forgetful functor.

Definition 3.3. Let var(A) : K −→ Sets be the restriction of spec(A) to K. We define avariety to be a functor X : K −→ Sets such that X ' var(A) for some A. Thus, everyscheme X has an underlying variety; this will often also be called X, but we will writeXvar where it is necessary to emphasise the distinction.

We say that a map f : X −→ Y of schemes is a V -isomorphism if the induced mapXvar −→Yvar is an isomorphism, and that a map A −→B of rings is a V -isomorphism if theinduced map spec(A)←−spec(B) is so.

This turns out to be closely related (see Proposition 3.8) to the following more elementarycondition. We say that a map u : A −→ B of Fp-algebras is an F -isomorphism if everyelement of the kernel is nilpotent, and for each b ∈ B there is an integer k ≥ 0 such thatbpk ∈ u(A).


We next define a number of useful properties of (formal) schemes and maps betweenthem. We refer the reader to [19] (for example) for the basic facts about these concepts.

Definition 3.4. 1. A (formal) scheme X is connected if it is not the disjoint union oftwo nonempty (formal) schemes, equivalently if OX is not the product of two nonzerorings.

2. A (formal) scheme X is reduced if OX has no nonzero nilpotents. The reduced partXred of a scheme X is spec(OX/

√0).

3. A connected formal scheme X is smooth if OX is a regular local ring.4. A closed subscheme of a scheme X is a scheme of the form V (I) = spec(OX/I) for

some ideal I ≤ OX , and similarly for formal schemes. We say thatX is an infinitesimalthickening of Y if Y = V (I) for some nilpotent ideal I ≤ OX .

5. A map f : W −→X of (formal) schemes is finite if the corresponding map OX −→OWmakes OW into a finitely generated module over OX . The degree of f is the smallestpossible number of generators. When working over a fixed base scheme X, we writedeg[W ] for the degree of the given map W −→X.

6. A map f : X −→Y of (formal) schemes is dominant if the corresponding map OY −→OXhas nilpotent kernel. If f is finite and dominant, then f induces a surjective mapXvar −→Yvar.

7. A map f : X −→Y of (formal) schemes is flat if it makes OX into a flat module overOY , and similarly for faithful flatness. If Y is a connected formal scheme and f isfinite and flat then OX is a free module over OY .

It is more usual to define spec(A) to be the space of prime ideals of A, but this isnot convenient for our purposes. It turns out that a V -isomorphism of schemes gives acontinuous bijection of prime ideal spaces, which is a homeomorphism if the map is finite;as we shall not use this, we leave the proof to the reader.

We next state various results about V -isomorphisms, mostly proven in [21, AppendixB].

Proposition 3.5. An infinitesimal thickening is a V -isomorphism.

Proposition 3.6. Every V -isomorphism is dominant.

Proof. Let f : X −→Y be a V -isomorphism. Given a prime ideal p ≤ OY , we can embedthe integral domain OY /p in an algebraically closed field K. As f is a V -isomorphism, wecan find a map u : OX −→K such that u ◦ f ∗ is the composite OY � OY /p� K. It followsthat ker(f ∗) ≤ p. As the intersection of all primes in OY is the ideal of nilpotents, we seethat ker(f ∗) is nilpotent, as required.

Proposition 3.7 (Quillen). Let B be a ring, M a B-module, d : B −→M a derivation,and A the kernel of d (which is a subring of B). If B is a finitely generated A-module thenthe inclusion A −→B is a V -isomorphism.

Proof. This is Corollary B.6 of [21].

Proposition 3.8 (Quillen). A finite map of schemes over spec(Fp) is a V -isomorphism ifand only if it is an F -isomorphism.


Proof. This combines Propositions B.8 and B.9 of [21].

We next consider finite limits and colimits of (formal) schemes. We will often applythe following results to (co)ends, which can be converted to (co)limits as described in [18,Chapter IX]. Whenever we write something like lim

←- iWi, we implicitly refer to a finite

diagram. We first make the following convention:

Convention 3.9. When we talk about (co)limits of varieties, we mean (co)limits com-puted in the category of all functors from K to sets. It is easy to see that a limit ofvarieties is a variety, but a colimit of varieties need not be.

Most of the time, we will work over a fixed base scheme X. We write CX for the categoryof schemes W equipped with a finite map W −→X, and Γ(V,W ) = CX(V,W ) for the setof maps of schemes V −→W that commute with the given maps to X.

Proposition 3.10. The category CX has finite limits and colimits, which are converted bythe functor W 7→ OW into colimits and limits in the category of OX-algebras. In particular,the products are given by V ×X W = spec(OV ⊗OX OW ) and the coproducts are given byV qW = spec(OV × OW ). For any ring A ∈ R we have

(lim←-i

Wi)(A) = lim←-i

(Wi(A)),

where the limit on the right is computed in the category of sets over X(A). For anyalgebraically closed field K ∈ K, we have

(lim-→i

Wi)(K) = lim-→i

(Wi(K)).

Proof. It is not hard to see that CX is equivalent to the opposite of the category of finite OX-algebras. Given this, the only nontrivial point is the last sentence, which is Proposition B.1of [21].

Proposition 3.11. If X is a formal scheme, then CX is equivalent to the category offormal schemes W equipped with a finite map to X.

Proof. This just means that every finite OX-algebra is a complete semilocal Noetherianring, and that all maps of such algebras are continuous with respect to the Jacobsonradicals. This is a standard piece of commutative algebra.

Definition 3.12. Let {Wi} be a finite diagram of schemes, equipped with compatiblemaps Wi −→W . We say that W is a V -colimit of the diagram if the induced map lim

-→ iWi −→

W is a V -isomorphism.

Proposition 3.13. Any map of schemes X ′ −→X gives rise to a functor W 7→ W ×X X ′from CX to CX′; explicitly, W×XX ′ = spec(OW⊗OXOX′). This has the following properties:

(lim←-i

Wi)×X X ′ = lim←-i

(Wi ×X X ′).


The canonical map

lim-→i

(Wi ×X X ′) −→(lim-→i

Wi)×X X ′

is a V -isomorphism, and a genuine isomorphism if the colimit is a coproduct. We alsohave

(W ×X X ′)var = Wvar ×Xvar X′var,

so that the pullback functor preserves V -isomorphisms and V -colimits.

Proof. The only nontrivial point is the V -isomorphism. For any algebraically closed fieldK, it is easy to see that (Wi ×X X ′)(K) = Wi(K)×X(K) X

′(K) and

((lim-→i

Wi)×X X ′)(K) = (lim-→i

Wi)(K)×X(K) X′(K).

Proposition 3.10 gives (lim-→ i

Wi)(K) = lim-→ i

Wi(K) and (lim-→ i

X ′ ×X Wi)(K) = lim-→ i

(X ′ ×XWi)(K), and the claim follows.

Remark 3.14. In the light of Proposition 3.8, this has the following consequence. Ifwe have a Noetherian Fp-algebra A and a finite diagram of finite A-algebras Ai, and aNoetherian A-algebra B, then the natural map B ⊗A lim

←- iAi −→ lim

←- iB ⊗A Ai is an F -

isomorphism. It would be nice to have a more direct proof of this fact.

We conclude by showing that a splitting of varieties gives rise to a splitting of theunderlying schemes.

Proposition 3.15. Suppose that we have a finite collection of finite maps of schemesfi : Xi −→ Y , such that the combined map f :

∐iXi −→ Y is a V -isomorphism. Then

there are closed subschemes Yi ⊆ Y such that Y =∐

i Yi and fi factors through a finiteV -isomorphism Xi −→Yi.

Proof. Write B = OY and Ai = OXi and Ii = ker(B −→Ai). Because f is a V -isomorphismand thus dominant, we see that J =

⋂i Ii is nilpotent. We claim that if i 6= j then

Ii+Ij = B. If not, we can find a map B/(Ii+Ij) −→K for some algebraically closed field K.As the map B/Ii −→Ai is finite and injective, we can factor the map B/Ii � B/Ii+Ij −→Kthrough Ai. Similarly, we can factor it through Aj, so the map B −→B/Ii+Ij −→K definesa point of Xi(K) ∩Xj(K), which contradicts the assumption that Y (K) =

∐iXi(K).

We can now apply the Chinese Remainder Theorem to see that the natural map B/J −→∏iB/Ii is an isomorphism. This means that there are elements ei ∈ B such that ei = 1

(mod Ii) and ei ∈⋂j 6=i Ij , so that e2

i − ei lies in the nilpotent ideal J . By the Idempotent

Lifting Theorem, we can modify the ei modulo J in a unique way to ensure that e2i = ei,

and then∑

i ei = 1. This means that B =∏

iCi as rings where Ci = Bei. It is easy tocheck that the map B −→Ai factors through a V -isomorphism Ci −→Ai, so we can putYi = spec(Ci).


4. Admissible cohomology theories

In this section we prove some basic facts about admissible cohomology theories. For easeof reference, we repeat the definition.

Definition 4.1. Let pCP∞ : CP∞ −→CP∞ be the map classifying the p’th tensor power ofthe canonical line bundle. A p-local commutative ring spectrum E is admissible if

(a) E0 is a complete local Noetherian ring.(b) E1 = 0.(c) E2 contains a unit.

(d) pCP∞ induces a nonzero self-map of E0(CP∞).

It follows from (b) and (c) that E∗ is concentrated in even degrees, so the Atiyah-Hirzebruch spectral sequence

H∗(CP∞;E∗) =⇒ E∗CP∞

collapses. This means that E∗CP∞ = E∗[[x]] for suitable x ∈ E∗CP∞, in other words,E is complex-orientable. Because of (c), we can choose such an x in degree zero, so thatE0(CP∞) = E0[[x]]. In particular, this is a complete Noetherian local ring. By a similarargument, we see that

E0(CP∞ × CP∞) = E0(CP∞)⊗E0E0(CP∞) = E0[[x ⊗ 1, 1⊗ x]].

For convenience, we shall assume that we have chosen an element x as above once and forall, but as far as possible we shall state our results in a form which is independent of thischoice.

We now define formal schemes X = spf(E0) and G = spf(E0(CP∞)). Because CP∞ isa commutative H-space, we get a multiplication map

µ : G×X G = spf(E0(CP∞ × CP∞)) −→spf(E0(CP∞)) = G.

This makes G into a group object in the category of formal schemes over X, or in otherwords, a formal group scheme over X. The basic theory of formal groups from this pointof view is developed in [23]. Condition (d) has a natural reformulation in terms of G. AsG is an Abelian group object, its endomorphisms form an Abelian group under addition,so it makes sense to consider

pkG = pk.1G = spf(E0(pkCP∞)) : G −→G.

Condition (d) says that pG is not the zero homomorphism. We can of course translate backto the more traditional language: there is a unique formal group law F (s, t) over OX suchthat

µ∗(x) = F (x⊗ 1, 1⊗ x) ∈ E0(CP∞ ×CP∞) = E0[[x⊗ 1, 1⊗ x]].

In terms of this formal group law, condition (d) simply says that [p]F (x) 6= 0.We will often consider the following subgroups of G.


Definition 4.2.

G(k) = ker(pkG : G −→G) = spf E0[[x]]/[pk](x).

The next two results are well-known. Proofs are given under more restrictive hypothesesin [5], but they generalise readily.

Proposition 4.3. There is an integer k ≥ 0 (called the strict height of G) such that [p](x)

has the form f(xpk) with f ′(0) 6= 0. There is an integer n ≥ 0 (called the height of G)

such that [p](x) has the form g(xpn) modulo m[[x]], with g′(0) ∈ (E0/m)×.

Proposition 4.4. The maps pkG : G −→G and G(k) −→X are flat of degree pnk.

Note that the strict height of G may change if we pull back along a finite map Y −→X,but the height does not. Note also that [p](x) has Weierstrass degree pn, where n is theheight.

Definition 4.5. For 0 ≤ k ≤ n, let uk be the coefficient of xpk

in [p](x). Note that u0 = p,that uk ∈ m for 0 < k < n, and that un 6∈ m. Write

Ik = (u0, . . . , uk−1) ≤ E0

Xk = spf(E0/Ik) ≤ X

X ′k = spec(u−1k E0/Ik)

X ′k = spec((u−1k E0)

∧Ik

).

One can check quite directly that these objects do not depend on the choice of coordinatex. Moreover, Ik is also the ideal generated by the coefficients of xi in [p](x) for 0 < i < pk.Thus G has strict height ≥ k over Xk.

Some of our results will only be valid if E is Landweber exact. We recall the definition,adapted to take account of our other assumptions.

Definition 4.6. An admissible cohomology theory E is Landweber exact if the the se-quence (u0, u1, . . . , un−1) is regular in E0.

The main example is Morava E-theory: all other examples are very closely related to it.Before proceeding to the next definition, we must clear up a possible ambiguity. Given

a space Y , there are two possible meanings for u−1k E0(Y ). On the one hand, we can form

the module E0(Y ) and then invert uk algebraically; we call the result u−1k (E0(Y )). On the

other hand, multiplication by uk gives a self-map of the spectrum E, whose telescope is aspectrum u−1

k E, so we can form (u−1k E)0(Y ). These two objects can be very different. For

example, let E be Morava E-theory and let k = 0. Then u−1k E is a rational spectrum, so

(u−1k E)0(BG) = u−1

k E0, but u−1k (E0(BG)) is much more complicated, as described in [14].

Convention 4.7. The symbol u−1k E0(Y ) will always mean u−1

k (E0(Y )), as discussed inthe previous paragraph.


Definition 4.8. Let G be a finite group, and Z a G-space. We write E∗G(Z) = E∗(EG×GZ), and

X(Z,G) = spf(E0G(Z))

Xk(Z,G) = spf(E0G(Z)/Ik) = X(Z,G)×X Xk

X ′k(Z,G) = spec(u−1k E0

G(Z)/Ik) = X(Z,G)×X X ′kX ′k(Z,G) = spec((u−1

k E0G(Z))

∧Ik

)

Remark 4.9. It is often the case that there is an admissible ring spectrum F and a ringmap E −→F inducing an isomorphism π∗(E)/Ik ' π∗(F ) (so F could reasonably be calledE/Ik). If so, the map E0

G(Z)/Ik −→F 0G(Z) need not be an isomorphism, but it will induce

an isomorphism of varieties. This can be proved by applying Theorem 2.5 to both E and F .There is a much more direct proof using Bockstein spectral sequences in any context wherethese spectral sequences exist, but one needs either the technology of highly structured ringspectra [3, Chapter V] or the technology of generalised Moore spectra [15] to set them up.We give this argument in Appendix C.

Proposition 4.10. Let A be a finite Abelian group. Then there is a scheme Hom(A,G)over X such that

Γ(W,Hom(A,G)) = Hom(A,Γ(W,G))

for all schemes W ∈ CX. Moreover, Hom(A,G) is finite and flat over X, with degree|A(p)|n.

Proof. If A has order prime to p then Hom(A,Γ(W,G)) = 0, so we may assume that A isa p-group, say

A = 〈a0, . . . , ar−1 | pdiai = 0〉 =r−1⊕k=0

Z/pdk .

We then have Hom(A,G) =∏

kG(dk). This means that

OHom(A,G) = OX [[x0, . . . , xr−1]]/([pd0 ](x0), . . . , [pdr−1](xr−1));

by Proposition 4.4 this is a free module of rank∏

i pndi = |A|n over E0 = OX .

Convention 4.11. We write φ : A −→G when we mean there is a homomorphism φ : A −→Γ(W,G) for some scheme W over X that is to be understood from the context.

Proposition 4.12. If A is a finite Abelian group, then there is a natural isomorphism

X(A) = Hom(A∗,G),

where A∗ = Hom(A, S1) is the character group of A. In particular, X(A) is finite over X.

Proof. We may assume that A is a p-group. By construction, maps X(A) −→Hom(A∗,G)over X biject with homomorphisms φ : A∗ −→ Γ(X(A),G), and Γ(X(A),G) is the set ofcontinuous E0-algebra maps E0(CP∞) −→E0(BA). By regarding CP∞ as BS1, we get a


canonical map φ : A∗ −→Γ(X(A),G) defined by φ(α) = E0(Bα), and thus a canonical mapX(A) −→Hom(A∗,G). One can check that there is a natural commutative square

X(A⊕B) Hom((A⊕B)∗,G)

X(A)×X X(B) Hom(A∗,G)×X Hom(B∗,G).u

w

w

Next, recall the well-known cofibration

BZ/pk −→CP∞ −→(CP∞)Lpk

.

Using the Thom isomorphism, we get a long exact sequence

· · · ←−E0BZ/pk ←−OX [[x]][pk](x)←−−−− OX [[x]]←−· · · .

As [pk](x) is not a zero-divisor in OX [[x]], we see that

E0(BZ/pk) = OX [[x]]/[pk](x) = OG(k) = OHom((Z/pk)∗,G).

One can check that this is the same as the map considered previously. Similarly, we seethat E1(BZ/pk) = 0. As E0(BZ/pk) is free of finite rank over E0, we have a Kunnethisomorphism

E0(B(Z/pk)× Z) = E0(BZ/pk)⊗E0 E0(Z).

It follows easily that X(A⊕B) = X(A)×XX(B), and that our map X(A) −→Hom(A∗,G)is an isomorphism for all A.

5. The proof of Theorem 2.4

In this section, we prove Theorem 2.4. The method is essentially due to Quillen, but weadapt some of his ideas to our particular technical context, and we present the argumentin the language of equivariant topology.

We begin by outlining the argument. Consider the space T of complete flags in a faithfulcomplex representation of G; we show that

(a) T has Abelian isotropy groups.(b) There is an equaliser diagram

E0G(Z) −→E0

G(Z × T )⇒ E0G(Z × T × T ).

(c) E0G(Z) is a finitely generated module over E0.

Next, we try to extract information from the filtration of Z by G-skeleta, or equivalentlyfrom the equivariant Atiyah-Hirzebruch spectral sequence. The conclusion is that E0

G(Z)has the same variety as the ring H0

G(Z;E), where H0G(·;E) denotes ordinary cohomology

in the sense of Bredon, for the coefficient system E : G/H 7→ E∗G(G/H) = E∗H .


We next consider the category A of Abelian p-subgroups of G, and the functor

EA(Z) =

∫A∈A

Map(π0(ZA), E0A)

considered by Quillen (he denotes it AG(Z) and writes the end as a limit). We show thatthis is the same as H0

G(Z;E) if Z has Abelian isotropy groups. By combining this with (a),(b) and (c) above, we shall see that

X(Z,G) =

∫ A

π0(ZA)×X(A)

as varieties.We now start work on the proof. Choose a faithful complex representation V of G, of

dimension d, say. Let T = Flag(V ) be the space of complete flags of subspaces of V , so apoint of V is a chain

V = (0 = V0 < V1 < . . . < Vd = V )

where dim(Vk) = k for all k. As T is a compact G-manifold, it can be made into a finiteG-CW complex.

Lemma 5.1. Every point in T has Abelian isotropy group.

Proof. If H ≤ G stabilises a flag 0 = V0 < V1 < · · · < Vd = V then we have an isomor-phism V '

⊕i Vi/Vi−1 of H-representations. Since V is faithful, H embeds in the group∏

i Aut(Vi/Vi−1), and since Vi/Vi−1 is one dimensional, Aut(Vi/Vi−1) is Abelian.

We will need the following well-known calculation.

Proposition 5.2. Let Y be a space, and W a complex vector bundle of dimension d overY . Let π : Flag(W ) −→Y be the associated flag bundle, so a point of π−1y is a completeflag in the vector space Wy. There is an evident filtration

0 = W0 < W1 < . . . < Wd = π∗W

of bundles over Flag(W ). We let xk ∈ E0(Flag(W )) denote the Euler class of Wk/Wk−1,and write σk for the k’th symmetric function in the variables xi. We also let ck ∈ E0(Y )denote the k’th Chern class of W . There is a natural isomorphism

E∗(Flag(W )) = E∗(Y )[[x1, . . . , xd]]/(σk − ck | 0 ≤ k ≤ d).

Moreover, this is a free module over E∗(Y ) of rank d!, spanned by the monomials xα =∏di=1 x

αii for which 0 ≤ αi < i for all i.

Corollary 5.3. For any finite G-complex Z, there is an equaliser diagram

E∗G(Z) −→E∗G(Z × T )⇒ E∗G(Z × T × T ).

Proof. We use an explicit calculation to replace Quillen’s use of faithfully flat descent; thisavoids a discussion of Kunneth isomorphisms.

Let S be the set of multiindices α = (α1, . . . , αd) such that 0 ≤ αi < i. By applyingProposition 5.2 to the bundle V −→EG ×G (Z × V ) −→EG ×G Z, whose associated flag


bundle is EG×G (Z ×T ), we find that E∗G(Z ×T ) is the free module E∗G{xα | α ∈ S}. Onreplacing Z by Z × T we find that

E∗G(Z × T × T ) = E∗G(Z × T ){yβ | β ∈ S} = E∗G(Z){xαyβ | α, β ∈ S}.The two projections Z×T ×T ⇒ Z×T induce maps sending xα to xα and yα respectively.It follows immediately that the given diagram is an equaliser.

Corollary 5.4. E∗G(Z) is a finitely generated module over E∗ (and thus a complete semilo-cal Noetherian ring).

Proof. First suppose that Z has Abelian isotropy groups, so it is built from cells of theform Bk × G/A with A Abelian. We know from Proposition 4.12 that E∗G(G/A) = E∗A isfinitely generated over E∗. An evident induction on the number of cells shows that E∗G(Z)is also finitely generated.

Even if Z does not have Abelian isotropy, the product Z × T does, so E∗G(Z × T ) isfinitely generated. We know from Proposition 5.2 that E∗G(Z) is a submodule of E∗G(Z×T ),so it is also finitely generated.

Remark 5.5. The finite generation does not hold without some completeness hypothesison E. For example it fails for uncompleted K-theory.

We now look at E0G(Z) through the eyes of the equivariant Atiyah-Hirzebruch spectral

sequence. This is the spectral sequence arising from the filtration of Z by skeleta, and ithas

Es,t1 = Es+t

G (Z(s), Z(s−1)).

If we let Et denote the coefficient system G/A 7→ EtG(G/A) = Et(BA), then we have

Es,t2 = Hs

G(Z;Et).

We need to know this spectral sequence is multiplicative from E2 onwards: this followsby using the fact that the diagonal map Z −→ Z × Z is skeletal if Z × Z is given theproduct filtration. An alternative is to use the Postnikov filtration of E to obtain a spectralsequence: it is proved in [8, Appendix B] that this agrees with the first from the E2 termonwards.

Lemma 5.6. The edge homomorphism

i : E0G(Z) −→H0

G(Z;E0)

of the Atiyah-Hirzebruch spectral sequence is a V -isomorphism.

Proof. Let d be the dimension of Z, so that Es,∗1 = 0 for s > d. We first show that the

kernel of i consists of nilpotents. Indeed, if i(x) = 0 then x has positive filtration, so xd+1

has filtration at least d + 1 and thus is zero. Thus, E0G(Z) has the same variety as its

image under i, which is precisely E0,0∞ = E0,0

d+1. Next, observe that E0,0r+1 is the kernel of

a derivation from the ring E00r to the module Er+1,−r

r , and that everything in sight is afinitely generated module over E0 (by Corollary 5.4). By Proposition 3.7, we see that E0,0

r+1

has the same variety as E0,0r , and the claim follows.


We now appeal to an explicit description of H0G(Z;E), which is valid when Z has Abelian

isotropy. Recall that A is the category whose objects are the Abelian p-subgroups of G,with morphisms

A(A,B) = MapG(G/A,G/B) ' {g ∈ G | Ag ≤ B}/B.We have two contravariant functors from A to sets, given by A 7→ π0(ZA) and A 7→ E0

A =E0G(G/A). We define

EA(Z) = NatA∈A(π0(ZA), E0A) =

∫A

Map(π0(ZA), E0A).

Lemma 5.7. If H ≤ G is Abelian (but not necessarily a p-group) then EA(G/H) = E0H .

Proof. We can write H = B × C where B is a p-group and C is a p′-group. Note thatthe action of C on E0

B by conjugation is trivial because H is commutative. Since B is aretract of H, E0

B is a retract of E0H ; by a transfer argument E0

H = E0B. Note that if A is

a p-group, a coset g(B × C) is A-fixed only if the corresponding coset gB is A-fixed, so(G/(B × C))A = (G/B)A/C, so that π0((G/H)A) = A(A,H) = A(A,B)/C. By Yoneda’slemma, we have NatA(A(A,B), E0

A) = E0B, so NatA(A(A,B)/C,E0

A) = (E0B)C = E0

B. ThusEA(G/H) is just E0

B = E0H .

Lemma 5.8. There is a natural map H0G(Z;E) −→EA(Z), which is an isomorphism if all

isotropy groups of points of Z are Abelian.

Proof. From the definitions one sees that there are natural maps

H0G(Z;E) −→H0

A(ZA;E) −→H0(ZA;E0A) = Map(π0(ZA), E0

A),

and that these assemble to give a natural map

αZ : H0G(Z;E) −→EA(Z).

Now suppose that Z is a union of two subcomplexes Z0 and Z1. We have a Mayer-Vietorissequence

0 −→H0G(Z;E) −→H0

G(Z0;E)⊕H0G(Z1;E) −→H0

G(Z0 ∩ Z1;E).

Similarly, for each A we have ZA = ZA0 ∪ ZA

1 , so there is a coequaliser of sets

π0(ZA0 ∩ ZA

1 )⇒ π0(ZA0 )q π0(ZA

1 ) −→π0(ZA).

By applying Map(−, E0A), we get an equaliser diagram. It is easy to see that equalisers

commute with ends, so we get a left exact sequence

0 −→EA(Z) −→EA(Z0)⊕EA(Z1) −→EA(Z0 ∩ Z1).

By comparing these two sequences, we see that αZ is an isomorphism provided that αZ0 ,αZ1 and αZ0∩Z1 are isomorphisms. As Z has Abelian isotropy groups, it is built from cellsof the form Bm×G/H, with H Abelian. By induction on the number of cells, it is enoughto check that αZ is an isomorphism when Z = Sm−1 × G/H or Z = Bm × G/H. Thisreduces easily to a check that α : H0

G(G/H;E) −→EA(G/H) is an isomorphism. The lefthand side is just E0

H , which is the same as the right hand side by Lemma 5.7.


Lemma 5.9. For any Z and T , the evident diagram

EA(Z) −→EA(Z × T )⇒ EA(Z × T × T )

is an equaliser.

Proof. Suppose that A ∈ A. Because π0((Z × T )A) = π0(ZA) × π0(TA) and so on, it isclear that the diagram

π0(ZA)←−π0((Z × T )A)⇔ π0((Z × T × T )A)

is a coequaliser. By applying Map(−, E0A) we therefore get an equaliser. As equalisers

commute with ends, the claim follows.

We now restate and prove Theorem 2.4.

Theorem 5.10. There is a natural map of formal schemes∫ A

π0(ZA)×Hom(A∗,G) −→X(Z,G),

which gives an isomorphism of the underlying varieties. Thus, there are also V -isomorphisms∫ A


and ∫ A

π0(ZA)×Hom′k(A∗,G) −→X ′k(Z,G).

Proof. As previously, we let T denote the space of flags in a faithful complex representationof G. By Lemma 5.1, we know that T has only Abelian isotropy groups. It follows thatthe same is true of Z × T and Z × T × T . Consider the diagram

EG(Z) EG(Z × T ) EG(Z × T × T )

H0G(Z;E) H0

G(Z × T ;E) H0G(Z × T × T ;E)

EA(Z) EA(Z × T ) EA(Z × T × T )

u

i0

v w

u

i1

w

w

u

i2

u

j0

w

u

j1

w

w

u

j2

v ww

w

The first and third rows are exact by Corollary 5.3 and Lemma 5.9 respectively. The mapsi0, i1 and i2 are V -isomorphisms by Lemma 5.6. The maps j1 and j2 are isomorphisms byLemma 5.8. Everything in sight is a finitely generated module over the Noetherian ringE0, by Corollary 5.4. We now apply the functor var(−) to get a diagram of varieties withthe arrows reversed. The first and third lines are coequalisers by Proposition 3.10, andall vertical maps except for var(j0) are known to be isomorphisms. It follows immediately


that var(j0) is also an isomorphism, so that X(Z,G) = var(EA(Z)). We can rewrite theend that defines EA(Z) as the inverse limit of a finite diagram of rings that are finitelygenerated as modules over E0. By Proposition 3.10, this becomes colimit diagram whenwe apply var(−). This shows that

(5.0.1) X(Z,G)var = var

(∫A

Map(π0(ZA), E0A)

)=∫ A

var(Map(π0(ZA), E0A)) =

∫ A

π0(ZA)×X(A).

Using Proposition 4.10, we can rewrite this as

X(Z,G)var =

∫ A

π0(ZA)×Hom(A∗,G),

as claimed. The other two claims follow at once using Proposition 3.13.

6. Multiple level structures

In this section, we study the notion of a multiple level structure on a formal group. Thistheory is closely related to questions studied in [23], and we will assume some familiaritywith that paper. For the purposes of the next three sections, we do not need to startwith an admissible cohomology theory. We merely assume that we have a formal groupG of finite height n over a connected formal scheme X. We also fix an integer k with0 ≤ k ≤ n, and assume that G has strict height at least k. Thus, to apply this section tothe topological questions studied in previous sections, we need to replace X by Xk and Gby G×X Xk.

We now define the main object of interest.

Definition 6.1. Let k ≥ 0 be an integer, A a finite Abelian p-group, and Y a scheme overX. We say that a homomorphism φ : A∗ −→ Γ(Y,G) is a pk-fold level-A∗ structure if wehave an inequality of divisors

pk[φA∗(1)] ≤ G(1) ∈ Γ(Y,Div(G)).

Here as usual A∗(1) = ker(p : A∗ −→ A∗) = (A/p)∗ and G(1) = ker(p : G −→ G). It isequivalent to require that [p](x) be divisible by∏

α∈A∗(1)

(x− x(φ(α)))pk

in OY [[x]].

¿From a purely algebraic point of view, there is no reason to consider maps A∗ −→Grather than maps A −→G, but it is convenient for the topological applications.

The case k = 0 just gives level structures as studied in [23]. We shall usually ignore thiscase.

The following result follows easily from [23, Proposition 16].


Proposition 6.2. The functor from schemes over X to sets defined by

Y 7→ { pk-fold level-A∗ structures on G over Y }is represented by a scheme Levelk(A

∗,G) over X. Moreover, Levelk(A∗,G) is a closed

subscheme of Hom(A∗,G) and thus is finite over X.

The next introduce notation for a group that will occur in many places.

Definition 6.3. We write Λk = (Qp/Zp)n−k.

The main result is as follows.

Theorem 6.4. The scheme Levelk(A∗,G) is flat over X, of degree |A|k|Mon(A∗,Λk)|. If

(G, X) is of universal type (see Definition 6.5) then Levelk(A∗,G) is smooth. For each

B ≤ A, there is a natural closed inclusion Levelk(B∗,G) −→Hom(A∗,G). Together these

maps give a map ∐B

Levelk(B∗,G) −→Hom(A∗,G),

and the resulting map ∐B

Level′k(B∗,G) −→Hom′(A∗,G)

is an infinitesimal thickening (and thus a V -isomorphism).

We shall prove this theorem at the end of this section, except that we will only obtainan inequality

deg[Levelk(A∗,G)] ≤ |A|k|Mon(A∗,Λk)|.

The reverse inequality will be proved as Corollary 8.4.For the rest of this section, A will be a finite Abelian p-group, and B will be a subgroup

of A. There is an evident restriction map π : A∗ � B∗; the kernel is the annihilator of B,which we write as B◦. We can choose a presentation

A = 〈a0, . . . , ar−1 | pdiai = 0 for all i〉.We shall order the generators so that 0 < d0 ≤ . . . ≤ dr−1. We write αi for the element ofA∗ that is dual to ai in the evident sense, so that A∗ = 〈αi | pdiαi = 0〉. Given a schemeY over X and a map φ : A∗ −→Γ(Y,G), we write xα = x(φ(α)) and xi = xαi = x(φ(αi)).

Let φ be a pk-fold level-A∗ structure on G over Y . Over the special fibre of Y we haveφ = 0 so xp

r+kdivides [p](x), which is a unit multiple of xp

n. It follows that r ≤ n − k

(unless Y is empty). Thus, Levelk(A∗,G) = ∅ if rankp(A) > n− k.

Many questions about multiple level structures become simpler if the base scheme Xhas good properties. Fortunately, we can often assume this, as we now explain.

Definition 6.5. We shall say that the pair (G, X) is of universal type if the sequence ofelements (uk, . . . , un−1) is regular on OX and generates the maximal ideal (and thus X isa smooth scheme).


If k > 0 and (G, X) has universal type then OX = K[[uk, . . . , un−1]] for some field Kof characteristic p, and if k = 0 then OX = R[[u1, . . . , un−1]] for some discrete valuationring R whose maximal ideal is generated by p. In either case, OX is a unique factorisationdomain and has many other good properties. We can often assume that X has universaltype, and deduce the general case by base change using the following proposition. We willgive this argument in detail in Corollary 6.11, and subsequently leave similar argumentsto the reader.

Proposition 6.6. Let G be a formal group of height n and strict height at least k over aformal scheme X. Then there are pullback diagrams of formal groups

G H K

X W Vu

u w

u uu

fw

g

where f is faithfully flat and (K, V ) has universal type.

Proof. Let m < OX be the maximal ideal, and let K be the perfect closure of OX/m. ByEGA 0III10.3.1 [12] there is a complete Noetherian local ring OW and a faithfully flat mapOX −→OW such that the associated extension of residue fields is the inclusion OX/m� K.We define H = G×X W . By the Lubin-Tate deformation theory (see [17], or [23, Section6] for an account in geometric language) there is a smooth scheme U , a formal group Lover U , and a pullback diagram

H L

W U

w

u uw

Moreover, we have OU = R[[w1, . . . , wn−1]] for some complete discrete valuation ring R

with R/p = K, where wi is the coefficient of ypi

in the [p](y) for some coordinate y on L.It follows that

V = spf(OU/(p, w1, . . . , wk−1)) = spf(K[[wk, . . . , wn−1]])

is the closed subscheme where L has strict height at least k. As H has strict height at mostk, we see that the map W −→U must factor through V . We thus get a diagram as claimedwith K = L×U V .

Definition 6.7. We use the following notation.

D = OX

R(A) = OHom(A∗,G) = D[[x0, . . . , xr−1]]/([pd0 ](x0), . . . , [pdr−1 ](xr−1))

D(A) = OLevelk(A∗,G).


We also write D′ = u−1k D, R′(A) = u−1

k R(A) and so on, and X ′ = spec(D′). Note that D′

is not a complete local ring, so our theorems about functors represented by various ringscannot be applied directly to D′.

Proposition 6.8. If φ : A∗ −→ Γ(X,G) is such that for all 0 6= α ∈ A∗(1) the elementxα ∈ OX is not a zero-divisor, then φ is a pk-fold level-A∗ structure. In particular, thisholds if OX is an integral domain and φ is injective.

Proof. We may assume that A∗ = A∗(1) (which just simplifies notation). Let g(x) bethe Weierstrass polynomial that is a unit multiple in OX [[x]] of [p](x), and write f(x) =∏

α(x− xα)pk. We need to prove that f divides g in OX [x]. Because divisibility of monic

polynomials is a closed condition, it is enough to prove this over any extension ring of OX .Note that ∆ =

∏α6=β(xα − xβ) is a unit multiple of

∏α6=β xα−β (because x − y is a

unit multiple of x −F y), so ∆ is not a zero-divisor, so the natural map OX −→∆−1OX isinjective. If α 6= β then x−xα and x−xβ generate the unit ideal in ∆−1OX [x], so the same

is true of (x− xα)pk

and (x− xβ)pk. It follows from the Chinese Remainder Theorem that

the intersection of the ideals ((x − xα)pk) is the same as their product. It is thus enough

to check that (x − xα)pk

divides f(x) in OX [x], or equivalently that (x −F xα)pk

divides

[p](x) in OX [[x]]. As G has strict height at least k, we know that xpk

divides [p](x) and

thus that (x −F xα)pk

divides [p](x −F xα). However, [p](x −F xα) = [p](x) −F [p](xα) =

[p](x)−F xpα = [p](x), so (x−F xα)pk

divides [p](x) as required.

The following result is a partial converse to the above.

Proposition 6.9. Let G be a formal group of strict height precisely k over an integraldomain OX . Then a map φ : A∗ −→Γ(X,G) is a pk-fold level structure if and only if it isinjective.

Proof. The hypothesis on G is that xpk+1 does not divide [p](x). If φ is a pk-fold level

structure and 0 6= α ∈ A∗(1) then xpk(x − xα) divides [p](x) so we must have xα 6= 0. It

follows that ker(φ) ∩ A∗(1) = 0 and thus that φ is injective. The converse is covered bythe previous proposition.

Proposition 6.10. If X is of universal type then the scheme Levelk(A∗,G) is smooth and

is flat over X.

Proof. Choose m so large that pmA = 0, and let g(x) be the Weierstrass polynomial thatis a unit multiple of [pm](x), so g is monic and has degree pnm. Let L be the splittingfield of g over the field of fractions of OX , and R the subring of L generated by OX andthe roots of g. These roots form a group A′ under formal addition. It is easy to see thatthey all have the same multiplicity, and that the multiplicity of zero is pmk; it followsthat |A′| = p(n−k)m. A similar argument shows that the subgroup A′(j) has order p(n−k)j

for 0 ≤ j ≤ m, and it follows from the structure theory of finite Abelian groups thatA′ ' (Z/pm)n−k. Write Y = spf(R), so A′ is naturally a subgroup of Γ(Y,G). Becauserankp(A

∗) ≤ n− k and pmA∗ = 0, we can choose a monomorphism φ : A∗ −→A′ ≤ Γ(Y,G).By Proposition 6.8, this is a pk-fold level structure (because R is an integral domain). The


evident map Y −→X (which is finite and dominant) thus factors through Levelk(A∗,G). It

follows that Levelk(A∗,G) −→X is dominant, so that dim(Levelk(A

∗,G)) ≥ dim(X) = n−k.Note that Levelk(A

∗,G) is by construction a closed subscheme of Hom(A∗,G) =∏

iG(di).It follows that the maximal ideal in D(A) is generated by x0, . . . , xr−1, uk, . . . , un−1, andalso that Levelk(A

∗,G) is finite over X. Let S be the set {x0, . . . , xr−1, uk+r, . . . , un−1},so |S| = n − k. Write D(A) = D(A)/(S). Over this ring we know that φ(αi) = 0 for all

i and thus that φ(α) = 0 for all α ∈ A∗. It follows that∏

α∈A∗(1)(x − xα)pk

= xpr+k

. We

also know that this divides [p](x), so that G has strict height at least r + k over D(A).This implies that uk = . . . = ur+k−1 = 0 in D(A). The rest of the u’s vanish by definition.It follows that the maximal ideal in D(A) is zero, or equivalently, that S generates themaximal ideal in D(A). As |S| is the same as the Krull dimension of D(A), we see thatD(A) is a regular local ring, or in other words that Levelk(A

∗,G) is a smooth scheme. Asa finite dominant map of smooth schemes is flat, we see that Levelk(A

∗,G) is flat overX.

Corollary 6.11. Even when X is not universal, the map Levelk(A∗,G) −→X is flat.

Proof. Choose a diagram as in Proposition 6.6. By looking at the represented functors, wesee easily that

Levelk(A∗,H) = Levelk(A

∗,G)×X W = Levelk(A∗,K)×V W.

As (K, V ) has universal type, we see that Levelk(A∗,K) is flat over V . As flatness

is preserved by pullbacks, we see that Levelk(A∗,H) is flat over W . Thus, the map

Levelk(A∗,G) −→ X becomes flat after pullback along the faithfully flat map W −→ X;

this means that it is itself flat, by standard properties of flatness.

We next prove a bound on the degree of Levelk(A∗,G).

Lemma 6.12. If A =⊕r−1

i=0 Z/pdi with di > 0 then

|Mon(A∗,Λk)| =r−1∏i=0

p(n−k)(di−1)(pn−k − pi) = |A∗/A∗(1)|n−k|Mon(A∗(1),Λk)|.

It follows that

|A|k|Mon(A∗,Λk)| =r−1∏i=0

pn(di−1)(pn − pi+k).

Proof. Write βi = pdi−1αi ∈ A∗(1). Let ψ be a map A∗ −→Λk. If ψ is not injective, sayψ(α) = 0 with α 6= 0, then piα ∈ ker(ψ) ∩ A∗(1) for some i. Thus, ψ is injective if andonly if ψ|A∗(1) is injective, if and only if we have

ψ(βi) 6∈ 〈ψ(β0), . . . , ψ(βi−1)〉for all i. Suppose that we have chosen ψ(αj) for j < i; there are then pn−k − pi possiblechoices for ψ(βi) = pdi−1ψ(αi). As |Λk(di−1)| = p(n−k)(di−1), there are p(n−k)(di−1)(pn−k−pi)


possible choices for ψ(αi). It follows that

|Mon(A∗,Λk)| =∏i

p(n−k)(di−1)(pn−k − pi)

as claimed. The rest is trivial.

Proposition 6.13. deg[Levelk(A∗,G)] ≤ |A|k|Mon(A∗,Λk)|.

Proof. In view of Lemma 6.12, it is enough to show that xi has degree at most pn(di−1)(pn−pi+k) over OX [x0, . . . , xi−1]. Because φ is a pk-fold level structure, we see that [pdi−1](xi)

is a root of f(x) = [p](x)/∏

β(x − xβ)pk, where β runs over 〈βj | j < i〉. As f(x) has

Weierstrass degree pn − pi+k and [pdi−1](x) has Weierstrass degree pn(di−1), we see thatxk is a root of a power series f([pdi−1](x)) of Weierstrass degree pn(di−1)(pn − pi+k) asrequired.

Proof of Theorem 6.4. We proved as Corollary 6.11 that Levelk(A∗,G) is flat over X,

and as Proposition 6.13 that the degree is at most |A|k|Mon(A∗,Λk)|. The proof ofthe other inequality is postponed to Corollary 8.4. We proved as Proposition 6.10 thatLevelk(A

∗,G) is smooth in the universal case. All that is left is to define and study themap

∐B Levelk(B

∗,G) −→Hom(A∗,G).By the usual argument, we may assume that (G, X) is of universal type, so that D is a

regular local ring.For any B ≤ A, it is easy to see that the epimorphism π : A∗ −→B∗ gives a closed embed-

ding Hom(B∗,G)� Hom(A∗,G), or equivalently a surjective mapR(A)� R(B). By com-posing this with the closed embedding Levelk(B

∗,G)� Hom(B∗,G), we get a closed em-bedding Levelk(B

∗,G)� Hom(A∗,G), corresponding to a surjective map R(A)� D(B).Write pB for the kernel; this is prime because D(B) is a regular local ring and thus an in-tegral domain. Because D(B) is flat (and thus free) over D, it is also clear that pB∩D = 0and in particular uk 6∈ pB. We also write p′B = pBR

′(A), so that R′(A)/p′B = D′(B).Now let p be any prime ideal in R(A) such that uk 6∈ p (these biject with prime ideals

in R′(A), of course). For some B ≤ A, the map A∗φ−→ Γ(R(A)/p,G) can be factored as

the projection π : A∗ � B∗ followed by a monomorphism ψ : B∗ � Γ(R(A)/p,G), andProposition 6.9 tells us that ψ is a pk-fold level-B∗ structure. It follows that pB ≤ p, sothat the intersection of the p′B’s is the same as the intersection of all primes in R′(A). Itis well-known that this intersection is just

√0 = {x ∈ R′(A) | x is nilpotent }.

We next consider pB + pC , where B and C are distinct subgroups of A, say C 6≤ B.There is an element γ ∈ B◦ \C◦, so xγ ∈ pB. On the other hand, there is an integer j suchthat pjγ has exact order p modulo C◦ and thus [pj ](xγ) divides uk mod pC by the argumentof Proposition 6.9. As xγ divides [pj ](xγ), we deduce that uk = 0 (mod pB +pC), and thusthat p′B + p′C = R′(A). The Chinese remainder theorem now tells us that

R′(A)/√

0 =∏B≤A

R′(A)/p′B =∏B≤A

D′(B).


In geometric terms, this means that the map∐B≤A

Level′k(B∗,G) −→Hom′(A∗,G)

is an infinitesimal thickening, as claimed.

7. The geometric Frobenius map

We next discuss an interesting reformulation of the definition of a multiple level structure.The concepts involved will be useful later. Assuming that k > 0, we have p = 0 in OYfor all schemes Y under consideration, so the map x 7→ xp is a ring endomorphism of OY ,giving rise to a map FY : Y −→Y of schemes, called the geometric Frobenius. For any d ≥ 0,we let G〈d〉 be the pullback of G along the map F kd

X : X −→X. The map F kG : G −→G covers

F kX and thus induces a morphism f : G −→G〈1〉 of formal groups over X. A coordinate x

on G gives rise to a coordinate y on G〈d〉 such that y(fd(a)) = x(a)pkd

for any point a ofG. If the formal group law arising from x is

∑ij cijx

i0x

j1, then the formal group law arising

from y is∑

ij cpkd

ij xi0x

j1.

Because G has strict height at least k, we see that pG factors through f , say pG = q ◦ ffor some map q : G〈1〉 −→G. The kernel of q is a subgroup divisor K < G〈1〉 of degree pn−k.

More explicitly, we have [p](x) = h(xpk) for some power series h over OX , and q is given

by x(q(b)) = h(y(b)) for any point b of G〈1〉. The subgroup K is just spf(OX [[y]]/h(y)).

Proposition 7.1. A homomorphism φ : A∗ −→ Γ(Y,G) is a pk-fold level structure if andonly if we have an inequality of divisors

[fφA∗(1)] ≤ K ∈ Γ(Y,Div(G〈1〉)).

Proof. Let g : H −→K be any isogeny of formal groups. Let D and D′ be divisors on K.Because g is faithfully flat, it is easy to see that D ≤ D′ if and only if the pullbacks satisfyg∗D ≤ g∗D′. It is also easy to see that for any point a of H, the divisor g∗[g(a)] is thetranslate by a of the divisor ker(g). In the case g = f we have ker(f) = pk[0], so therelevant translate is just pk[a]. It follows that

f ∗[fφA∗(1)] =∑

α∈A∗(1)

f ∗[fφ(α)] = pk[φA∗(1)].

On the other hand, we have

f ∗K = f ∗ ker(q) = ker(q ◦ f) = ker(pG) = G(1).

It follows that [fφA∗(1)] ≤ K if and only if pk[φA∗(1)] ≤ G(1), which means precisely thatφ is a pk-fold level structure.

We can pull the maps p = pG, q and f back along the map F ekX : X −→X to obtain maps

which we still call f , q and p. These fit into an infinite diagram whose top left corner looks


like this:

G G〈1〉 G〈2〉

G〈1〉 G〈2〉 G〈3〉

G〈2〉 G〈3〉 G〈4〉

u

p

w

f

u

p

w

f

��

��

q

u

p

��

��

q

u

p

w

f

u

p

w

f

��

��

q

u

p

��

��

q

w

fw

f

(Half of the triangles commute because p = q ◦ f ; the squares commute because f is ahomomorphism of formal groups; because all maps are isogenies and thus epimorphisms, itfollows that the remaining triangles commute.) It follows from this that we have a filtration

ker(fd) ≤ ker(fd−1 ◦ p) ≤ . . . ≤ ker(f ◦ pd−1) ≤ ker(pd) = G(1).

On the other hand, ker(fd−i ◦pi) is the pullback along piG of the divisor ker(fd−i) = p(d−i)k,which is just p(d−i)kG(i). Thus, the above filtration has the form

pdk[0] ≤ p(d−1)kG(1) ≤ . . . ≤ pkG(d− 1) ≤ G(d).

We can thus define divisors

Di = p(d−i)kG(i)− p(d−i+1)G(i− 1),

and we find that G(d) =∑d

i=0Di.Algebraically, this decomposition takes the form of the following lemma, whose proof we

leave to the reader.

Lemma 7.2. We can factor [p](x) as xpkg(x) for some power series g of Weierstrass

degree pn − pk. We write g0(x) = x and gi(x) = g([pi−1](x)) for i > 0. We then have

[pd](x) =d∏i=0

gi(x)pk(d−i)

,

and the divisor Di is just spf(OX [[x]]/gi(x)pk(d−i)

).

8. Thickenings

In this section we “thicken up” the subschemes Levelk(B∗,G) of Hom(A∗,G) to get

schemes Y (A,B) with nice properties such that Hom′(A∗,G) =∐

B Y′(A,B) as schemes

(where Y ′(A,B) means Y (A,B)×X X ′k as usual).

Definition 8.1. Let A be a finite Abelian p-group, B a subgroup of A, and π : A∗ � B∗

the canonical map. Let W be a scheme over X, and φ a homomorphism A∗ −→Γ(W,G).We say that φ is a pk-fold level-(A∗, B∗) structure if


(I) If C∗ is a subgroup of ker(π) with pdC∗ = 0 (such as B◦(d)) then the composite

C∗φ−→G fd−→ G〈d〉

is zero.(II) If C∗ is a subgroup of A∗ with pdC∗ = 0 and ker(π)∩C∗ = pC∗ then there is a pk-fold

level structure ψ : πC∗ −→Γ(W,G〈d−1〉) such that the following diagram commutes.

C∗ G

πC∗ G〈d− 1〉

w

φ

uu

π

u

fd−1

w

ψ

The following result follows easily from [23, Proposition 16].

Proposition 8.2. The functor from schemes over X to sets defined by

Y 7→ { pk-fold level-(A∗, B∗) structures on G over Y }is represented by a scheme Y (A,B) over X. This is a closed subscheme of Hom(A∗,G)and thus is finite over X.

The main result is as follows.

Theorem 8.3. (1) Y (A,B) is flat over X, of degree |A|k|Hom(B∗,Λk)|.(2) Y (A,B) is an infinitesimal thickening of Levelk(B

∗,G).(3) Hom′(A∗,G) =

∐B Y

′(A,B).(4) Y (A,A) = Levelk(A

∗,G).(5) Y ′(A,B) ∩Hom(B∗,G) = Levelk(B

∗,G).

Corollary 8.4. By parts (1) and (4), we see that

deg[Levelk(A∗,G)] = |A|k|Mon(A∗,Λk)|.

Part (5) will be proved separately as Proposition 8.12. The proof of the rest will rely onthe following three results.

Proposition 8.5. deg[Y (A,B)] ≤ |A|k|Mon(B∗,Λk)|.

Proposition 8.6. There are closed subschemes Z(A,B) ⊆ Hom(A∗,G) such that Z(A,B)is an infinitesimal thickening of Levelk(B

∗,G), and Hom′(A∗,G) =∐

B Z′(A,B). More-

over, uk is not a zero-divisor on OZ(A,B). (Warning: unlike almost all other constructionswe use, the formation of Z(A,B) is not compatible with base change.)

Proposition 8.7. If Z(A,B) is as in Proposition 8.6, then Z(A,B) ⊆ Y (A,B).

We also record some notation


Definition 8.8.

D = OX

R(A) = OHom(A∗,G)

D(B) = OLevelk(B∗,G)

D(A,B) = OY (A,B)

E(A,B) = OZ(A,B).

We now prove Theorem 8.3 assuming the above propositions.

Proof of Theorem 8.3. Write dB = |A|k|Mon(B∗,Λk)| and d = |A|n = |A|k|Hom(A∗,Λk)|.Note that the obvious decomposition Hom(A∗,Λk) =

∐B Mon(B∗,Λk) gives d =

∑B dB,

and recall from Proposition 4.10 that R(A) is a free module of rank d over D. Proposi-tion 8.5 gives us an epimorphism tB : DdB � D(A,B) of D-modules, and putting thesetogether gives an epimorphism t : Dd �

∏BD(A,B). The closed inclusions Z(A,B) �

Y (A,B)� Hom(A∗,G) give epimorphisms R(A)rA−→ D(A,B)

sA−→ E(A,B) and thus maps

R(A)r−→∏

BD(A,B)s−→∏

B E(A,B) with s surjective.We now invert uk everywhere, and denote this by adding a dash. We may assume that

we are in the universal case, so that uk acts injectively on free D-modules.By Proposition 8.6, we know that s′r′ is an isomorphism, so

∏B E

′(A,B) is free of rankd over D′. This means that s′t′ is an epimorphism between free modules of the same rankd over D′; by a general fact about Noetherian rings, we conclude that it is an isomorphism.As s′ and t′ are epimorphisms and s′t′ is an isomorphism, we conclude that s′ and t′ areisomorphisms. As s′r′ is an isomorphism, we now see that r′ is an isomorphism.

As t′ is iso, we see that the kernel of t : Dd �∏

B D(A,B) is a uk-torsion module, butuk is regular on Dd, so t is injective. It is also surjective, so each tB is an isomorphism,so D(A,B) is a free module over D of rank dB. This proves (1). It also shows that ukis regular on D(A,B); as s′B : D(A,B) −→E(A,B) is an isomorphism, the same argumentshows that sB is an isomorphism, so Y (A,B) = Z(A,B). Given this, part (2) of thetheorem follows from Proposition 8.6.

Part (3) of the theorem says that r′ is an isomorphism, which we have already seen.Part (4) holds by inspection of the definitions. As mentioned previously, part (5) will beproved separately as Proposition 8.12.

The proof of Proposition 8.5 relies on three lemmas.

Lemma 8.9. Let B be a subgroup of A, so that (A/B)∗ ≤ A∗. Then the natural map

r : Hom(A∗,G) −→Hom((A/B)∗,G)

is flat of degree |B|n.

Proof. Write d = deg[Hom(A∗,G) −→Hom((A/B)∗,G)]. By definition, d is the same as thedegree of the restriction of r over the special fibre of Hom((A/B)∗,G). The special fibre iscontained in the closed subscheme of Hom((A/B)∗,G) where the universal homomorphism(A/B)∗ −→G vanishes, which is a copy of X. The inverse image of this copy of X under


r is just Hom(B∗,G). Thus, d = deg[Hom(B∗,G)] = |B|n. We can choose generatorsx1, . . . , xd giving a surjective map f : OdHom((A/B)∗ ,G) −→ OHom(A∗,G). We also know that

OHom((A/B)∗ ,G) and OHom(A∗,G) are free over OX , with ranks e = |A/B|n and ed = |A|nrespectively. Thus, f is an epimorphism between free modules of the same finite rank overthe Noetherian ring OX , so it must be an isomorphism. Thus, r is a flat map of degreed = |Bn|, as claimed.

Lemma 8.10. There is a pullback diagram of the following form:

Y (A,B) Y (A/pB,B/pB)

Hom(A∗,G) Hom((A/pB)∗,G)

ww

v

u

v

uww

Proof. Note that (A/pB)∗ is the preimage under π : A∗ −→B∗ of B∗(1). Thus, in condi-tions (I) and (II) of Definition 8.1, the subgroup C∗ is necessarily contained in (A/pB)∗.This means that a map φ : A∗ −→G is a pk-fold level-(A∗, B∗) structure if and only if therestriction of φ to (A/pB)∗ is a pk-fold level-((A/pB)∗, (B/pB)∗) structure, as required.

Lemma 8.11. If pB = 0 then we can choose a presentation of A∗ of the form

A∗ = 〈β1, . . . , βs | pdiβi = 0〉 ⊕ 〈γ1, . . . , γt | pejγj = 0〉,where π(γj) = 0 for all j, and {π(β1), . . . , π(βs)} is a basis for B∗. We may also assumethat 0 < d1 ≤ . . . ≤ ds.

Proof. Choose a presentation

A∗ = 〈α0, . . . , αr−1 | pciαi = 0 for all i〉,where 0 < c0 ≤ . . . ≤ cr−1. Write

S = {i | π(αi) ∈ 〈π(αj) | j < i〉}.If i ∈ S, then we can choose an element α′i which is congruent to αi mod 〈αj | j < i〉, suchthat π(α′i) = 0. Because we have cj ≤ ci when j < i, we see that α′i has order precisely pci.We relabel {αi | i 6∈ S} as {β1, . . . , βs} and {α′i | i ∈ S} as {γ1, . . . , γt}. It is easy to seethat these elements give a presentation of the required type.

We can now prove that deg[Y (A,B)] ≤ |A|k|Mon(B∗,Λk)|.Proof of Proposition 8.5. For the moment we assume that pB = 0, and we choose a pre-sentation of A as in Lemma 8.11. Write A∗j = 〈βi | i ≤ j〉 ≤ A∗, and B∗j = π(A∗j) ' A∗j/p.We also write xi = x(φ(βi)) ∈ OY (A,B) and yj = x(φ(γj)), and note that these generateOY (A,B) as an algebra over OX . We let Ri be the subalgebra generated by {x1, . . . , xi}.

For each β ∈ B∗j we choose a lift β ∈ A∗j , and we write

hj(x) =∏β∈B∗j

(x− x(φ(β))).


We now apply condition (II) of Definition 8.1 to the group C∗ = B∗j , giving a diagram

A∗j B∗j

G G〈dj − 1〉

ww

π

u

φ

u

ψ

w

fdj−1

in which ψ is a pk-fold level-B∗j structure. This means that (hj(x)pk)pk(dj−1)

= hj(x)pkdj

divides [p](x)pk(dj−1)

. It follows that

kj(x) = [p](x)pk(dj−1)

/hj−1(x)pkdj

is a power series of Weierstrass degree pkdj(pn−k − pj−1) over Rj−1, and that kj(xj) = 0.As Rj is generated by xj over Rj−1 we see that the degree of Rs over R0 = OX is at most∏s

j=1 pkdj(pn−k − pj−1).

Next, we apply condition (I) of Definition 8.1 to the subgroup spanned by γj to see that

fejφ(γj) = 0 and thus that ypkej

j = 0. It follows that the degree of OY (A,B) over OX is atmost

t∏i=1

pkeis∏j=1

pkdj(pn−k − pj−1).

A comparison with Lemma 6.12 shows that this is just |A|k|Mon(B∗,Λk)|, as required.We now remove the assumption that pB = 0. We can still apply the above argument to

see that

deg[Y (A/pB,B/pB)] ≤ |A/pB|k|Mon((B/pB)∗,Λk)|.On the other hand, Lemma 8.10 and Lemma 8.9 give

deg[Y (A,B) −→Y (A/pB,B/pB)] = deg[Hom(A∗,G) −→Hom((A/pB)∗,G)] = |pB|n.It follows that

deg[Y (A,B)] ≤ |pB|n|A/pB|k|Mon((B/pB)∗,Λk)|.Using (B/pB)∗ = B∗(1) and (pB)∗ = B∗/B∗(1) and Lemma 6.12, we can rewrite this as

|A|k|B∗/B∗(1)|n−k|Mon(B∗(1),Λk)| = |A|k|Mon(B∗,Λk)|,as required.

We next prove, as promised, that Z(A,B) ⊆ Y (A,B).

Proof of Proposition 8.7. Let φ be the tautological map A∗ −→ Γ(Z(A,B),G). We needto prove that this satisfies conditions (I) and (II) of Definition 8.1. We may assume that(G, X) is of universal type, and we reuse the notation of Definition 8.8.


We first address condition (I). Suppose that α ∈ A∗ satisfies pdα = 0 and π(α) = 0;

it will be enough to prove that apkd

= 0, where a = x(φ(α)). If we restrict to the closedsubscheme Levelk(B

∗,G) then φ factors through π so a becomes zero, in other words a isin the kernel of the map E(A,B) −→D(B). As Z(A,B) is an infinitesimal thickening ofLevelk(B

∗,G), this means that a is nilpotent. Using the factorisation in Lemma 7.2, wealso see that

d∏i=0

gi(a)pk(d−i)

= 0.

Suppose that i > 0. In D(B) we have gi(a) = gi(0) = uk, so gi(a) becomes a unit in D′(B).As ker(E′(A,B) � D′(B)) is nilpotent, it follows that gi(a) is a unit in E′(A,B). Thus,

only the i = 0 term in the above product is not a unit, so apkd

= 0 in E′(A,B). As E(A,B)

has no uk-torsion, we conclude that apkd

= 0 in E(A,B), as required.We next prove condition (II). Let C∗ be a subgroup of A∗ such that pdC∗ = 0 and

ker(π)∩C∗ = pC∗. Suppose that α ∈ C∗ \pC∗, and write a = x(φ(α)). As before, we have

d∏i=0

gi(a)pk(d−i)

= 0.

Because π(α) has exact order p and φ becomes a pk-fold level-B∗ structure over D(B),

we see that xpk(x − a)p

kdivides [p](x) in D(B)[[x]] and thus that g(a) becomes zero in

D(B). This means that g0(a) = a divides the constant term g(0) = uk in D(B), so thatg0(a) becomes a unit in D′(B). It also means that in D(B) we have [p](a) = 0 and thusgi(a) = g([pi−1](a)) = g(0) = uk for i > 1, so that gi(a) is also a unit in D′(B). Arguing as

before, we see that g(a)pk(d−1)

= 0 in E(A,B).

As g(a) divides [p](a), we conclude that [p](a)pk(d−1)

= 0, which implies that the map

pC∗� C∗φ−→Γ(Z(A,B),G)

fd−1−−→ Γ(Z(A,B),G〈d− 1〉)

is zero. This means that there is a map ψ : π(C∗) = C∗/pC∗ −→Γ(Z(A,B),G〈g− 1〉) suchthat fd−1φ = ψπ. In the previous paragraph we saw that a was a unit in E′(A,B) and thusnot a zero-divisor in E(A,B). It follows by Proposition 6.9 that ψ is a pk-fold level-πC∗

structure.

Proposition 8.12. Y ′(A,B) ∩ Hom(B∗,G) = Level′k(B∗,G).

Proof. It is clear that Levelk(B∗,G) is a closed subscheme of W = Y (A,B)∩Hom(B∗,G) ⊆

Hom(A∗,G), so that Level′k(B∗,G) ≤ W ′ = Y ′(A,B) ∩ Hom(B∗,G). For the converse, let

OV be the image of OHom(B∗,G) in OW ′ , or equivalently the quotient of OW by the idealof elements annihilated by a power of uk. Note that V is a closed subscheme of W withV ′ = W ′, so it is enough to show that V ≤ Levelk(B

∗,G). Over W , the map φ : A∗ −→Gfactors through a map ψ : B∗ −→G; we need to show that this becomes a level structurewhen we restrict over V .


Let β be a nonzero element of B∗(1), and let α be a preimage of β in A∗, of order pd say.

By the definition of Y (A,B) we see that (x−x(φ(α)))pkd

divides [p](x)pk(d−1)

over Y (A,B).Thus, over W we see that x(ψ(β)) = x(φ(α)) divides a power of uk. It follows that over V ,the element x(ψ(β)) is not a zero-divisor. The proposition follows by Proposition 6.8.

9. Pure strata

In this section, we return to the framework of Sections 4 and 5, so we again have anadmissible ring spectrum E, a finite group G, a finite G-complex Z, and a category Awhose objects are the Abelian p-subgroups of G. We define a new functor L from A toschemes as follows. Each object A ∈ A is sent to the scheme

MA =∐B≤A

Levelk(B∗,G).

Consider a morphism u : B −→A in A, so u is really a map of G-sets from G/B to G/A,sending B to gA say. As this is a map of G-sets we must have Bg ≤ A. The mapMu : MB −→MA is defined to send each component Levelk(C

∗,G) of MB to the componentLevelk((C

g)∗,G) of MA by the obvious map induced by g. It is easy to see that this iswell-defined.

Another point of view is to consider the category A′, whose objects are the same asthe objects of A, and whose maps are the isomorphisms in A. The assignment A 7→Levelk(A

∗,G) gives a functor from A′ to schemes, and M is the left Kan extension of thisfunctor along the inclusion A′ −→A.

Lemma 9.1. There is an isomorphism of formal schemes∫ A∈A(π0(ZA)×

∐B≤A

Levelk(B∗,G)

)=

(∐A∈A


)/G,

and similarly with Levelk(A∗,G) replaced by Level′k(A

∗,G).

Proof. Both statements are formalities: we prove the first statement for definiteness.For brevity, we write PA = π0(ZA) and LA = Levelk(A

∗,G). By definition, (∐

A PA ×LA)/G is the initial example of a scheme Y with maps kA : PA × LA −→Y such that

kAg = kA ◦ (Pg × Lg−1) : PAg × LAg −→Y.

Thus, to construct a map k′ : (∐PA × LA)/G −→

∫ APA ×MA, we need to construct maps

k′A : PA × LA −→∫ A

PA ×MA with k′Ag = k′A ◦ (Pg × Lg−1).

Similarly, the coend∫ A

PA×LA is the initial example of a scheme Y equipped with mapsjA : PA ×MA −→ Y such that for each morphism u : B −→A in A, the following diagram


commutes.

PA ×MB PA ×MA

PB ×MB Y

w

1×Mu

u

Pu×1

u

jA

w

jB

Each such morphism u : B −→ A is given by an element g ∈ G such that Bg ≤ A;the element g is not well-defined, but its coset gB is well-defined. By looking at the

components of the schemes in the above diagram, we see that∫ A

PA ×MA is the initialexample of a scheme Y with maps jA,B : PA×LB −→Y whenever B ≤ A, such that wheneverCg ≤ Bg ≤ A we have a commutative diagram as follows.

PA × LC PA × LCg

PB × LC Y

w

1×Lg

u

Pu×1

u

jA,Cg

w

jB,C

Thus, to construct a map j′ :∫ A

PA×MA −→(∐PA×LA)/G, we need to construct maps

j′A,B : PA × LB −→(∐PA × LA)/G making diagrams like the above commute.

We define

k′A = jA,A : PA × LA −→∫ A

PA ×MA

and

j′A,B = (PA × LB Pu×1−−−→ PB × LBkB−→ (

∐A

PA × LA)/G).

(Here Pu is the evident morphism PA −→PB induced by the inclusion B −→A.) We leave itto the reader to check that these maps have the necessary compatibilities to induce maps

(∐A

PA × LA)/Gk′−→∫ A

PA ×MAj′−→ (∐A

PA × LA)/G,

and that these maps are mutually inverse.

Theorem 9.2. There is a natural map of formal schemes(∐A


)/G −→Xk(Z,G),

which induces a V -isomorphism(∐A


)/G −→X ′k(Z,G).


Proof. When B ≤ A ∈ A, we have an inclusion

Levelk(B∗,G) −→Homk(B

∗,G) −→Homk(A∗,G).

These can be assembled to give a map

MA =∐B≤A

Levelk(B∗,G) −→Homk(A

∗,G),

and one can check that this is natural for maps A −→A′ in A. We therefore get an inducedmap of coends

f :

(∐A


)/G =

∫ A

π0(ZA)×MA −→∫ A

π0(ZA)× Homk(A∗,G).

We compose this with the map

g :

∫ A


provided by Theorem 2.4 to get the map claimed in the Proposition.General nonsense provides a map(∐

A


)/G −→X ′ ×X

(∐A


)/G,

which is a V -isomorphism by Proposition 3.13. By composing this with the pullback of fand the isomorphism of Lemma 9.1, we get a map

f ′ :

(∐A


)/G −→

∫ A

π0(ZA)× Hom′k(A∗,G).

We know by Theorem 6.4 that the maps X ′k ×X MA −→Hom′k(A∗,G) are V -isomorphisms,

and it follows using Proposition 3.13 that f ′ is a V -isomorphism. We also know fromTheorem 2.4 that g′ = 1X′k ×X g is a V -isomorphism, so g′f ′ is a V -isomorphism, asrequired.

Remark 9.3. One way to think about this result is that it guarantees that the intersectionof any two irreducible components of Xk(Z,G) lies over an infinitesimal thickening of Xk+1.There is another way that one might hope to prove this, at least in the case where E isMorava E-theory. As the components are distinct and have Krull dimension n − k, theimage of the intersection in X will have dimension strictly less than n− k. Because of theway that the components arise from Abelian subgroups, this image will also be invariantunder the action of the Morava stabiliser group. It is widely believed that the only primeideals of E0 that are invariant under this action are the ideals Ik, and that this is a simpleconsequence of the classification of invariant primes in MU∗. We suspect that the first ofthese beliefs is true, but the second seems to be false. We also suspect that the Ik arethe only invariant radical ideals, or equivalently that they are the only primes that areinvariant under an open subgroup of the stabiliser group. If we assume this, we see easily


that the reduced part of the image of the intersection of two irreducible components is Xj

for some j > k.

We now prove Theorem 2.7. We repeat the statement for ease of reference.

Theorem 9.4. Let A ≤ G be Abelian and z an element of π0(ZA). The pair (A, z) definesa point of the finite G-set

∐A π0(ZA). We write NG(A, z) for its stabiliser and [A, z] for

its orbit, and put WG(A, z) = NG(A, z)/A.

There is a closed subscheme Y ′k(Z,G,A, z) ≤ X ′k(Z,G) depending only on the orbit of(A, z), such that

X ′k(Z,G) =∐[A,z]

Y ′k(Z,G,A, z)

(where the coproduct is indexed by orbits). Moreover, there is a map of schemes

Level′k(A∗,G)/WG(A, z) −→Xk ×X Y ′k(Z,G,A, z),

which is an isomorphism on the underlying varieties.

Proof. It is not hard to see that(∐A


)/G =

∐(A,z)

Level′k(A∗,G)

/G

=∐[A,z]

Level′k(A∗,G)/WG(A, z).

This scheme maps by a V -isomorphism to X ′k(Z,G) = spec(u−1k E0

G(Z)/Ik), which can beinfinitesimally thickened to get the scheme spec(u−1

k E0G(Z)/Imk ) for any m > 0. Applying

Proposition 3.15, we get an induced splitting of the ring u−1k E0

G(Z)/Imk into pieces indexedby the orbits in

∐A π0(ZA). Moreover, the piece indexed by (A, z) is V -isomorphic to

Level′k(A∗,G). All this is canonical and thus compatible as m varies, and by taking the

inverse limit we get the theorem.

We can be a little more explicit in the Abelian case. We write Y ′k(A,B) for Y ′k(∗, A,B, ∗).

Theorem 9.5. If A is an Abelian p-group and B ≤ A then OY ′(A,B) is a free module of

rank |A|k|Mon(B∗,Λk)| over OX′k.

Proof. The decomposition X ′k(A) =∐

B Y′k(A,B) gives a decomposition

Hom′k(A∗,G) = X ′k(A) = X ′k(A)×X Xk =

∐B

Y ′k(A,B)×X Xk.

It is not hard to see that this must coincide with the splitting in Theorem 8.3, so thatOY ′k(A,B)/Ik = OY ′(A,B). We also know that OY ′(A,B) can be generated over u−1

k E0/Ik =

OX′k/Ik by dB elements, where dB = |A|k|Mon(B∗,Λk)|. It follows easily that we can


choose an epimorphism tB : OdBX′k� OY ′k(A,B). The direct sum of all these is an epimorphism

OdX′k� OX′k(A) where d =

∑B dB = |A|n. In other words, it is an epimorphism between

free modules of the same finite rank over the Noetherian ring OX′k, hence an isomorphism.

This implies that tB is an isomorphism, as required.

10. The E-homology of BG.

In this section we use our results as input to the calculation of E∗(BG) by equivariantmeans. We have been much concerned with the cohomology theoryE∗G(Z) = E∗(EG×GZ),and now we consider the calculation of EG

∗ (EG×Z) = E∗(EG×GZ) for finite G-complexesZ, particularly if Z = ∗ when we obtain EG

∗ (EG) = E∗(BG). It is necessary to remind thereader that EG

∗ (Z) is usually not the homology of the Borel construction unless Z is free.For this section we assume that

(1) E is Landweber exact, and(2) E∗G(Z) is a free module over E∗, concentrated in even degrees.

Condition (2) is known to hold for Z = ∗ for a large class of groups.Let J be the augmentation ideal of E0

G, so J = ker(E0G � E0). We will apply

Theorem 2.7 to the study of the local cohomology H∗J(E0G(Z)) and the local Tate co-

homology H∗J(E0G(Z)), which is the same as the Cech cohomology H∗J(E0

G(Z)) becauseE0G(Z) is complete at J . In good cases, these will be relevant to the calculation of

EG∗ (EG × Z) = E∗(EG ×G Z) and the E-Tate cohomology groups t(E)∗G(Z) of [8]; in-

deed, we expect to have spectral sequences

E∗∗2 = H∗J(E∗G(Z)) =⇒ EG∗ (EG× Z) = E∗(EG×G Z)

and

E∗∗2 = H∗J(E∗G(Z)) =⇒ t(E)∗G(Z).

These spectral sequences can be constructed by elementary means if G is a p-group [11, 7],but for a general finite group we rely on the theory of highly structured ring spectra [9, 7].The particular way that the algebra is reduced to the ungraded ring E0

G is discussed indetail in [11].

This approach is a continuation of the progression starting with [10, 11], although theanswer is necessarily less explicit in the higher dimensional cases.

Recall that for a finitely generated ideal J = (α1, . . . , αd) of E0G we may define the flat

stable Koszul complex

K•(J) = (A −→A[1/α1])⊗ · · · ⊗ (A −→A[1/αd]);

this is independent of the generators up to quasi-isomorphism and the local cohomologyof a module M can be defined by

H∗J(M) = H∗(K•(J)⊗M).

Note that

H0J(M) = ΓJ(M) := {x ∈M | JNx = 0 for N sufficiently large};


geometrically, ΓJ(M) is the module of sections of the sheaf M with support in V (J).Since E0

G is a Noetherian ring, a result of Grothendieck states that local cohomologycalculates the right derived functors of the left exact functor ΓJ . Note that in generalthe local cohomology of a ring depends on the nilpotents, so that it is not a functor ofthe variety var(R). However the J-local cohomological dimension only depends on thevariety, and in particular H i

J(R) = 0 for i > 0 if and only if J is nilpotent, or equivalentlyvar(R) = V (J) = var(R/J).

For any module M over E0G we may form the Cousin complex corresponding to the

chromatic filtration

C•(M) =(p−1M −→u−1

1 M/p∞ −→u−12 M/p∞, u∞1 −→· · · −→u−1

n−1M/I∞n−1 −→M/I∞n),

where the terms are defined recursively by the exact sequences

M/I∞k � u−1k M/I∞k �M/I∞k+1.

If p, u1, . . . , un−1 is an M-regular sequence the complex is acyclic and H∗(C•(M)) = M .Our main result is as follows.

Theorem 10.1. Assuming that E is Landweber exact and E∗G(Z) is a free module overE∗ concentrated in even degrees, there is a natural isomorphism

H∗J(E0G(Z)) = H∗ΓJC

•(E0G(Z)).

Proof. This follows immediately from Lemma 10.2 and Proposition 10.3.

Lemma 10.2. Let M be a module over E0G. If p, u1, . . . , un−1 is a regular sequence for M

then there is a spectral sequence

H∗H∗J(C•(M)) =⇒ H∗J(M).

Proof. Consider the double complex

T •,• = C•(M)⊗K•(J).

If p, u1, . . . , un−1 is a regular sequence for M then if we take Cousin cohomology of T •,• weobtain M ⊗K•(J) whose Koszul cohomology is H∗J(M); this is therefore the cohomologyof the total complex. The spectral sequence is the one for calculating the cohomology ofthe total complex obtained by taking Koszul homology first.

For any module M over E0G, it is easy to see that Ck(M) = u−1

k M/I∞k admits a unique

structure as a module over (u−1k E0

G/Ik)∧Ik

= OX′k(G) =∏

(A)OY ′k(G,A) extending its structure

as an E0G-module. It follows that Ck(M) has a canonical splitting as a direct sum of pieces

CkA(M), where Ck

A(M) is a module over OY ′k(G,A). In particular, we have a piece Ck1 (M)

corresponding to the trivial subgroup A = 1.

Proposition 10.3. For any module M over E0G we have

ΓJCk(M) = H0

JCk(M) = Ck

1 (M)

and H iJ(Ck(M)) = 0 for i > 0.


Proof. Write V = V (J). Note that the maps 1 � G � 1 induce an isomorphism V 'X, and thus V ′k = X ′k = Level′k(1,G). Recall that X ′k(G) =

∐(A) Y

′k(G,A), where the

map Level′k(A∗,G) −→Y ′k(G,A) factors through a V -isomorphism Level′k(A

∗,G)/WG(A) −→Y ′k(G,A). In particular, we have V ′k ⊆ Y ′k(G, 1).

The splitting gives an idempotent e ∈ u−1k E0

G/Ik such that e 7→ 0 in OY ′k(G,1) and e 7→ 1

in OY ′k(G,A) for all A 6= 1. For large enough N we have a = uNk e ∈ E0G/Ik. As V ′k ⊆ Y ′k(G, 1),

we see that a becomes zero in u−1k E0/Ik, but E0/Ik has no uk-torsion so a becomes zero

in E0/Ik. We can thus choose a lift b ∈ E0G of a such that b becomes zero in E0, in other

words b ∈ J . When A 6= 1, it is clear that b becomes a unit in OY ′k(G,A) and thus also inOY ′k(G,A).

Now let M be an E0G-module. If A 6= 1 then b acts as an isomorphism on Ck

A(M) andb ∈ J ; by standard properties of local cohomology, we see that H∗J(Ck

A(M)) = 0.Next, recall that V ′k −→ Y ′k(G,A) is a V -isomorphism and thus dominant; this means

that the image of J in OY ′k(G,A) = OY ′k(G,A)/Ik is nilpotent. It follows that for each m ≥ 1

the image of J in OY ′k(G,A)/Imk is nilpotent. As every element of Ck

1 (M) is annihilated

by Imk for some m, we see that it is also annihilated by a power of J . The definition oflocal cohomology now tell us that H0

J(Ck1 (M)) = Ck

1 (M), and since ΓJ preserves injectives,H iJ(Ck

1 (M)) = 0 for i > 0.

The geometric content of the proof is that the components Y ′k(G,A) do not meet V (J)

unless A = 1 and that Y ′k(G, 1) is essentially a colimit of infinitesimal thickenings of V (J).We may argue rather similarly for Cech cohomology, and obtain a chromatic route to

calculating it, and hence to calculating the Tate cohomology t(E)∗G(Z) via the local Tatespectral sequence. Note that there is a fibre sequence K•(J) −→A −→ C•(J), where C•(J)is the usual flat complex for calculating Cech cohomology; therefore we may argue exactlyas in 10.2 to obtain a spectral sequence for calculating the Cech cohomology. Furthermorein case H1

J(M) = 0 we find H0J(M) = M/ΓJM , so that by 10.3 we obtain the following in

which ΨJ(M) := M/ΓJM .

Corollary 10.4. Provided that p, u1, . . . , un−1 is a regular sequence for E0G(Z), the Cech

cohomology groups H∗J(E0G(Z)) are the cohomology groups of the complex ΨJC

•(E0G(Z)).

11. Some examples.

Let us consider some special cases. We have p-complete K-theory with n = 1, where thesequence reads

ΓJ(p−1R(G)∧p ) −→ΓJ(R(G)∧p /p∞)

By character theory we see that J-power torsion in the zero’th term consists of multiplesof the regular representation, and if G is a p-group Jn ⊆ (p) for some n, so all of the firstterm is J-power torsion. The sequence is thus

Z∧p [1/p] −→R(G)/p∞;


we recover the p-complete version of the calculation [6] H0J(R(G)) = Z and H1

J(R(G)) =(R(G)/Z) ⊗ Z/p∞. Geometrically, there are components Y0(1), and Y0(A) when A is acyclic p-subgroup. The statement that the local cohomology of p−1R(G)∧p is concentratedin degree 0, corresponds to the fact that Y ′0(1) does not meet Y ′0(A) if A 6= 1, or thatY0(1) only meets other components over the point (p) ∈ Spec(Z∧p ). The statement thatthere is a single component Y1(1) mod p suggests that the components Y0(A) meet Y0(1)‘tangentially’ in an infinitesimal neighbourhood of p, and the rank of the answer suggeststhat Y0(A) contributes with multiplicity equal to the number of generators of the groupA. In this case one may make a more precise integral analysis using Segal’s description ofSpec(R(G)).

The results of [11] concern the admissible theory

En,i = En/(p, u1, . . . , ui−1, ui+1, . . . , un−1),

and give a precise calculation for an elementary Abelian group of rank r, fitting into thefollowing picture. Again the local cohomology is given by the complex

ΓJ(v−1i E0

n,i(BG)) −→ΓJ(E0n,i(BG)/v∞i ).

If i = n−1 these terms correspond to the nonzero terms of the complex of ΓJC•(E0

n,i(BG))in degrees n − 1 and n. If G is elementary Abelian of order pr, the (n − 1)’st term is afree v−1

i E0-module of rank pir, and the nth term is a direct sum of pnr copies of E0/v∞i .This again suggests that X0(G) has components Y0(1) and Y0(A) for A a cyclic p-subgroup,and that Y0(1) only meets the other components over (vi). The difference is that it nowhas multiplicity pir. Again, in an infinitesimal neighbourhood of (vi), there is a singlecomponent Yi(1), but now with multiplicity pnr. This suggests that the intersection ofY0(A) with Y0(1) is ‘tangential’, but now the multiplicity contributed by each group elementof order p is (pnr − pir)/(pn − 1).

Now consider the case of an Abelian p-group G = A, and E = En. For each m ≥ 1 thereare rings D′k(A,B,m), where D′k(A,B, 1) = D′k(A,B), and there is a splitting

u−1k E0(BA)/Imk

∼=∏B

D′k(A,B,m).

Furthermore, D′k(A,B,m) is a free u−1k E0/Imk -module of rank |A|k|Mon(B, (Z/p∞)n−k)|,

independent of m. Thus we find that H∗J(E∗(BA)) is the cohomology of a ‘chromaticcomplex with multiplicities’

|A|0[p−1E∗] −→|A|1[u−11 E∗/p∞] −→|A|2[u−1

2 E∗/p∞, u∞1 ] −→· · ·· · · −→|A|n−1[u−1

n−1M/I∞n−1] −→|A|n[E∗/I∞n ].

This then suggests the general picture that each ΓJ(u−1k E0(BG)/I∞k ) is a direct sum of

lk(G) copies of u−1k E0/I∞k . However, the differential

ΓJ(u−1k E0(BG)/I∞k ) −→ΓJ(u−1

k+1E0(BG)/I∞k+1)


is not given by just projecting onto lk+1(G) factors and then composing with the naturalmap

u−1k E0/I∞k −→u−1

k+1E0/I∞k+1

in each one, contrary to what one might guess from the one-dimensional case. To see this,consider the case where G is elementary Abelian. The naive guess would give E0 in degree0, and a direct sum of pkr − p(k−1)r copies of E0/I∞k in degree k ≥ 1. This is non-zerofor 0 ≤ k ≤ n. However, the local cohomology vanishes above dimension r if J can begenerated as a radical ideal by r elements (for example if G is Abelian of rank r); this givesa contradiction.

Appendix A. Proof of the theorem of Hopkins-Kuhn-Ravenel

In this appendix we prove Theorem 2.6. The proof we give here is essentially that ofHopkins-Kuhn-Ravenel; we record it partly because of notational differences, and partlybecause [14] is not easily accessible.

Proof of Theorem 2.6. First we construct a natural map

ν : p−1E∗G(Z) −→(∏

A

p−1E∗(ZA)⊗E0 D(A)

)G

.

Indeed, it suffices to construct the components

νA : E∗G(Z) −→E∗(ZA)⊗E0 D(A)

in a suitably natural way, since G acts trivially on the domain. For this we use therestriction maps:

E∗G(Z) −→E∗A(Z) −→E∗A(ZA) = E∗(ZA ×BA) = E∗(ZA)⊗E0 E0(BA)

together with the natural map E0(BA) −→D(A). It remains to show that ν is an isomor-phism for all finite G-complexes Z.

The domain is evidently a cohomology theory, and the codomain is too, because D(A) isflat and the group order is inverted before we take invariants. Thus, if ν is an isomorphismfor Z = G/B whenever B is an Abelian subgroup, it is an isomorphism whenever Z hasfinite Abelian isotropy groups. The general case follows since both sides give equaliserdiagrams when applied to Z × T × T ⇒ Z × T −→Z. For the domain this is Corollary 5.3.For the codomain, it suffices to show that for each A when we apply E∗(·) to the diagramZA×TA×TA ⇒ ZA×TA −→ZA we obtain an equaliser. Indeed, the subsequent operationspreserve equalisers: D(A) is flat, localisation and products are exact, and passage to G-invariants is exact since the group order is invertible. However TA is the space of A-invariant complete flags in V , and this is a disjoint union of products of the flag spacesof the A-isotypical parts of V . The result therefore again follows from the nonequivariantinstance of Corollary 5.3.


It remains to show ν is an isomorphism if Z = G/B, which we do by direct calculationand the theory of level structures. A simple argument reduces us to the case where B is ap-group. In this case the codomain of ν is(∏

A

p−1 Hom(A(A,B), D(A))

)G

in each even degree. By Lemma 9.1 and the Yoneda Lemma, this coincides with∫A∈A

Hom(A(A,B),∏C≤A

p−1D(C)) = p−1∏C≤B

D(C).

By Theorem 6.4, the natural map

p−1EB = p−1OHom(B∗,G) −→p−1∏C≤B

D(C)

is surjective (with nilpotent kernel). Both sides are free modules over p−1E. The domainhas rank |Hom(B∗,Λ0)|, and the codomain has rank

∑C≤B |Mon(C∗,Λ0)|, which is the

same. An epimorphism of free modules of the same rank over a Noetherian ring is anisomorphism, so p−1EB = p−1

∏C≤BD(C), as required.

Appendix B. The Evens norm map

In this section we outline an alternative approach to Theorem 2.5, parallel to that ofEvens [4] for the case of ordinary cohomology. This has the advantage that it constructssome useful elements of E0

G explicitly. However, we need to assume that E has the structureof an H∞ ring spectrum [1]. There are essentially only two admissible theories for whichthere is a published construction of an H∞ structure (rational cohomology and p-adic K-theory). However, it is widely believed that Morava E-theory admits even an E∞ structure.The key ideas needed to prove this are due to Mike Hopkins and Haynes Miller, but thefoundations needed to support them have turned out to be unexpectedly hard to construct.We understand that an account will appear in a paper currently being written by PaulGoerss and Mike Hopkins.

For this appendix we assume that E has an H∞ structure. For simplicity, we alsoassume that k > 0. The H∞ structure allows us to construct an Evens norm map E0

H −→E0G whenever H ≤ G, which should be thought of as a multiplicative analogue of the

transfer. The action of G on G/H together with a choice of coset representatives G/H ={x1H, . . . , xdH} gives rise to a homomorphism G −→ Σd o H, which is canonical up toconjugacy. If u ∈ E0

H = [BH+, E] then we define

normGH(u) = (BG+ −→B(Σd oH) = Dd(BH+)

Dd(u)−−−→ DdEθd−→ E),

where Dd is the d’th extended power construction, and θd is provided by the H∞ structure.This norm map has the following properties:

1. It is natural for isomorphisms of groups, and thus behaves in the obvious way underconjugation.


2. It is multiplicative: normGH(uv) = normG

H(u) normGH(v).

3. There is a double coset formula: For any two subgroups H and K of G we have

resGK normGH(u) =

∏KgH∈K\G/H

normKK∩gH res

gHK∩gHcg(u).

Definition B.1. For any Abelian p-subgroup A of G we write χ(A) ∈ E0A for the Eu-

ler class of the reduced regular representation of A. Under the identification X(A) =Hom(A∗,G), this becomes the natural transformation Hom(A∗,G) −→ Forget given byφ 7−→

∏α∈A∗\0 x(φ(α)). This is a unit over Level′k(A

∗,G) and nilpotent over Level′k(B∗,G)

when B < A. It is also invariant under Aut(A). We write |WGA| = pab where p does notdivide b, and we note that g(x) = (1 + x)1/b − 1 is a power series over Zp. We define

zA = g(normGA(1 + χ(A))− 1).

Using the above properties of the norm map, we see that

resGB(zA) =

{0 if B contains no conjugate of A

χ(A)pa

if B = A.

If A and B are non-conjugate Abelian p-subgroups, with |B| ≤ |A| say, then B cannotcontain a conjugate of A. Thus, zA is zero over the image of Level′k(B

∗,G) in X ′k(G), andinvertible over the image of Level′k(A

∗,G), so these images are disjoint.Consider the natural map

f : u−1k E0

G/Ik = OX′k(G) −→D′k(A)WGA = OLevel′k(A∗,G)/WGA.

We know that χ(A)pa

lies in the image of f and that it is invertible. The inverse is integralover the image of f (because everything is finite over X ′k). After multiplying a monicequation of integral dependence by a suitable power of χ(A)p

a, we find that χ(A)−p

aalso

lies in the image of f , say χ(A)−pa

= f(ξ(A)). Now suppose that y ∈ Dk(A)WGA, andchoose an element y ∈ E0

A lifting y. One can check that

f(g(normGA(1 + χ(A)y))ξ(A)) = yp

a

.

Thus, f is F -surjective in the evident sense. A little more work in this direction recoversTheorem 2.5, at least in the case Z = ∗. For general Z, we can observe that F = F (Z+, E)is again an H∞ ring spectrum, with F 0

G = E0G(Z).

Appendix C. Varieties and reduction mod Ik.

In this appendix we give the elementary argument that E∗(BG)/Ik and (E/Ik)∗(BG)

have isomorphic varieties, assuming that Bockstein spectral sequences with appropriatemultiplicative properties exist, and we sketch a construction under certain circumstances.

Proposition C.1. For any finite group G there is an F-isomorphism

E∗(BG)/Ik −→(E/Ik)∗(BG).


Proof. We shall in fact prove the corresponding result for any space X with K(n)∗(X)finitely generated over K(n)∗; by a result of Ravenel [22] or by 5.4 above, this applies toX = BG.

By a Bockstein spectral sequence argument given in [11] or by 5.4 above, since E is

complete, E∗(X) is finitely generated over E

∗for E = E/Ik for any k ≤ n. We argue

by induction on k, supposing that we have constructed an F-isomorphism E∗(X)/Ik −→(E/Ik)

∗(X), using the identity if k = 0. Now let E = E/Ik and v = uk, and consider the

cofibre sequence Ev−→E −→E/v; this gives a ring map E

∗(X)/v −→(E/v)∗(X). Combining

this with the F-isomorphism already constructed, we obtain

E∗(X)/Ik+1 = (E∗(X)/Ik)/v −→{E/Ik}∗(X)/v −→{E/(Ik + (v))}∗(X) = {E/Ik+1}∗(X).

The fact that the first map is an F-isomorphism is an amusing exercise.

Lemma C.2. If θ : A −→B is an F-isomorphism and a ∈ A then θ : A/a −→B/θ(a) is alsoan F-isomorphism.

It therefore suffices to deal with the principal case.

Lemma C.3. If E∗(X) is finitely generated over the characteristic p ring E

∗then for any

v the map E∗(X)/v −→(E/v)∗(X) is an F-isomorphism.

Proof. Of course the map is injective by construction, so it suffices to show that there isan integer t such that for any y ∈ (E/v)∗(X) the power yp

tis the reduction of some class

x ∈ E∗(X). This is a job for the Bockstein spectral sequence arising from the diagram

E E E E · · ·

E/v E/v E/v

w

vw

v

uρ

w

v

uρ

w

v

uρ[

[[[

β[[[[

β[

[[[

β

by applying [X, ·]∗. Here β is the connecting homomorphism (E/v)∗(X) −→E∗(Σ−1X).

Since E∗(X) is Noetherian it has bounded v-torsion in the sense that there is an integer

t so that if vt+1y = 0 then also vty = 0. Now by definition, if dtz = 0 we have βz = vtztfor some zt ∈ E

∗(X), and since 0 = vβz = vt+1zt it follows that 0 = vtzt = βz, and hence

z is the reduction of an element of E∗(X). It thus suffices to show that any ptth power

ypt ∈ (E/v)∗(X) is a t-cycle. But the spectral sequence is multiplicative by assumption,

and so for any s-cycle z, ds+1zp = 0, thus yp

tis a t-cycle by induction.

If we assume E is an S-algebra in the sense of [3] then the tower can be constructedin the category of highly structured E-modules, since v comes from the coefficients of E.This gives the first property we required. The second property is that the differentials are


derivations. For this we first need to know that E/v is an E-algebra up to homotopy, andthat the diagram

E/v ∧E E/v E/v

ΣE ∧E E/v ∨ E/v ∧E ΣE ΣE

ΣE/v ∨ ΣE/v ΣE/v

u

(β∧11∧β

)w

µ

u

β

u

'

u

Σρ

w

∇

commutes. This follows by elementary obstruction theory as in Chapter V of [3] providedE∗ is concentrated in degrees divisible by 4, and Ik is generated by a regular sequence (andalso under slightly weaker hypotheses). Indeed, E is the E-smash product of the spectraE/u for the elements u in the regular sequence, so it suffices to deal with the case E = E.Here it is easy to be explicit since E/v ∧E E/v = E/v ∨ ΣE/v, and the product µ is theidentity on the first summand and zero on the second. The clockwise composite is zero

on the second factor, and E/vβ−→ ΣE −→ΣE/v on the first. Consider the anticlockwise

contribution from the second factor: the components into each of the factors at the bottomleft are ±1 and ∓1, and thus cancel. On the first factor, one of the terms is zero and the

other is E/vβ−→ΣE −→ΣE/v, depending on the isomorphisms used.

References

[1] R. R. Bruner, J. P. May, J. E. McClure, and M. Steinberger. H∞ Ring Spectra and their Applications,volume 1176 of Lecture Notes in Mathematics. Springer–Verlag, 1986.

[2] V. G. Drinfel’d. Elliptic modules. Math. USSR Sb., 23:561–592, 1974.[3] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May. Rings, Modules and Algebras in Stable

Homotopy Theory, Volume 47 of Amer. Math. Soc. Surveys and Monographs. American MathematicalSociety, 1996.

[4] L.Evens A generalization of the transfer map in the cohomology of groups. Trans. American Math.Soc. 108(1963) 54-65.

[5] A. Frohlich. Formal Groups, volume 74 of Lecture Notes in Mathematics. Springer–Verlag, 1968.[6] J. P. C. Greenlees. K-homology of universal spaces and local cohomology of the representation ring.

Topology 32 (1993) 295-308.[7] J. P. C. Greenlees. Augmentation ideals of equivariant cohomology rings. Preprint, 1996 (submitted

for publication).[8] J. P. C. Greenlees and J. P. May. Generalized Tate Cohomology, Number 543 of Memoirs Of The

American Mathematical Society. American Mathematical Society, 1995.[9] J. P. C. Greenlees and J. P. May. Localization and completion theorems for MU -module spectra.

Preprint, 1995.[10] J. P. C. Greenlees and H. Sadofsky. The Tate spectrum of vn-periodic complex oriented theories.

Math. Zeits. 222 (1996) 391-405


[11] J. P. C. Greenlees and H. Sadofsky. Tate cohomology of theories with one dimensional coefficient ring.Preprint, 1995 (submitted for publication).

[12] A. Grothendieck and J. Dieudonne. Elements de Geometrie Algebrique III (i). IHES Pub. Math. 11(1961)

[13] M. J. Hopkins, N. J. Kuhn, and D. C. Ravenel. Morava K-theories of classifying spaces and generalisedcharacters for finite groups. Proceedings of 1990 Barcelona Conference on Algebraic Topology. LectureNotes in Mathematics 1509 Springer-Verlag (1992) 186–209

[14] M. J. Hopkins, N. J. Kuhn, and D. C. Ravenel. Generalised group characters and complex orientedcohomology theories. Preprint, (various editions, since late 1980’s).

[15] M. J. Hopkins and J. H. Smith. Nilpotence and stable homotopy theory II. (to appear).[16] I. Kriz. Morava K-theory of classifying spaces: some calculations. Preprint, 1995.[17] J. Lubin and J. Tate. Formal moduli for one parameter formal lie groups. Bull. Soc. Math. France,

94:49–60, 1966.[18] S. MacLane. Categories for the Working Mathematician, Volume 5 of Graduate Texts in Mathematics.

Springer–Verlag, 1971.[19] H. Matsumura. Commutative Ring Theory, Volume 8 of Cambridge Studies in Advanced Mathematics.

Cambridge University Press, 1986.[20] D. G. Quillen. The spectrum of an equivariant cohomology ring I. Annals of Mathematics, 94:549–572,

1971.[21] D. G. Quillen. The spectrum of an equivariant cohomology ring II. Annals of Mathematics, 94:573–602,

1971.[22] D.C.Ravenel Morava K theories and finite groups. Contemporary Mathematics, 12 (1982) 289-292.[23] N. P. Strickland. Finite subgroups of formal groups. Journal of Pure and Applied Algebra, (to appear)

1996.[24] N. P. Strickland. Functorial philosophy for formal phenomena. In preparation, 1994.

School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH, UK.

E-mail address : [email protected]

Trinity College, Cambridge CB2 1TQ, UK.

E-mail address : [email protected]

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