VARIETIES AND LOCAL COHOMOLOGY FOR CHROMATIC GROUP COHOMOLOGY RINGS.

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

Abstract. Following Quillen [20, 21], we use the methods of algebraic geometry to study the ring E^{∗}(BG) where E is a suitable complete periodic complex oriented theory and G is a finite group: we describe its variety in terms of the formal group associated to E, and the category of Abelian p-subgroups of G. This also gives information about the associated 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)/I_{k} by the invariant prime ideal I_{k} = (p, v_{1}, …, v_{k−1}) correspond to conjugacy classes of Abelian p-subgroups of rank ≤ n − k. Furthermore, if we invert v_{k} the decomposition of the variety into irreducible pieces corresponding to minimal primes becomes a decomposition into connected components, corresponding to the fact that the ring splits as a product.

Contents

1. Informal introduction 2 2. Description of results. 4 3. Schemes and varieties 9 4. Admissible cohomology theories 14 5. The proof of Theorem 2.4 17 6. Multiple level structures 22 7. The geometric Frobenius map 28 8. Thickenings 29 9. Pure strata 35 10. The E-homology of BG. 39 11. Some examples. 41 Appendix A. Proof of the theorem of Hopkins-Kuhn-Ravenel 43 Appendix B. The Evens norm map 44 Appendix C. Varieties and reduction mod I_{k}. 45 References 47

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

1. Informal introduction

The article investigates the interface between equivariant topology and the chromatic approach 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 = E_{n} of the Johnson-Wilson theories E(n) with coefficients E(n)_{∗} = ℤ_{(p)}[v_{1}, v_{2}, …, v_{n−1}, v_{n}, v_{n}^{−1}] for some prime p, and we restrict attention to this case in the introduction. To obtain a general description, we follow the example of Quillen [20, 21], and concentrate on the geometric properties of the ring E^{∗}(BG). We use Quillen’s descent argument to reduce to the study of E^{∗}(BA) for Abelian p-subgroups A. The Abelian case is translated into the theory of formal groups, and the resulting questions studied by extending the second author’s theory of multiple level structures [23]. The work is also strongly influenced by that of Hopkins-Kuhn-Ravenel [14], discussed in [13].

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

Quillen’s method involves considering the cohomology E^{∗}(EG ×_{G} Z) of the Borel construction for a finite G-complex Z, even if one is only concerned with the special case Z = ∗. Our results apply to all such Z, but it may help the reader if we spend the rest of the introduction summarising the highlights when Z = ∗. We explain the geometric language 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 information, and varieties, which ignore nilpotents and only capture a crude picture of the ring. Nonetheless many important features, such as the dimension, connected components and irreducible components are visible at the level of varieties. Readers comfortable with the formal framework may wish to skip directly to the next section, where our results are stated precisely and in appropriate generality.

Following Morava’s chromatic philosophy we concentrate on the invariant prime ideals I_{k} = (p, u_{1}, …, u_{k−1}) of the complete local ring E^{0}. Indeed, if we let X = spf(E^{0}) denote its formal scheme, the geometric counterpart of the filtration

0 = I_{0} ≤ I_{1} ≤ I_{2} ≤ ⋯ ≤ I_{n},

is the filtration

X = X_{0} ≥ X_{1} ≥ X_{2} ≥ ⋯ ≥ X_{n},

where X_{k} = spf(E^{0}/I_{k}) is the formal subscheme defined by I_{k}. Evidently X_{k} is a formal affine space of dimension n − k. One of the themes will be that we can understand phenomena over X by restricting to the subschemes X_{k}, and that they will be especially simple over the pure strata X′_{k} = X_{k} \ X_{k−1}. The notational convention that a subscript k denotes restriction to the kth chromatic stratum and that a dash denotes restriction to a pure stratum will remain in force throughout the paper.

Now consider the formal scheme X(G) = spf(E^{0}(BG)), which is finite over X; we shall give a description of the primary features of its underlying variety. Let X_{k}(G) denote the restriction of X(G) to the part over X_{k}, and note that this is spf(E^{0}(BG)/I_{k}). On the other hand, we are often interested in the rings (E/I_{k})^{0}(BG); for example if k = n it is simply the Morava K-theory K(n)^{0}(BG) made 2-periodic. If, as often happens, E^{0}(BG) is free over E^{0} then E^{0}(BG)/I_{k} is actually equal to (E/I_{k})^{0}(BG); we shall see that in any case their varieties agree.

Next we describe the irreducible components of X_{k}(G), and how they behave over pure strata. In fact there is a decomposition

X_{k}(G) = ⋃_{(A)} Y_{k}(G, A)

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

X′_{k}(G) = ∐ Y′_{k}(G, A).

Indeed, there is a single formal scheme Level_{k}(A^{∗}) for each Abelian p-group of rank ≤ n − k so that at the level of varieties, Y′_{k}(G, A) = Level′_{k}(A^{∗})/W_{G}(A) where W_{G}(A) = N_{G}(A)/A is the Weyl group of A in G. At one extreme, spf(E^{0}(BG)) has one irreducible component for each conjugacy class of Abelian p-subgroups of rank ≤ n whilst at the other, spf(K(n)^{0}(BG)) is itself irreducible. The schemes Level_{k}(A^{∗}) classify ‘multiple level structures’ (see Section 6) for the formal group 𝔾 of E; they are formal spectra of complete regular local rings D_{k}(A) of dimension n − k.

Translating geometric statements back into ring theory, our decompositions state that the minimal primes of E^{0}(BG)/I_{k} are in bijective correspondence with the conjugacy classes of Abelian p-subgroups of rank ≤ n − k. Moreover, if we invert u_{k} the ring splits up to F -isomorphism as a product of rings of invariants of well-understood rings:

u_{k}^{−1}E^{0}(BG)/I_{k} ⟶^{F −iso} ∏_{(A)} D′_{k}(A)^{W_{G}(A)}.

Here W_{G}(A) = N_{G}(A)/A, but the action factors through N_{G}(A)/C_{G}(A), so that in the Abelian case the action is trivial. In the Abelian case G = A, we identify u_{k}^{−1}E^{0}(BA)/I_{k} exactly, and it decomposes in the same way:

u_{k}^{−1}E^{0}(BA)/I_{k} = ∏_{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|^{k} times 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 11 before 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 reduced cohomology of a based space will be indicated by a tilde.

Definition 2.1. Let p_{ℂP^{∞}}: ℂP^{∞} ⟶ℂP^{∞} be the map classifying the p’th tensor power of the canonical line bundle. A p-local commutative ring spectrum E is admissible if

(a) E^{0} is a complete local Noetherian ring.

(b) E^{1} = 0.

(c) E^{2} contains a unit.

(d) p_{ℂP^{∞}} induces a nonzero self-map of Ẽ^{0}(ℂP^{∞}).

These conditions will be discussed further in Section 4. Examples include the cohomology theory obtained from { \atop E(n)} by adjoining a (p^{n} −1)’st root of v_{n}, or various spectra obtained from this by tensoring with the Witt ring of 𝔽_{p^{n}} or \overline{𝔽}_{p}, or killing some generators (provided that p > 2). For suitable versions of elliptic cohomology, the spectrum L_{K(2)}Ell will be a finite product of admissible ring spectra. The version of integral Morava K-theory used by Igor Kriz in [16] is also admissible. It can be shown that every admissible ring spectrum E is K(n)-local for some n, called the height of E (this is the same as the height of the formal group law associated to E).

We will specify one example more precisely. Let W be the Witt ring of 𝔽_{p^{n}}, and consider the following graded ring:

E^{∗} = W [[u_{1}, …, u_{n−1}]][u, u^{−1}].

The generators u_{k} have degree 0, and u has degree −2. We take u_{0} = p and u_{n} = 1 and u_{k} = 0 for k > n. There is a map BP^{∗} ⟶E^{∗} sending v_{k} to u^{p^{k}−1}u_{k}. Using this, we define a 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 we shall 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 approach is influenced heavily by the work of Quillen [20, 21] and Hopkins-Kuhn-Ravenel [13,

Because of axiom (c), the ring E_{G}^{∗} has period two. Our methods cannot see E_{G}^{1}(Z), and anyway it is zero in many cases, so we focus attention on E_{G}^{0}(Z), which is an ungraded commutative ring. We shall prove in Corollary 5.4 that it is a finitely generated module over E^{0}, and thus a complete semilocal Noetherian ring.

It will be convenient to give our results in geometric language. We will set up a suitable technical 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 variety is a functor from algebraically closed fields to sets that is represented by a Noetherian ring. We will define formal schemes in Section 3. We write X(Z, G) for the formal scheme spf(E_{G}^{0}(Z)) represented by E_{G}^{0}(Z), or for its underlying scheme spec(E_{G}^{0}(Z)), or for its underlying variety. We omit Z from the notation when it is a point, so X(G) means X(∗, G) = spec(E^{0}(BG)).

We also write X for spf(E^{0}), and 𝔾 for spf(E^{0}(ℂP^{∞})), which turns out to be a formal group. We will construct a “chromatic” filtration of X by closed formal subschemes:

X = X_{0} ≥ X_{1} ≥ … ≥ X_{n}.

In terms more familiar to topologists, we have X_{k} = spec(E^{0}/I_{k}), where I_{k} is the ideal generated by the coefficients of x, x^{p}, …, x^{p^{k−1}} in the p-series of a suitable formal group law over E^{0}. We also consider

X′_{k} = X_{k} \ X_{k+1} = spec(u_{k}^{−1}E^{0}/I_{k}) = k’th pure stratum of the filtration,

which are schemes, but not formal schemes. One of the themes will be that we can understand phenomena over X by restricting to the subschemes X_{k}, and that they will be especially simple over the pure strata X′_{k}. We adopt the permanent notational convention that a subscript k denotes restriction to the k’th chromatic stratum and that a dash denotes restriction to a pure stratum. Thus

X_{k}(Z, G) = X(Z, G) ×_{X} X_{k}

(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. This can be described completely in terms of the formal group 𝔾. Suitably interpreted (see Proposition 4.10), the answer is that

X(A) = Hom(A^{∗}, 𝔾),

where A^{∗} is the character group Hom(A, S^{1}). We define

Hom_{k}(A^{∗}, 𝔾) = Hom(A^{∗}, 𝔾) ×_{X} X_{k} = spf(E^{0}(BA)/I_{k}) = X_{k}(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 a quotient 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 does not quite work out as expected. The right substitute for the notion of a monomorphism A^{∗} ⟶ 𝔾 turns out to be a level-A^{∗} structure on 𝔾. This concept is essentially due to Drinfel’d [2], and is studied in detail in the present context in [23]. When working over the subscheme X_{k}, it is useful to refine this notion and use p^{k}-fold level structure instead; the relevant theory is developed in Section 6, where we prove the following theorem.

Theorem 2.3. There are formal schemes Level_{k}(A^{∗}, 𝔾) = spf(D_{k}(A)) (for 0 ≤ k < n) with the following properties.

1. Level_{k}(A^{∗}, 𝔾) is a closed subscheme of Hom_{k}(A^{∗}, 𝔾).

2. Level_{k}(A^{∗}, 𝔾) is flat of degree |A|^{k}| Mon(A^{∗}, (ℚ_{p}/ℤ_{p})^{n−k})| over X_{k}. In particular, it is empty if rank_{p}(A) > n − k.

3. The evident map

Level_{k}(B^{∗}, 𝔾) ↣ Hom_{k}(B^{∗}, 𝔾) ↣ Hom_{k}(A^{∗}, 𝔾)

is a closed embedding.

4. The evident map of formal schemes

∐_{B≤A} Level_{k}(B^{∗}, 𝔾) ⟶Hom_{k}(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) = Map_{G}(G/A, G/B) ≃ {g ∈ G | A^{g} ≤ B}/B.

We have a functor

𝒜^{op} × 𝒜 ⟶ Formal Schemes

given by (A, B) ↦ π_{0}(Z^{A}) × Hom(B^{∗}, 𝔾), where Z^{A} is the subspace of A-fixed points. We denote its coend by ∫^{A} π_{0}(Z^{A}) × Hom(A^{∗}, 𝔾) (see [18, Chapter IX] for the theory of ends and coends).

Theorem 2.4. There is a natural map of formal schemes

∫ ^{A} π_{0}(Z^{A}) × Hom(A^{∗}, 𝔾) ⟶X(Z, G),

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

∫ ^{A} π_{0}(Z^{A}) × Hom_{k}(A^{∗}, 𝔾) ⟶X_{k}(Z, G)

and a natural map of ordinary schemes

∫ ^{A} π_{0}(Z^{A}) × 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 Level_{k}(A^{∗}, 𝔾) is empty unless rank_{p}(A) ≤ n − k.

Theorem 2.5. There is a natural map of formal schemes

( ∐_{A∈𝒜} π_{0}(Z^{A}) × Level_{k}(A^{∗}, 𝔾)) /G ⟶X_{k}(Z, G)

such that the induced map of schemes

( ∐_{A∈𝒜} π_{0}(Z^{A}) × 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(Z^{A}) ×_{X} Level′_{0}(A^{∗}, 𝔾)) /G ⟶X′_{0}(Z, G),

where X(Z^{A}) = spf(E^{0}(Z^{A})). Moreover, it is easy to see that X(Z^{A}) has the same variety as π_{0}(Z^{A}) × X.

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

One can deduce from Theorem 2.5 that the underlying variety of X′_{k}(Z, G) splits as a disjoint union of pieces indexed by (∐_{A} π_{0}(Z^{A}))/G where the coproduct runs over Abelian subgroups of p-rank ≤ n − k, and that the irreducible components of X_{k}(Z, G) are indexed by the same set. In particular, when Z = ∗ the set of pieces correspond to conjugacy classes of Abelian subgroups of p-rank ≤ n − k. For some applications, we need to know that the scheme itself splits. Moreover, we would like a splitting that is valid after completing at I_{k}, rather than just reducing modulo I_{k}. In other words, we would like to study the scheme

X̂′_{k}(Z, G) = spec((u_{k}^{−1}E_{G}^{0}(Z))_{I_{k}}^{∧}).

For this we need some further notation. Let A ≤ G be Abelian and z an element of π_{0}(Z^{A}). The pair (A, z) defines a point of the finite G-set ∐_{A} π_{0}(Z^{A}). We write N_{G}(A, z) for its stabiliser and [A, z] for its orbit, and put W_{G}(A, z) = N_{G}(A, z)/A.

We will prove the following theorem.

Theorem 2.7. There is a closed subscheme Ŷ′_{k}(Z, G, A, z) ≤ X̂′_{k}(Z, G) depending only on the orbit of (A, z), such that

X̂′_{k}(Z, G) = ∐_{[A,z]} Ŷ′_{k}(Z, G, A, z),

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

Level′_{k}(A^{∗}, 𝔾)/W_{G}(A, z) ⟶X_{k} ×_{X} Ŷ′_{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 Ŷ′_{k}(G, A) = Ŷ′_{k}(∗, G, A, ∗), so that X̂′_{k}(G) = ∐_{(A)} Ŷ′_{k}(G, A). Using the material developed in Section 8, we can make this somewhat more explicit when G is Abelian. In particular, we have the following theorem.

Theorem 2.8. If A is a finite Abelian p-group and B ≤ A then Ŷ′_{k}(A, B) is flat over X̂′_{k}, with degree |A|^{k}| Mon(B^{∗}, (ℚ_{p}/ℤ_{p})^{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 π_{0}L_{K(k)}F (BG_{+}, E) ⟶π_{0}F (BG_{+}, L_{K(k)}E) is the projection 𝒪_{X̂′_{k}(G)} ↠ 𝒪_{Ŷ′_{k}(G,1)}. It is probably enough for E to be Landweber exact and E_{G}^{∗} to be free over E^{∗} and concentrated in even degrees.

One application of these results is to the study of E_{∗}^{G}(EG) = E_{∗}BG, in the spirit of [10, 11], and equally to E_{∗}^{G}(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 even degrees. 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 in odd degrees. Usually it can be calculated using local cohomology at the augmentation ideal J = ker(E_{G}^{0} ↠ E^{0}); in fact, if G is a p-group or E admits an equivariant E_{∞} structure, there is a spectral sequence

E_{2}^{∗} = H_{J}^{∗}(E_{G}^{0}) ⟹ E_{∗}^{G}(EG) = E_{∗}(BG).

The groups H_{J}^{∗}(M) occuring here are the derived functors of the functor

Γ_{J}M = H_{J}^{0}(M) = {m ∈ M | J^{N}m = 0 for N ≫ 0}.

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

C^{•}(E_{G}^{0}) = (p^{−1}E_{G}^{0} ⟶u_{1}^{−1}E_{G}^{0}/p^{∞} ⟶⋯ ⟶u_{n−1}^{−1}E_{G}^{0}/I_{n−1}^{∞} ⟶E_{G}^{0}/I_{n}^{∞}).

Note that the k’th term is a module over (u_{k}^{−1}E_{G}^{0})_{I_{k}}^{∧} = 𝒪_{X̂′_{k}(G)}. There is a spectral sequence

E_{1}^{s,t} = H^{s}H_{J}^{t}C^{•}(E_{G}^{0}) ⟹ H_{J}^{s+t}(E_{G}^{0}).

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

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

H_{J}^{∗}(E_{G}^{0}(Z)) = H^{∗}Γ_{J}C^{•}(E_{G}^{0}(Z)).

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

X̂′_{k}(G) = ∐_{(A)} Ŷ′_{k}(G, A),

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

V (J) ×_{X(G)} X̂′_{k}(G) = Ŷ′_{k}(G, 1),

and the local cohomology of this one factor is concentrated in degree zero.

We should point out that the complex Γ_{J}C^{•}(E_{G}^{0}) 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 obvious that 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 4 develops 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 with Theorem 2.4 to prove Theorem 2.5 and Theorem 2.7. The applications to the calculation of E_{∗}(BG) are developed in Section 10. In Appendix B, we outline an alternative approach to some of our results (under some quite restrictive hypotheses) using a variant of the Evens norm 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 J_{A} of a ring A is the intersection of its maximal ideals, and that a ring is said to be semilocal if it has only finitely many maximal ideals. A semilocal 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:

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

In ℛ̂, we consider only maps f: A ⟶ B such that f(J_{A}) ≤ J_{B}; equivalently, f must be continuous with respect to the J-adic topologies on A and B. In the other two cases, we consider all ring maps. Of course, any ring map between fields is injective.

Definition 3.1. If A ∈ ℛ̂, we define a functor spf(A): ℛ̂ ⟶Sets by

spf(A)(B) = ℛ̂(A, B).

We define a formal scheme to be a functor X: ℛ̂ ⟶ Sets such that X ≃ spf(A) for some A.

If X is a formal scheme, we write 𝒪_{X} for the set of natural maps from X to the forgetful functor. These form a ring, and an isomorphism X ≃ spf(A) gives an isomorphism 𝒪_{X} ≃ A by Yoneda’s lemma. Thus, the functors A ↦ spf(A) and X ↦ 𝒪_{X} give an equivalence between ℛ̂^{op} and the category of formal schemes. We prefer to think about the functor category because many examples arise in topology where the functor is easier to describe than 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 topological motivation, in a rather more general technical framework. We will make heavy use of the functorial 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. If X is a scheme, we write 𝒪_{X} 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 a variety to be a functor X: 𝒦 ⟶ Sets such that X ≃ var(A) for some A. Thus, every scheme X has an underlying variety; this will often also be called X, but we will write X_{var} 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 map X_{var} ⟶Y_{var} is an isomorphism, and that a map A ⟶B of rings is a V -isomorphism if the induced map spec(A) ⟵spec(B) is so.

This turns out to be closely related (see Proposition 3.8) to the following more elementary condition. We say that a map u: A ⟶ B of 𝔽_{p}-algebras is an F -isomorphism if every element of the kernel is nilpotent, and for each b ∈ B there is an integer k ≥ 0 such that b^{p^{k}} ∈ u(A).

We next define a number of useful properties of (formal) schemes and maps between them. 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 of two nonempty (formal) schemes, equivalently if 𝒪_{X} is not the product of two nonzero rings.

2. A (formal) scheme X is reduced if 𝒪_{X} has no nonzero nilpotents. The reduced part X_{red} of a scheme X is spec(𝒪_{X}/\sqrt{0}).

3. A connected formal scheme X is smooth if 𝒪_{X} is a regular local ring.

4. A closed subscheme of a scheme X is a scheme of the form V (I) = spec(𝒪_{X}/I) for some ideal I ≤ 𝒪_{X}, and similarly for formal schemes. We say that X is an infinitesimal thickening of Y if Y = V (I) for some nilpotent ideal I ≤ 𝒪_{X}.

5. A map f: W ⟶X of (formal) schemes is finite if the corresponding map 𝒪_{X} ⟶𝒪_{W} makes 𝒪_{W} into a finitely generated module over 𝒪_{X}. The degree of f is the smallest possible number of generators. When working over a fixed base scheme X, we write deg[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 𝒪_{Y} ⟶𝒪_{X} has nilpotent kernel. If f is finite and dominant, then f induces a surjective map X_{var} ⟶Y_{var}.

7. A map f: X ⟶Y of (formal) schemes is flat if it makes 𝒪_{X} into a flat module over 𝒪_{Y}, and similarly for faithful flatness. If Y is a connected formal scheme and f is finite and flat then 𝒪_{X} is a free module over 𝒪_{Y}.

It is more usual to define spec(A) to be the space of prime ideals of A, but this is not convenient for our purposes. It turns out that a V -isomorphism of schemes gives a continuous 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, Appendix B].

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 ≤ 𝒪_{Y}, we can embed the integral domain 𝒪_{Y} /𝔭 in an algebraically closed field K. As f is a V -isomorphism, we can find a map u: 𝒪_{X} ⟶K such that u ∘ f^{∗} is the composite 𝒪_{Y} ↠ 𝒪_{Y} /𝔭 ↣ K. It follows that ker(f^{∗}) ≤ 𝔭. As the intersection of all primes in 𝒪_{Y} is the ideal of nilpotents, we see that 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 then the 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(𝔽_{p}) is a V -isomorphism if and 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 apply the following results to (co)ends, which can be converted to (co)limits as described in [18, Chapter IX]. Whenever we write something like lim_{←-}_{i} W_{i}, 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 computed in the category of all functors from 𝒦 to sets. It is easy to see that a limit of varieties 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 𝒞_{X} for the category of schemes W equipped with a finite map W ⟶ X, and Γ(V, W) = 𝒞_{X}(V, W) for the set of maps of schemes V ⟶W that commute with the given maps to X.

Proposition 3.10. The category 𝒞_{X} has finite limits and colimits, which are converted by the functor W ↦ 𝒪_{W} into colimits and limits in the category of 𝒪_{X}-algebras. In particular, the products are given by V ×_{X} W = spec(𝒪_{V} ⊗_{𝒪_{X}} 𝒪_{W}) and the coproducts are given by V ⨿ W = spec(𝒪_{V} × 𝒪_{W}). For any ring A ∈ ℛ we have

(lim_{←-}_{i} W_{i})(A) = lim_{←-}_{i}(W_{i}(A)),

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

(lim_{-→}_{i} W_{i})(K) = lim_{-→}_{i}(W_{i}(K)).

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

Proposition 3.11. If X is a formal scheme, then 𝒞_{X} is equivalent to the category of formal schemes W equipped with a finite map to X.

Proof. This just means that every finite 𝒪_{X}-algebra is a complete semilocal Noetherian ring, and that all maps of such algebras are continuous with respect to the Jacobson radicals. This is a standard piece of commutative algebra.

Definition 3.12. Let {W_{i}} be a finite diagram of schemes, equipped with compatible maps W_{i} ⟶W. We say that W is a V -colimit of the diagram if the induced map lim_{-→}_{i} W_{i} ⟶ W is a V -isomorphism.

Proposition 3.13. Any map of schemes X′ ⟶X gives rise to a functor W ↦ W ×_{X} X′ from 𝒞_{X} to 𝒞_{X′}; explicitly, W ×_{X}X′ = spec(𝒪_{W} ⊗_{𝒪_{X}} 𝒪_{X′}). This has the following properties:

(lim_{←-}_{i} W_{i}) ×_{X} X′ = lim_{←-}_{i}(W_{i} ×_{X} X′).

The canonical map

lim_{-→}_{i}(W_{i} ×_{X} X′) ⟶(lim_{-→}_{i} W_{i}) ×_{X} X′

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

(W ×_{X} X′)_{var} = W_{var} ×_{X_{var}} 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 field K, it is easy to see that (W_{i} ×_{X} X′)(K) = W_{i}(K) ×_{X(K)} X′(K) and

((lim_{-→}_{i} W_{i}) ×_{X} X′)(K) = (lim_{-→}_{i} W_{i})(K) ×_{X(K)} X′(K).

Proposition 3.10 gives (lim_{-→}_{i} W_{i})(K) = lim_{-→}_{i} W_{i}(K) and (lim_{-→}_{i} X′ ×_{X} W_{i})(K) = lim_{-→}_{i}(X′ ×_{X}

W_{i})(K), and the claim follows.

Remark 3.14. In the light of Proposition 3.8, this has the following consequence. If we have a Noetherian 𝔽_{p}-algebra A and a finite diagram of finite A-algebras A_{i}, and a Noetherian A-algebra B, then the natural map B ⊗_{A} lim_{←-}_{i} A_{i} ⟶ lim_{←-}_{i} B ⊗_{A} A_{i} 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 the underlying schemes.

Proposition 3.15. Suppose that we have a finite collection of finite maps of schemes f_{i}: X_{i} ⟶ Y, such that the combined map f: ∐_{i} X_{i} ⟶ Y is a V -isomorphism. Then there are closed subschemes Y_{i} ⊆ Y such that Y = ∐_{i} Y_{i} and f_{i} factors through a finite V -isomorphism X_{i} ⟶Y_{i}.

Proof. Write B = 𝒪_{Y} and A_{i} = 𝒪_{X_{i}} and I_{i} = ker(B ⟶A_{i}). Because f is a V -isomorphism and thus dominant, we see that J = ⋂_{i} I_{i} is nilpotent. We claim that if i ≠ j then I_{i}+I_{j} = B. If not, we can find a map B/(I_{i}+I_{j}) ⟶K for some algebraically closed field K. As the map B/I_{i} ⟶A_{i} is finite and injective, we can factor the map B/I_{i} ↠ B/I_{i}+I_{j} ⟶K through A_{i}. Similarly, we can factor it through A_{j}, so the map B ⟶B/I_{i} +I_{j} ⟶K defines a point of X_{i}(K) ∩ X_{j}(K), which contradicts the assumption that Y (K) = ∐_{i} X_{i}(K).

We can now apply the Chinese Remainder Theorem to see that the natural map B/J ⟶ ∏_{i} B/I_{i} is an isomorphism. This means that there are elements e_{i} ∈ B such that e_{i} = 1 (mod I_{i}) and e_{i} ∈ ⋂_{j≠i} I_{j}, so that e_{i}^{2} − e_{i} lies in the nilpotent ideal J. By the Idempotent Lifting Theorem, we can modify the e_{i} modulo J in a unique way to ensure that e_{i}^{2} = e_{i}, and then ∑_{i} e_{i} = 1. This means that B = ∏_{i} C_{i} as rings where C_{i} = Be_{i}. It is easy to check that the map B ⟶ A_{i} factors through a V -isomorphism C_{i} ⟶ A_{i}, so we can put Y_{i} = spec(C_{i}).

4. Admissible cohomology theories

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

Definition 4.1. Let p_{ℂP^{∞}}: ℂP^{∞} ⟶ℂP^{∞} be the map classifying the p’th tensor power of the canonical line bundle. A p-local commutative ring spectrum E is admissible if

(a) E^{0} is a complete local Noetherian ring.

(b) E^{1} = 0.

(c) E^{2} contains a unit.

(d) p_{ℂP^{∞}} induces a nonzero self-map of Ẽ^{0}(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 ∈ Ẽ^{∗}ℂP^{∞}, in other words, E is complex-orientable. Because of (c), we can choose such an x in degree zero, so that E^{0}(ℂP^{∞}) = E^{0}[[x]]. In particular, this is a complete Noetherian local ring. By a similar argument, we see that

E^{0}(ℂP^{∞} × ℂP^{∞}) = E^{0}(ℂP^{∞})ˆ⊗_{E^{0}}E^{0}(ℂP^{∞}) = E^{0}[[x ⊗ 1, 1 ⊗ x]].

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

We now define formal schemes X = spf(E^{0}) and 𝔾 = spf(E^{0}(ℂP^{∞})). Because ℂP^{∞} is a commutative H-space, we get a multiplication map

μ: 𝔾 ×_{X} 𝔾 = spf(E^{0}(ℂP^{∞} × ℂP^{∞})) ⟶spf(E^{0}(ℂP^{∞})) = 𝔾.

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

p_{𝔾}^{k} = p^{k}.1_{𝔾} = spf(E^{0}(p_{ℂP^{∞}}^{k})): 𝔾 ⟶𝔾.

Condition (d) says that p_{𝔾} is not the zero homomorphism. We can of course translate back to the more traditional language: there is a unique formal group law F (s, t) over 𝒪_{X} such that

μ^{∗}(x) = F (x ⊗ 1, 1 ⊗ x) ∈ E^{0}(ℂP^{∞} × ℂP^{∞}) = E^{0}[[x ⊗ 1, 1 ⊗ x]].

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

Definition 4.2.

𝔾(k) = ker(p_{𝔾}^{k}: 𝔾 ⟶𝔾) = spf E^{0}[[x]]/[p^{k}](x).

The next two results are well-known. Proofs are given under more restrictive hypotheses in [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(x^{p^{k}}) with f′(0) ≠ 0. There is an integer n ≥ 0 (called the height of 𝔾) such that [p](x) has the form g(x^{p^{n}}) modulo 𝔪[[x]], with g′(0) ∈ (E^{0}/𝔪)^{×}.

Proposition 4.4. The maps p_{𝔾}^{k}: 𝔾 ⟶𝔾 and 𝔾(k) ⟶X are flat of degree p^{nk}.

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 p^{n}, where n is the height.

Definition 4.5. For 0 ≤ k ≤ n, let u_{k} be the coefficient of x^{p^{k}} in [p](x). Note that u_{0} = p, that u_{k} ∈ 𝔪 for 0 < k < n, and that u_{n} ∉ 𝔪. Write

I_{k} = (u_{0}, …, u_{k−1}) ≤ E^{0}
X_{k} = spf(E^{0}/I_{k}) ≤ X
X′_{k} = spec(u_{k}^{−1}E^{0}/I_{k})
X̂′_{k} = spec((u_{k}^{−1}E^{0})_{I_{k}}^{∧}).

One can check quite directly that these objects do not depend on the choice of coordinate x. Moreover, I_{k} is also the ideal generated by the coefficients of x^{i} in [p](x) for 0 < i < p^{k}. Thus 𝔾 has strict height ≥ k over X_{k}.

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 sequence (u_{0}, u_{1}, …, u_{n−1}) is regular in E^{0}.

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_{k}^{−1}E^{0}(Y). On the one hand, we can form the module E^{0}(Y) and then invert u_{k} algebraically; we call the result u_{k}^{−1}(E^{0}(Y)). On the other hand, multiplication by u_{k} gives a self-map of the spectrum E, whose telescope is a spectrum u_{k}^{−1}E, so we can form (u_{k}^{−1}E)^{0}(Y). These two objects can be very different. For example, let E be Morava E-theory and let k = 0. Then u_{k}^{−1}E is a rational spectrum, so (u_{k}^{−1}E)^{0}(BG) = u_{k}^{−1}E^{0}, but u_{k}^{−1}(E^{0}(BG)) is much more complicated, as described in [14].

Convention 4.7. The symbol u_{k}^{−1}E^{0}(Y) will always mean u_{k}^{−1}(E^{0}(Y)), as discussed in the previous paragraph.

Definition 4.8. Let G be a finite group, and Z a G-space. We write E_{G}^{∗}(Z) = E^{∗}(EG ×_{G} Z), and

X(Z, G) = spf(E_{G}^{0}(Z))
X_{k}(Z, G) = spf(E_{G}^{0}(Z)/I_{k}) = X(Z, G) ×_{X} X_{k}
X′_{k}(Z, G) = spec(u_{k}^{−1}E_{G}^{0}(Z)/I_{k}) = X(Z, G) ×_{X} X′_{k}
X̂′_{k}(Z, G) = spec((u_{k}^{−1}E_{G}^{0}(Z))_{I_{k}}^{∧})

Remark 4.9. It is often the case that there is an admissible ring spectrum F and a ring map E ⟶F inducing an isomorphism π_{∗}(E)/I_{k} ≃ π_{∗}(F) (so F could reasonably be called E/I_{k}). If so, the map E_{G}^{0}(Z)/I_{k} ⟶F_{G}^{0}(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 where these spectral sequences exist, but one needs either the technology of highly structured ring spectra [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 ∈ 𝒞_{X}. 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 is a p-group, say

A = ⟨a_{0}, …, a_{r−1} | p^{d_{i}}a_{i} = 0⟩ = ⨁_{k=0}^{r−1} ℤ/p^{d_{k}}.

We then have Hom(A, 𝔾) = ∏_{k} 𝔾(d_{k}). This means that

𝒪_{Hom(A,𝔾)} = 𝒪_{X}[[x_{0}, …, x_{r−1}]]/([p^{d_{0}}](x_{0}), …, [p^{d_{r−1}}](x_{r−1}));

by Proposition 4.4 this is a free module of rank ∏_{i} p^{nd_{i}} = |A|^{n} over E^{0} = 𝒪_{X}.

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, S^{1}) 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 of continuous E^{0}-algebra maps E^{0}(ℂP^{∞}) ⟶ E^{0}(BA). By regarding ℂP^{∞} as BS^{1}, we get a canonical map ϕ: A^{∗} ⟶Γ(X(A), 𝔾) defined by ϕ(α) = E^{0}(Bα), and thus a canonical map X(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/p^{k} ⟶ℂP^{∞} ⟶(ℂP^{∞})^{L^{p^{k}}} .

Using the Thom isomorphism, we get a long exact sequence

⋯ ⟵E^{0}Bℤ/p^{k} ⟵𝒪_{X}[[x]] ⟵^{[p^{k}](x)} 𝒪_{X}[[x]] ⟵· · · .

As [p^{k}](x) is not a zero-divisor in 𝒪_{X}[[x]], we see that

E^{0}(Bℤ/p^{k}) = 𝒪_{X}[[x]]/[p^{k}](x) = 𝒪_{𝔾(k)} = 𝒪_{Hom((ℤ/p^{k})^{∗},𝔾)}.

One can check that this is the same as the map considered previously. Similarly, we see that E^{1}(Bℤ/p^{k}) = 0. As E^{0}(Bℤ/p^{k}) is free of finite rank over E^{0}, we have a Künneth isomorphism

E^{0}(B(ℤ/p^{k}) × Z) = E^{0}(Bℤ/p^{k}) ⊗_{E^{0}} E^{0}(Z).

It follows easily that X(A ⊕ B) = X(A) ×_{X} X(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 we adapt some of his ideas to our particular technical context, and we present the argument in the language of equivariant topology.

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

(a) T has Abelian isotropy groups.

(b) There is an equaliser diagram

E_{G}^{0}(Z) ⟶E_{G}^{0}(Z × T) ⇉ E_{G}^{0}(Z × T × T).

(c) E_{G}^{0}(Z) is a finitely generated module over E^{0}.

Next, we try to extract information from the filtration of Z by G-skeleta, or equivalently from the equivariant Atiyah-Hirzebruch spectral sequence. The conclusion is that E_{G}^{0}(Z) has the same variety as the ring H_{G}^{0}(Z; \underline{E}), where H_{G}^{0}(·; \underline{E}) denotes ordinary cohomology in the sense of Bredon, for the coefficient system \underline{E} : G/H ↦ E_{G}^{∗}(G/H) = E_{H}^{∗}.

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

E_{𝒜} (Z) = ∫_{A∈𝒜} Map(π_{0}(Z^{A}), E_{A}^{0})

considered by Quillen (he denotes it 𝒜_{G}(Z) and writes the end as a limit). We show that this is the same as H_{G}^{0}(Z; \underline{E}) if Z has Abelian isotropy groups. By combining this with (a),

(b) and (c) above, we shall see that

X(Z, G) = ∫ ^{A} π_{0}(Z^{A}) × 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 a point of V is a chain

\underline{V} = (0 = V_{0} < V_{1} < … < V_{d} = V)

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

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

Proof. If H ≤ G stabilises a flag 0 = V_{0} < V_{1} < ⋯ < V_{d} = V then we have an isomorphism V ≃ ⨁_{i} V_{i}/V_{i−1} of H-representations. Since V is faithful, H embeds in the group ∏_{i} Aut(V_{i}/V_{i−1}), and since V_{i}/V_{i−1} is one dimensional, Aut(V_{i}/V_{i−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 over Y. Let π: Flag(W) ⟶ Y be the associated flag bundle, so a point of π^{−1}y is a complete flag in the vector space W_{y}. There is an evident filtration

0 = W_{0} < W_{1} < … < W_{d} = π^{∗}W

of bundles over Flag(W). We let x_{k} ∈ E^{0}(Flag(W)) denote the Euler class of W_{k}/W_{k−1}, and write σ_{k} for the k’th symmetric function in the variables x_{i}. We also let c_{k} ∈ E^{0}(Y) denote the k’th Chern class of W. There is a natural isomorphism

E^{∗}(Flag(W)) = E^{∗}(Y)[[x_{1}, …, x_{d}]]/(σ_{k} − c_{k} | 0 ≤ k ≤ d).

Moreover, this is a free module over E^{∗}(Y) of rank d!, spanned by the monomials x^{α} = ∏_{i=1}^{d} x_{i}^{α_{i}} for which 0 ≤ α_{i} < 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; this avoids a discussion of Künneth isomorphisms.

Let S be the set of multiindices α = (α_{1}, …, α_{d}) such that 0 ≤ α_{i} < i. By applying Proposition 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}^{∗}{x^{α} | α ∈ S}. On replacing Z by Z × T we find that

E_{G}^{∗}(Z × T × T) = E_{G}^{∗}(Z × T){y^{β} | β ∈ S} = E_{G}^{∗}(Z){x^{α}y^{β} | α, β ∈ S}.

The two projections Z ×T ×T ⇉ Z ×T induce maps sending x^{α} to x^{α} and y^{α} 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 semilocal Noetherian ring).

Proof. First suppose that Z has Abelian isotropy groups, so it is built from cells of the form B^{k} × G/A with A Abelian. We know from Proposition 4.12 that E_{G}^{∗}(G/A) = E_{A}^{∗} is finitely 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) is finitely 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 hypothesis on E. For example it fails for uncompleted K-theory.

We now look at E_{G}^{0}(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 it has

E_{1}^{s,t} = E_{G}^{s+t}(Z^{(s)}, Z^{(s−1)}).

If we let \underline{E}^{t} denote the coefficient system G/A ↦ E_{G}^{t}(G/A) = E^{t}(BA), then we have

E_{2}^{s,t} = H_{G}^{s}(Z; \underline{E}^{t}).

We need to know this spectral sequence is multiplicative from E_{2} onwards: this follows by using the fact that the diagonal map Z ⟶ Z × Z is skeletal if Z × Z is given the product filtration. An alternative is to use the Postnikov filtration of E to obtain a spectral sequence: it is proved in [8, Appendix B] that this agrees with the first from the E_{2} term onwards.

Lemma 5.6. The edge homomorphism

i: E_{G}^{0}(Z) ⟶H_{G}^{0}(Z; \underline{E}^{0})

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

Proof. Let d be the dimension of Z, so that E_{1}^{s,∗} = 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 x^{d+1} has filtration at least d + 1 and thus is zero. Thus, E_{G}^{0}(Z) has the same variety as its image under i, which is precisely E_{∞}^{0,0} = E_{d+1}^{0,0}. Next, observe that E_{r+1}^{0,0} is the kernel of a derivation from the ring E_{r}^{00} to the module E_{r}^{r+1,−r}, and that everything in sight is a finitely generated module over E^{0} (by Corollary 5.4). By Proposition 3.7, we see that E_{r+1}^{0,0} has the same variety as E_{r}^{0,0}, and the claim follows.

We now appeal to an explicit description of H_{G}^{0}(Z; \underline{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) = Map_{G}(G/A, G/B) ≃ {g ∈ G | A^{g} ≤ B}/B.

We have two contravariant functors from 𝒜 to sets, given by A ↦ π_{0}(Z^{A}) and A ↦ E_{A}^{0} = E_{G}^{0}(G/A). We define

E_{𝒜} (Z) = Nat_{A∈𝒜} (π_{0}(Z^{A}), E_{A}^{0}) = ∫_{A} Map(π_{0}(Z^{A}), E_{A}^{0}).

Lemma 5.7. If H ≤ G is Abelian (but not necessarily a p-group) then E_{𝒜} (G/H) = E_{H}^{0}.

Proof. We can write H = B × C where B is a p-group and C is a p′-group. Note that the action of C on E_{B}^{0} by conjugation is trivial because H is commutative. Since B is a retract of H, E_{B}^{0} is a retract of E_{H}^{0}; by a transfer argument E_{H}^{0} = E_{B}^{0}. 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 π_{0}((G/H)^{A}) = 𝒜(A, H) = 𝒜(A, B)/C. By Yoneda’s lemma, we have Nat_{A}(𝒜(A, B), E_{A}^{0}) = E_{B}^{0}, so Nat_{A}(𝒜(A, B)/C, E_{A}^{0}) = (E_{B}^{0})^{C} = E_{B}^{0}. Thus E_{𝒜} (G/H) is just E_{B}^{0} = E_{H}^{0}.

Lemma 5.8. There is a natural map H_{G}^{0}(Z; \underline{E}) ⟶E_{𝒜} (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

H_{G}^{0}(Z; \underline{E}) ⟶H_{A}^{0}(Z^{A}; \underline{E}) ⟶H^{0}(Z^{A}; E_{A}^{0}) = Map(π_{0}(Z^{A}), E_{A}^{0}),

and that these assemble to give a natural map

α_{Z}: H_{G}^{0}(Z; \underline{E}) ⟶E_{𝒜} (Z).

Now suppose that Z is a union of two subcomplexes Z_{0} and Z_{1}. We have a Mayer-Vietoris sequence

0 ⟶H_{G}^{0}(Z; \underline{E}) ⟶H_{G}^{0}(Z_{0}; \underline{E}) ⊕ H_{G}^{0}(Z_{1}; \underline{E}) ⟶H_{G}^{0}(Z_{0} ∩ Z_{1}; \underline{E}).

Similarly, for each A we have Z^{A} = Z_{0}^{A} ∪ Z_{1}^{A}, so there is a coequaliser of sets

π_{0}(Z_{0}^{A} ∩ Z_{1}^{A}) ⇉ π_{0}(Z_{0}^{A}) ⨿ π_{0}(Z_{1}^{A}) ⟶π_{0}(Z^{A}).

By applying Map(−, E_{A}^{0}), we get an equaliser diagram. It is easy to see that equalisers commute with ends, so we get a left exact sequence

0 ⟶E_{𝒜} (Z) ⟶E_{𝒜} (Z_{0}) ⊕ E_{𝒜} (Z_{1}) ⟶E_{𝒜} (Z_{0} ∩ Z_{1}).

By comparing these two sequences, we see that α_{Z} is an isomorphism provided that α_{Z_{0}}, α_{Z_{1}} and α_{Z_{0}∩Z_{1}} are isomorphisms. As Z has Abelian isotropy groups, it is built from cells of the form B^{m} × G/H, with H Abelian. By induction on the number of cells, it is enough to check that α_{Z} is an isomorphism when Z = S^{m−1} × G/H or Z = B^{m} × G/H. This reduces easily to a check that α: H_{G}^{0}(G/H; \underline{E}) ⟶ E_{𝒜} (G/H) is an isomorphism. The left hand side is just E_{H}^{0}, which is the same as the right hand side by Lemma 5.7.

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

E_{𝒜} (Z) ⟶E_{𝒜} (Z × T) ⇉ E_{𝒜} (Z × T × T)

is an equaliser.

Proof. Suppose that A ∈ 𝒜. Because π_{0}((Z × T)^{A}) = π_{0}(Z^{A}) × π_{0}(T^{A}) and so on, it is clear that the diagram

π_{0}(Z^{A}) ⟵π_{0}((Z × T)^{A}) ⇇ π_{0}((Z × T × T)^{A})

is a coequaliser. By applying Map(−, E_{A}^{0}) 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} π_{0}(Z^{A}) × Hom(A^{∗}, 𝔾) ⟶X(Z, G),

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

∫ ^{A} π_{0}(Z^{A}) × Hom_{k}(A^{∗}, 𝔾) ⟶X_{k}(Z, G)

and

∫ ^{A} π_{0}(Z^{A}) × Hom′_{k}(A^{∗}, 𝔾) ⟶X′_{k}(Z, G).

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

E_{G}(Z)  E_{G}(Z × T)  E_{G}(Z × T × T) H_{G}^{0}(Z; \underline{E})  H_{G}^{0}(Z × T; \underline{E})  H_{G}^{0}(Z × T × T; \underline{E}) E_{𝒜} (Z)  E_{𝒜} (Z × T)  E_{𝒜} (Z × T × T)_{_{i_{0}} u} v  w _{_{i_{1}} u}  w w  _{u_{i_{2}}}_{_{j_{0}} u}  w  _{_{j_{1}} u}  w w  _{u_{j_{2}}} v  w  w w

The first and third rows are exact by Corollary 5.3 and Lemma 5.9 respectively. The maps i_{0}, i_{1} and i_{2} are V -isomorphisms by Lemma 5.6. The maps j_{1} and j_{2} are isomorphisms by Lemma 5.8. Everything in sight is a finitely generated module over the Noetherian ring E^{0}, by Corollary 5.4. We now apply the functor var(−) to get a diagram of varieties with the arrows reversed. The first and third lines are coequalisers by Proposition 3.10, and all vertical maps except for var(j_{0}) are known to be isomorphisms. It follows immediately that var(j_{0}) is also an isomorphism, so that X(Z, G) = var(E_{𝒜} (Z)). We can rewrite the end that defines E_{𝒜} (Z) as the inverse limit of a finite diagram of rings that are finitely generated as modules over E^{0}. By Proposition 3.10, this becomes colimit diagram when we apply var(−). This shows that

(5.0.1) X(Z, G)_{var} = var (∫_{A} Map(π_{0}(Z^{A}), E_{A}^{0})) =

∫ ^{A} var(Map(π_{0}(Z^{A}), E_{A}^{0})) = ∫ ^{A} π_{0}(Z^{A}) × X(A).

Using Proposition 4.10, we can rewrite this as

X(Z, G)_{var} = ∫ ^{A} π_{0}(Z^{A}) × 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. This theory is closely related to questions studied in [23], and we will assume some familiarity with that paper. For the purposes of the next three sections, we do not need to start with an admissible cohomology theory. We merely assume that we have a formal group 𝔾 of finite height n over a connected formal scheme X. We also fix an integer k with 0 ≤ k ≤ n, and assume that 𝔾 has strict height at least k. Thus, to apply this section to the topological questions studied in previous sections, we need to replace X by X_{k} and 𝔾 by 𝔾 ×_{X} X_{k}.

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 over X. We say that a homomorphism ϕ: A^{∗} ⟶ Γ(Y, 𝔾) is a p^{k}-fold level-A^{∗} structure if we have an inequality of divisors

p^{k}[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 is equivalent to require that [p](x) be divisible by

∏_{α∈A^{∗}(1)}(x − x(ϕ(α)))^{p^{k}}

in 𝒪_{Y} [[x]].

¿From a purely algebraic point of view, there is no reason to consider maps A^{∗} ⟶ 𝔾 rather 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 this case.

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

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

Y ↦ { p^{k}-fold level-A^{∗} structures on 𝔾 over Y }

is represented by a scheme Level_{k}(A^{∗}, 𝔾) over X. Moreover, Level_{k}(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 Λ_{k} = (ℚ_{p}/ℤ_{p})^{n−k}.

The main result is as follows.

Theorem 6.4. The scheme Level_{k}(A^{∗}, 𝔾) is flat over X, of degree |A|^{k}| Mon(A^{∗}, Λ_{k})|. If (𝔾, X) is of universal type (see Definition 6.5) then Level_{k}(A^{∗}, 𝔾) is smooth. For each B ≤ A, there is a natural closed inclusion Level_{k}(B^{∗}, 𝔾) ⟶ Hom(A^{∗}, 𝔾). Together these maps give a map

∐_{B} Level_{k}(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 obtain an inequality

deg[Level_{k}(A^{∗}, 𝔾)] ≤ |A|^{k}| Mon(A^{∗}, Λ_{k})|.

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 = ⟨a_{0}, …, a_{r−1} | p^{d_{i}}a_{i} = 0 for all i⟩.

We shall order the generators so that 0 < d_{0} ≤ … ≤ d_{r−1}. We write α_{i} for the element of A^{∗} that is dual to a_{i} in the evident sense, so that A^{∗} = ⟨α_{i} | p^{d_{i}}α_{i} = 0⟩. Given a scheme Y over X and a map ϕ: A^{∗} ⟶Γ(Y, 𝔾), we write x_{α} = x(ϕ(α)) and x_{i} = x_{α_{i}} = x(ϕ(α_{i})).

Let ϕ be a p^{k}-fold level-A^{∗} structure on 𝔾 over Y. Over the special fibre of Y we have ϕ = 0 so x^{p^{r+k}} divides [p](x), which is a unit multiple of x^{p^{n}}. It follows that r ≤ n − k (unless Y is empty). Thus, Level_{k}(A^{∗}, 𝔾) = ∅ if rank_{p}(A) > n − k.

Many questions about multiple level structures become simpler if the base scheme X has 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 of elements (u_{k}, …, u_{n−1}) is regular on 𝒪_{X} and generates the maximal ideal (and thus X is a smooth scheme).

If k > 0 and (𝔾, X) has universal type then 𝒪_{X} = K[[u_{k}, …, u_{n−1}]] for some field K of characteristic p, and if k = 0 then 𝒪_{X} = R[[u_{1}, …, u_{n−1}]] for some discrete valuation ring R whose maximal ideal is generated by p. In either case, 𝒪_{X} is a unique factorisation domain and has many other good properties. We can often assume that X has universal type, and deduce the general case by base change using the following proposition. We will give this argument in detail in Corollary 6.11, and subsequently leave similar arguments to the reader.

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

𝔾  ℍ  𝕂 X  W  V_{u} _{u}  _{w}_{u}  _{u}_{u _{f}}  _{w_{g}}

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

Proof. Let 𝔪 < 𝒪_{X} be the maximal ideal, and let K be the perfect closure of 𝒪_{X}/𝔪. By EGA 0_{III}10.3.1 [12] there is a complete Noetherian local ring 𝒪_{W} and a faithfully flat map 𝒪_{X} ⟶𝒪_{W} such that the associated extension of residue fields is the inclusion 𝒪_{X}/𝔪 ↣ K. We define ℍ = 𝔾 ×_{X} W. By the Lubin-Tate deformation theory (see [17], or [23, Section 6] for an account in geometric language) there is a smooth scheme U, a formal group 𝕃 over U, and a pullback diagram

ℍ  𝕃 W  U_{w}_{u}  _{u}_{w}

Moreover, we have 𝒪_{U} = R[[w_{1}, …, w_{n−1}]] for some complete discrete valuation ring R with R/p = K, where w_{i} is the coefficient of y^{p^{i}} in the [p](y) for some coordinate y on 𝕃. It follows that

V = spf(𝒪_{U}/(p, w_{1}, …, w_{k−1})) = spf(K[[w_{k}, …, w_{n−1}]])

is the closed subscheme where 𝕃 has strict height at least k. As ℍ has strict height at most k, we see that the map W ⟶U must factor through V. We thus get a diagram as claimed with 𝕂 = 𝕃 ×_{U} V.

Definition 6.7. We use the following notation.

D = 𝒪_{X}
R(A) = 𝒪_{Hom(A^{∗},𝔾)} = D[[x_{0}, …, x_{r−1}]]/([p^{d_{0}}](x_{0}), …, [p^{d_{r−1}}](x_{r−1}))
D(A) = 𝒪_{Level_{k}(A^{∗},𝔾)}.

We also write D′ = u_{k}^{−1}D, R′(A) = u_{k}^{−1}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 rings cannot be applied directly to D′.

Proposition 6.8. If ϕ: A^{∗} ⟶ Γ(X, 𝔾) is such that for all 0 ≠ α ∈ A^{∗}(1) the element x_{α} ∈ 𝒪_{X} is not a zero-divisor, then ϕ is a p^{k}-fold level-A^{∗} structure. In particular, this holds if 𝒪_{X} is an integral domain and ϕ is injective.

Proof. We may assume that A^{∗} = A^{∗}(1) (which just simplifies notation). Let g(x) be the Weierstrass polynomial that is a unit multiple in 𝒪_{X}[[x]] of [p](x), and write f(x) = ∏_{α}(x − x_{α})^{p^{k}}. We need to prove that f divides g in 𝒪_{X}[x]. Because divisibility of monic polynomials is a closed condition, it is enough to prove this over any extension ring of 𝒪_{X}.

Note that Δ = ∏_{α≠β}(x_{α} − x_{β}) is a unit multiple of ∏_{α≠β} x_{α−β} (because x − y is a unit multiple of x −_{F} y), so Δ is not a zero-divisor, so the natural map 𝒪_{X} ⟶ Δ^{−1}𝒪_{X} is injective. If α ≠ β then x−x_{α} and x−x_{β} generate the unit ideal in Δ^{−1}𝒪_{X}[x], so the same is true of (x − x_{α})^{p^{k}} and (x − x_{β})^{p^{k}}. It follows from the Chinese Remainder Theorem that the intersection of the ideals ((x − x_{α})^{p^{k}}) is the same as their product. It is thus enough to check that (x − x_{α})^{p^{k}} divides f(x) in 𝒪_{X}[x], or equivalently that (x −_{F} x_{α})^{p^{k}} divides [p](x) in 𝒪_{X}[[x]]. As 𝔾 has strict height at least k, we know that x^{p^{k}} divides [p](x) and thus that (x −_{F} x_{α})^{p^{k}} divides [p](x −_{F} x_{α}). However, [p](x −_{F} x_{α}) = [p](x) −_{F} [p](x_{α}) = [p](x) −_{F} x_{pα} = [p](x), so (x −_{F} x_{α})^{p^{k}} 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 integral domain 𝒪_{X}. Then a map ϕ: A^{∗} ⟶ Γ(X, 𝔾) is a p^{k}-fold level structure if and only if it is injective.

Proof. The hypothesis on 𝔾 is that x^{p^{k}+1} does not divide [p](x). If ϕ is a p^{k}-fold level structure and 0 ≠ α ∈ A^{∗}(1) then x^{p^{k}}(x − x_{α}) divides [p](x) so we must have x_{α} ≠ 0. It follows that ker(ϕ) ∩ A^{∗}(1) = 0 and thus that ϕ is injective. The converse is covered by the previous proposition.

Proposition 6.10. If X is of universal type then the scheme Level_{k}(A^{∗}, 𝔾) is smooth and is flat over X.

Proof. Choose m so large that p^{m}A = 0, and let g(x) be the Weierstrass polynomial that is a unit multiple of [p^{m}](x), so g is monic and has degree p^{nm}. Let L be the splitting field of g over the field of fractions of 𝒪_{X}, and R the subring of L generated by 𝒪_{X} and the roots of g. These roots form a group A′ under formal addition. It is easy to see that they all have the same multiplicity, and that the multiplicity of zero is p^{mk}; it follows that |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 that A′ ≃ (ℤ/p^{m})^{n−k}. Write Y = spf(R), so A′ is naturally a subgroup of Γ(Y, 𝔾). Because rank_{p}(A^{∗}) ≤ n − k and p^{m}A^{∗} = 0, we can choose a monomorphism ϕ: A^{∗} ⟶A′ ≤ Γ(Y, 𝔾). By Proposition 6.8, this is a p^{k}-fold level structure (because R is an integral domain). The evident map Y ⟶X (which is finite and dominant) thus factors through Level_{k}(A^{∗}, 𝔾). It follows that Level_{k}(A^{∗}, 𝔾) ⟶X is dominant, so that dim(Level_{k}(A^{∗}, 𝔾)) ≥ dim(X) = n−k.

Note that Level_{k}(A^{∗}, 𝔾) is by construction a closed subscheme of Hom(A^{∗}, 𝔾) = ∏_{i} 𝔾(d_{i}). It follows that the maximal ideal in D(A) is generated by x_{0}, …, x_{r−1}, u_{k}, …, u_{n−1}, and also that Level_{k}(A^{∗}, 𝔾) is finite over X. Let S be the set {x_{0}, …, x_{r−1}, u_{k+r}, …, u_{n−1}}, so |S| = n − k. Write \overline{D}(A) = D(A)/(S). Over this ring we know that ϕ(α_{i}) = 0 for all i and thus that ϕ(α) = 0 for all α ∈ A^{∗}. It follows that ∏_{α∈A^{∗}(1)}(x − x_{α})^{p^{k}} = x^{p^{r+k}}. We also know that this divides [p](x), so that 𝔾 has strict height at least r + k over \overline{D}(A). This implies that u_{k} = … = u_{r+k−1} = 0 in \overline{D}(A). The rest of the u’s vanish by definition. It follows that the maximal ideal in \overline{D}(A) is zero, or equivalently, that S generates the maximal ideal in D(A). As |S| is the same as the Krull dimension of D(A), we see that D(A) is a regular local ring, or in other words that Level_{k}(A^{∗}, 𝔾) is a smooth scheme. As a finite dominant map of smooth schemes is flat, we see that Level_{k}(A^{∗}, 𝔾) is flat over X.

Corollary 6.11. Even when X is not universal, the map Level_{k}(A^{∗}, 𝔾) ⟶X is flat.

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

Level_{k}(A^{∗}, ℍ) = Level_{k}(A^{∗}, 𝔾) ×_{X} W = Level_{k}(A^{∗}, 𝕂) ×_{V} W.

As (𝕂, V) has universal type, we see that Level_{k}(A^{∗}, 𝕂) is flat over V. As flatness is preserved by pullbacks, we see that Level_{k}(A^{∗}, ℍ) is flat over W. Thus, the map Level_{k}(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 Level_{k}(A^{∗}, 𝔾).

Lemma 6.12. If A = ⨁_{i=0}^{r−1} ℤ/p^{d_{i}} with d_{i} > 0 then

| Mon(A^{∗}, Λ_{k})| = ∏_{i=0}^{r−1} p^{(n−k)(d_{i}−1)}(p^{n−k} − p^{i}) = |A^{∗}/A^{∗}(1)|^{n−k}| Mon(A^{∗}(1), Λ_{k})|.

It follows that

|A|^{k}| Mon(A^{∗}, Λ_{k})| = ∏_{i=0}^{r−1} p^{n(d_{i}−1)}(p^{n} − p^{i+k}).

Proof. Write β_{i} = p^{d_{i}−1}α_{i} ∈ A^{∗}(1). Let ψ be a map A^{∗} ⟶ Λ_{k}. If ψ is not injective, say ψ(α) = 0 with α ≠ 0, then p^{i}α ∈ ker(ψ) ∩ A^{∗}(1) for some i. Thus, ψ is injective if and only if ψ|_{A^{∗}(1)} is injective, if and only if we have

ψ(β_{i}) ∉ ⟨ψ(β_{0}), …, ψ(β_{i−1})⟩

for all i. Suppose that we have chosen ψ(α_{j}) for j < i; there are then p^{n−k} − p^{i} possible choices for ψ(β_{i}) = p^{d_{i}−1}ψ(α_{i}). As |Λ_{k}(d_{i}−1)| = p^{(n−k)(d_{i}−1)}, there are p^{(n−k)(d_{i}−1)}(p^{n−k}−p^{i}) possible choices for ψ(α_{i}). It follows that

| Mon(A^{∗}, Λ_{k})| = ∏_{i} p^{(n−k)(d_{i}−1)}(p^{n−k} − p^{i})

as claimed. The rest is trivial.

Proposition 6.13. deg[Level_{k}(A^{∗}, 𝔾)] ≤ |A|^{k}| Mon(A^{∗}, Λ_{k})|.

Proof. In view of Lemma 6.12, it is enough to show that x_{i} has degree at most p^{n(d_{i}−1)}(p^{n} − p^{i+k}) over 𝒪_{X}[x_{0}, …, x_{i−1}]. Because ϕ is a p^{k}-fold level structure, we see that [p^{d_{i}−1}](x_{i}) is a root of f(x) = [p](x)/ ∏_{β}(x − x_{β})^{p^{k}}, where β runs over ⟨β_{j} | j < i⟩. As f(x) has Weierstrass degree p^{n} − p^{i+k} and [p^{d_{i}−1}](x) has Weierstrass degree p^{n(d_{i}−1)}, we see that x_{k} is a root of a power series f([p^{d_{i}−1}](x)) of Weierstrass degree p^{n(d_{i}−1)}(p^{n} − p^{i+k}) as required.

Proof of Theorem 6.4. We proved as Corollary 6.11 that Level_{k}(A^{∗}, 𝔾) is flat over X, and as Proposition 6.13 that the degree is at most |A|^{k}| Mon(A^{∗}, Λ_{k})|. The proof of the other inequality is postponed to Corollary 8.4. We proved as Proposition 6.10 that Level_{k}(A^{∗}, 𝔾) is smooth in the universal case. All that is left is to define and study the map ∐_{B} Level_{k}(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 embedding Hom(B^{∗}, 𝔾) ↣ Hom(A^{∗}, 𝔾), or equivalently a surjective map R(A) ↠ R(B). By composing this with the closed embedding Level_{k}(B^{∗}, 𝔾) ↣ Hom(B^{∗}, 𝔾), we get a closed embedding Level_{k}(B^{∗}, 𝔾) ↣ Hom(A^{∗}, 𝔾), corresponding to a surjective map R(A) ↠ D(B). Write 𝔭_{B} for the kernel; this is prime because D(B) is a regular local ring and thus an integral domain. Because D(B) is flat (and thus free) over D, it is also clear that 𝔭_{B} ∩D = 0 and in particular u_{k} ∉ 𝔭_{B}. We also write 𝔭′_{B} = 𝔭_{B}R′(A), so that R′(A)/𝔭′_{B} = D′(B).

Now let 𝔭 be any prime ideal in R(A) such that u_{k} ∉ 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), and Proposition 6.9 tells us that ψ is a p^{k}-fold level-B^{∗} structure. It follows that 𝔭_{B} ≤ 𝔭, so that the intersection of the 𝔭′_{B}’s is the same as the intersection of all primes in R′(A). It is well-known that this intersection is just \sqrt{0} = \{x ∈ R′(A) | x is nilpotent \}.

We next consider 𝔭_{B} + 𝔭_{C}, where B and C are distinct subgroups of A, say C ≰ B. There is an element γ ∈ B^{∘} \ C^{∘}, so x_{γ} ∈ 𝔭_{B}. On the other hand, there is an integer j such that p^{j}γ has exact order p modulo C^{∘} and thus [p^{j}](x_{γ}) divides u_{k} mod 𝔭_{C} by the argument of Proposition 6.9. As x_{γ} divides [p^{j}](x_{γ}), we deduce that u_{k} = 0 (mod 𝔭_{B} + 𝔭_{C}), and thus that 𝔭′_{B} + 𝔭′_{C} = R′(A). The Chinese remainder theorem now tells us that

R′(A)/\sqrt{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 𝒪_{Y} for all schemes Y under consideration, so the map x ↦ x^{p} is a ring endomorphism of 𝒪_{Y}, giving rise to a map F_{Y} : 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_{X}^{kd}: X ⟶X. The map F_{𝔾}^{k}: 𝔾 ⟶𝔾 covers F_{X}^{k} 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(f^{d}(a)) = x(a)^{p^{kd}} for any point a of 𝔾. If the formal group law arising from x is ∑_{ij} c_{ij}x_{0}^{i}x_{1}^{j}, then the formal group law arising

from y is ∑_{ij} c_{ij}^{p^{kd}}x_{0}^{i}x_{1}^{j}.

Because 𝔾 has strict height at least k, we see that p_{𝔾} factors through f, say p_{𝔾} = q ∘ f for some map q: 𝔾⟨1⟩ ⟶𝔾. The kernel of q is a subgroup divisor K < 𝔾⟨1⟩ of degree p^{n−k}. More explicitly, we have [p](x) = h(x^{p^{k}}) for some power series h over 𝒪_{X}, and q is given by x(q(b)) = h(y(b)) for any point b of 𝔾⟨1⟩. The subgroup K is just spf(𝒪_{X}[[y]]/h(y)).

Proposition 7.1. A homomorphism ϕ: A^{∗} ⟶ Γ(Y, 𝔾) is a p^{k}-fold level structure if and only 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 satisfy g^{∗}D ≤ g^{∗}D′. It is also easy to see that for any point a of ℍ, the divisor g^{∗}[g(a)] is the translate by a of the divisor ker(g). In the case g = f we have ker(f) = p^{k}[0], so the relevant translate is just p^{k}[a]. It follows that

f^{∗}[fϕA^{∗}(1)] = ∑_{α∈A^{∗}(1)} f^{∗}[fϕ(α)] = p^{k}[ϕA^{∗}(1)].

On the other hand, we have

f^{∗}K = f^{∗} ker(q) = ker(q ∘ f) = ker(p_{𝔾}) = 𝔾(1).

It follows that [fϕA^{∗}(1)] ≤ K if and only if p^{k}[ϕA^{∗}(1)] ≤ 𝔾(1), which means precisely that ϕ is a p^{k}-fold level structure.

We can pull the maps p = p_{𝔾}, q and f back along the map F_{X}^{ek}: 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⟩_{_{p} u}  ^{f} w _{_{p} u}  ^{f} w ȷ ȷ ȷ ȷ ȷ fl_{q}  _{_{p} u} ȷ ȷ ȷ ȷ ȷ fl_{q}_{_{p} u}  _{^{f} w} _{_{p} u}  _{^{f} w} ȷ ȷ ȷ ȷ ȷ fl_{q}  _{_{p} u} ȷ ȷ ȷ ȷ ȷ fl_{q}_{^{f} w}  _{^{f} w}

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

ker(f^{d}) ≤ ker(f^{d−1} ∘ p) ≤ … ≤ ker(f ∘ p^{d−1}) ≤ ker(p^{d}) = 𝔾(1).

On the other hand, ker(f^{d−i} ∘ p^{i}) is the pullback along p_{𝔾}^{i} of the divisor ker(f^{d−i}) = p^{(d−i)k}, which is just p^{(d−i)k}𝔾(i). Thus, the above filtration has the form

p^{dk}[0] ≤ p^{(d−1)k}𝔾(1) ≤ … ≤ p^{k}𝔾(d − 1) ≤ 𝔾(d).

We can thus define divisors

D_{i} = p^{(d−i)k}𝔾(i) − p^{(d−i+1)}𝔾(i − 1),

and we find that 𝔾(d) = ∑_{i=0}^{d} D_{i}.

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 x^{p^{k}}g(x) for some power series g of Weierstrass degree p^{n} − p^{k}. We write g_{0}(x) = x and g_{i}(x) = g([p^{i−1}](x)) for i > 0. We then have

[p^{d}](x) = ∏_{i=0}^{d} g_{i}(x)^{p^{k(d−i)}},

and the divisor D_{i} is just spf(𝒪_{X}[[x]]/g_{i}(x)^{p^{k(d−i)}}).

8. Thickenings

In this section we “thicken up” the subschemes Level_{k}(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 p^{k}-fold level-(A^{∗}, B^{∗}) structure if (I) If C^{∗} is a subgroup of ker(π) with p^{d}C^{∗} = 0 (such as B^{∘}(d)) then the composite

C^{∗} ⟶^{ϕ} 𝔾 ⟶^{f^{d}} 𝔾⟨d⟩

is zero. (II) If C^{∗} is a subgroup of A^{∗} with p^{d}C^{∗} = 0 and ker(π) ∩ C^{∗} = pC^{∗} then there is a p^{k}-fold

level structure ψ: πC^{∗} ⟶Γ(W, 𝔾⟨d−1⟩) such that the following diagram commutes.

C^{∗}  𝔾 πC^{∗}  𝔾⟨d − 1⟩^{ϕ}  w_{_{π} u u}  _{u_{f^{d−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 ↦ { p^{k}-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^{∗}, Λ_{k})|.

(2) Y (A, B) is an infinitesimal thickening of Level_{k}(B^{∗}, 𝔾).

(3) Hom′(A^{∗}, 𝔾) = ∐_{B} Y′(A, B).

(4) Y (A, A) = Level_{k}(A^{∗}, 𝔾).

(5) Y′(A, B) ∩ Hom(B^{∗}, 𝔾) = Level_{k}(B^{∗}, 𝔾).

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

deg[Level_{k}(A^{∗}, 𝔾)] = |A|^{k}| Mon(A^{∗}, Λ_{k})|.

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

Proposition 8.5. deg[Y (A, B)] ≤ |A|^{k}| Mon(B^{∗}, Λ_{k})|.

Proposition 8.6. There are closed subschemes Z(A, B) ⊆ Hom(A^{∗}, 𝔾) such that Z(A, B) is an infinitesimal thickening of Level_{k}(B^{∗}, 𝔾), and Hom′(A^{∗}, 𝔾) = ∐_{B} Z′(A, B). Moreover, u_{k} is not a zero-divisor on 𝒪_{Z(A,B)}. (Warning: unlike almost all other constructions we 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 = 𝒪_{X}
R(A) = 𝒪_{Hom(A^{∗},𝔾)}
D(B) = 𝒪_{Level_{k}(B^{∗},𝔾)}
D(A, B) = 𝒪_{Y (A,B)}
E(A, B) = 𝒪_{Z(A,B)}.

We now prove Theorem 8.3 assuming the above propositions.

Proof of Theorem 8.3. Write d_{B} = |A|^{k}| Mon(B^{∗}, Λ_{k})| and d = |A|^{n} = |A|^{k}| Hom(A^{∗}, Λ_{k})|. Note that the obvious decomposition Hom(A^{∗}, Λ_{k}) = ∐_{B} Mon(B^{∗}, Λ_{k}) gives d = ∑_{B} d_{B}, and recall from Proposition 4.10 that R(A) is a free module of rank d over D. Proposition 8.5 gives us an epimorphism t_{B}: D^{d_{B}} ↠ D(A, B) of D-modules, and putting these together gives an epimorphism t: D^{d} ↠ ∏_{B} D(A, B). The closed inclusions Z(A, B) ↣ Y (A, B) ↣ Hom(A^{∗}, 𝔾) give epimorphisms R(A) ⟶^{r_{A}} D(A, B) ⟶^{s_{A}} E(A, B) and thus maps R(A) ⟶^{r}∏_{B} D(A, B) ⟶^{s}∏_{B} E(A, B) with s surjective.

We now invert u_{k} everywhere, and denote this by adding a dash. We may assume that we are in the universal case, so that u_{k} 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 rank d over D′. This means that s′t′ is an epimorphism between free modules of the same rank d 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′ are isomorphisms. 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: D^{d} ↠ ∏_{B} D(A, B) is a u_{k}-torsion module, but u_{k} is regular on D^{d}, so t is injective. It is also surjective, so each t_{B} is an isomorphism, so D(A, B) is a free module over D of rank d_{B}. This proves (1). It also shows that u_{k} is regular on D(A, B); as s′_{B}: D(A, B) ⟶E(A, B) is an isomorphism, the same argument shows that s_{B} is an isomorphism, so Y (A, B) = Z(A, B). Given this, part (2) of the theorem 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 be proved 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 the degree of the restriction of r over the special fibre of Hom((A/B)^{∗}, 𝔾). The special fibre is contained 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 generators x_{1}, …, x_{d} giving a surjective map f: 𝒪_{Hom((A/B)^{∗},𝔾)}^{d} ⟶ 𝒪_{Hom(A^{∗},𝔾)}. We also know that 𝒪_{Hom((A/B)^{∗},𝔾)} and 𝒪_{Hom(A^{∗},𝔾)} are free over 𝒪_{X}, with ranks e = |A/B|^{n} and ed = |A|^{n} respectively. Thus, f is an epimorphism between free modules of the same finite rank over the Noetherian ring 𝒪_{X}, so it must be an isomorphism. Thus, r is a flat map of degree d = |B^{n}|, 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 u}_{ww}

Proof. Note that (A/pB)^{∗} is the preimage under π: A^{∗} ⟶ B^{∗} of B^{∗}(1). Thus, in conditions (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 p^{k}-fold level-(A^{∗}, B^{∗}) structure if and only if the restriction of ϕ to (A/pB)^{∗} is a p^{k}-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^{∗} = ⟨β_{1}, …, β_{s} | p^{d_{i}}β_{i} = 0⟩ ⊕ ⟨γ_{1}, …, γ_{t} | p^{e_{j}}γ_{j} = 0⟩,

where π(γ_{j}) = 0 for all j, and {π(β_{1}), …, π(β_{s})} is a basis for B^{∗}. We may also assume that 0 < d_{1} ≤ … ≤ d_{s}.

Proof. Choose a presentation

A^{∗} = ⟨α_{0}, …, α_{r−1} | p^{c_{i}}α_{i} = 0 for all i⟩,

where 0 < c_{0} ≤ … ≤ c_{r−1}. Write

S = {i | π(α_{i}) ∈ ⟨π(α_{j}) | j < i⟩}.

If i ∈ S, then we can choose an element α′_{i} which is congruent to α_{i} mod ⟨α_{j} | j < i⟩, such that π(α′_{i}) = 0. Because we have c_{j} ≤ c_{i} when j < i, we see that α′_{i} has order precisely p^{c_{i}}. We relabel {α_{i} | i ∉ S} as {β_{1}, …, β_{s}} and {α′_{i} | i ∈ S} as {γ_{1}, …, γ_{t}}. It is easy to see that these elements give a presentation of the required type.

We can now prove that deg[Y (A, B)] ≤ |A|^{k}| Mon(B^{∗}, Λ_{k})|.

Proof of Proposition 8.5. For the moment we assume that pB = 0, and we choose a presentation of A as in Lemma 8.11. Write A_{j}^{∗} = ⟨β_{i} | i ≤ j⟩ ≤ A^{∗}, and B_{j}^{∗} = π(A_{j}^{∗}) ≃ A_{j}^{∗}/p. We also write x_{i} = x(ϕ(β_{i})) ∈ 𝒪_{Y (A,B)} and y_{j} = x(ϕ(γ_{j})), and note that these generate 𝒪_{Y (A,B)} as an algebra over 𝒪_{X}. We let R_{i} be the subalgebra generated by {x_{1}, …, x_{i}}.

For each β ∈ B_{j}^{∗} we choose a lift β̃ ∈ A_{j}^{∗}, and we write

h_{j}(x) = ∏_{β∈B_{j}^{∗}}(x − x(ϕ(β̃))).

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

A_{j}^{∗}  B_{j}^{∗} 𝔾  𝔾⟨d_{j} − 1⟩^{π}  ww_{_{ϕ} u}  _{u_{ψ}} w_{f^{d_{j} −1}}

in which ψ is a p^{k}-fold level-B_{j}^{∗} structure. This means that (h_{j}(x)^{p^{k}})^{p^{k(d_{j} −1)}} = h_{j}(x)^{p^{kd_{j}}} divides [p](x)^{p^{k(d_{j} −1)}}. It follows that

k_{j}(x) = [p](x)^{p^{k(d_{j} −1)}}/h_{j−1}(x)^{p^{kd_{j}}}

is a power series of Weierstrass degree p^{kd_{j}}(p^{n−k} − p^{j−1}) over R_{j−1}, and that k_{j}(x_{j}) = 0. As R_{j} is generated by x_{j} over R_{j−1} we see that the degree of R_{s} over R_{0} = 𝒪_{X} is at most ∏_{j=1}^{s} p^{kd_{j}}(p^{n−k} − p^{j−1}).

Next, we apply condition (I) of Definition 8.1 to the subgroup spanned by γ_{j} to see that f_{e_{j}}ϕ(γ_{j}) = 0 and thus that y_{j}^{p^{ke_{j}}} = 0. It follows that the degree of 𝒪_{Y (A,B)} over 𝒪_{X} is at most

∏_{i=1}^{t} p^{ke_{i}} ∏_{j=1}^{s} p^{kd_{j}}(p^{n−k} − p^{j−1}).

A comparison with Lemma 6.12 shows that this is just |A|^{k}| Mon(B^{∗}, Λ_{k})|, 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)^{∗}, Λ_{k})|.

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)^{∗}, Λ_{k})|.

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), Λ_{k})| = |A|^{k}| Mon(B^{∗}, Λ_{k})|,

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 need to 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 p^{d}α = 0 and π(α) = 0; it will be enough to prove that a^{p^{kd}} = 0, where a = x(ϕ(α)). If we restrict to the closed subscheme Level_{k}(B^{∗}, 𝔾) then ϕ factors through π so a becomes zero, in other words a is in the kernel of the map E(A, B) ⟶ D(B). As Z(A, B) is an infinitesimal thickening of Level_{k}(B^{∗}, 𝔾), this means that a is nilpotent. Using the factorisation in Lemma 7.2, we also see that

∏_{i=0}^{d} g_{i}(a)^{p^{k(d−i)}} = 0.

Suppose that i > 0. In D(B) we have g_{i}(a) = g_{i}(0) = u_{k}, so g_{i}(a) becomes a unit in D′(B). As ker(E′(A, B) ↠ D′(B)) is nilpotent, it follows that g_{i}(a) is a unit in E′(A, B). Thus, only the i = 0 term in the above product is not a unit, so a^{p^{kd}} = 0 in E′(A, B). As E(A, B) has no u_{k}-torsion, we conclude that a^{p^{kd}} = 0 in E(A, B), as required.

We next prove condition (II). Let C^{∗} be a subgroup of A^{∗} such that p^{d}C^{∗} = 0 and ker(π) ∩ C^{∗} = pC^{∗}. Suppose that α ∈ C^{∗} \ pC^{∗}, and write a = x(ϕ(α)). As before, we have

∏_{i=0}^{d} g_{i}(a)^{p^{k(d−i)}} = 0.

Because π(α) has exact order p and ϕ becomes a p^{k}-fold level-B^{∗} structure over D(B), we see that x^{p^{k}}(x − a)^{p^{k}} divides [p](x) in D(B)[[x]] and thus that g(a) becomes zero in D(B). This means that g_{0}(a) = a divides the constant term g(0) = u_{k} in D(B), so that g_{0}(a) becomes a unit in D′(B). It also means that in D(B) we have [p](a) = 0 and thus g_{i}(a) = g([p^{i−1}](a)) = g(0) = u_{k} for i > 1, so that g_{i}(a) is also a unit in D′(B). Arguing as before, we see that g(a)^{p^{k(d−1)}} = 0 in E(A, B).

As g(a) divides [p](a), we conclude that [p](a)^{p^{k(d−1)}} = 0, which implies that the map

pC^{∗} ↣ C^{∗} ⟶^{ϕ} Γ(Z(A, B), 𝔾) ⟶^{f_{d−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⟩) such that f^{d−1}ϕ = ψπ. In the previous paragraph we saw that a was a unit in E′(A, B) and thus not a zero-divisor in E(A, B). It follows by Proposition 6.9 that ψ is a p^{k}-fold level-πC^{∗} structure.

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

Proof. It is clear that Level_{k}(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 𝒪_{V} be the image of 𝒪_{Hom(B^{∗},𝔾)} in 𝒪_{W′}, or equivalently the quotient of 𝒪_{W} by the ideal of elements annihilated by a power of u_{k}. Note that V is a closed subscheme of W with V′ = W′, so it is enough to show that V ≤ Level_{k}(B^{∗}, 𝔾). Over W, the map ϕ: A^{∗} ⟶ 𝔾 factors through a map ψ: B^{∗} ⟶ 𝔾; we need to show that this becomes a level structure when 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 p^{d} say. By the definition of Y (A, B) we see that (x−x(ϕ(α)))^{p^{kd}} divides [p](x)^{p^{k(d−1)}} over Y (A, B). Thus, over W we see that x(ψ(β)) = x(ϕ(α)) divides a power of u_{k}. 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 an admissible ring spectrum E, a finite group G, a finite G-complex Z, and a category 𝒜 whose objects are the Abelian p-subgroups of G. We define a new functor L from 𝒜 to schemes as follows. Each object A ∈ 𝒜 is sent to the scheme

M_{A} = ∐_{B≤A} Level_{k}(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 B^{g} ≤ A. The map M_{u}: M_{B} ⟶M_{A} is defined to send each component Level_{k}(C^{∗}, 𝔾) of M_{B} to the component Level_{k}((C^{g})^{∗}, 𝔾) of M_{A} by the obvious map induced by g. It is easy to see that this is well-defined.

Another point of view is to consider the category 𝒜′, whose objects are the same as the objects of 𝒜, and whose maps are the isomorphisms in 𝒜. The assignment A ↦ Level_{k}(A^{∗}, 𝔾) gives a functor from 𝒜′ to schemes, and M is the left Kan extension of this functor along the inclusion 𝒜′ ⟶𝒜.

Lemma 9.1. There is an isomorphism of formal schemes

∫ ^{A∈𝒜} (π_{0}(Z^{A}) × ∐_{B≤A} Level_{k}(B^{∗}, 𝔾)) = ( ∐_{A∈𝒜} π_{0}(Z^{A}) × Level_{k}(A^{∗}, 𝔾)) /G,

and similarly with Level_{k}(A^{∗}, 𝔾) replaced by Level′_{k}(A^{∗}, 𝔾).

Proof. Both statements are formalities: we prove the first statement for definiteness.

For brevity, we write P_{A} = π_{0}(Z^{A}) and L_{A} = Level_{k}(A^{∗}, 𝔾). By definition, (∐_{A} P_{A} × L_{A})/G is the initial example of a scheme Y with maps k_{A}: P_{A} × L_{A} ⟶Y such that

k_{A^{g}} = k_{A} ∘ (P_{g} × L_{g^{−1}}): P_{A^{g}} × L_{A^{g}} ⟶Y.

Thus, to construct a map k′: (∐ P_{A} × L_{A})/G ⟶∫^{A} P_{A} × M_{A}, we need to construct maps k′_{A}: P_{A} × L_{A} ⟶∫^{A} P_{A} × M_{A} with k′_{A^{g}} = k′_{A} ∘ (P_{g} × L_{g^{−1}}).

Similarly, the coend ∫^{A} P_{A}×L_{A} is the initial example of a scheme Y equipped with maps j_{A}: P_{A} × M_{A} ⟶ Y such that for each morphism u: B ⟶ A in 𝒜, the following diagram commutes.

P_{A} × M_{B}  P_{A} × M_{A} P_{B} × M_{B}  Y^{1×M_{u}} w_{_{P_{u}×1} u}  _{u_{j_{A}}} w_{j_{B}}

Each such morphism u: B ⟶ A is given by an element g ∈ G such that B^{g} ≤ 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} P_{A} × M_{A} is the initial example of a scheme Y with maps j_{A,B}: P_{A}×L_{B} ⟶Y whenever B ≤ A, such that whenever C^{g} ≤ B^{g} ≤ A we have a commutative diagram as follows.

P_{A} × L_{C}  P_{A} × L_{C^{g}} P_{B} × L_{C}  Y^{1×L_{g}} w_{_{P_{u}×1} u}  _{u_{j_{A,C^{g}}}} w_{j_{B,C}}

Thus, to construct a map j′: ∫^{A} P_{A} × M_{A} ⟶(∐ P_{A} × L_{A})/G, we need to construct maps j′_{A,B}: P_{A} × L_{B} ⟶(∐ P_{A} × L_{A})/G making diagrams like the above commute.

We define

k′_{A} = j_{A,A}: P_{A} × L_{A} ⟶∫ ^{A} P_{A} × M_{A}

and

j′_{A,B} = (P_{A} × L_{B} ⟶^{P_{u}×1} P_{B} × L_{B} ⟶^{k_{B}} (∐_{A} P_{A} × L_{A})/G).

(Here P_{u} is the evident morphism P_{A} ⟶P_{B} induced by the inclusion B ⟶A.) We leave it to the reader to check that these maps have the necessary compatibilities to induce maps

(∐_{A} P_{A} × L_{A})/G ⟶^{k′} ∫ ^{A} P_{A} × M_{A} ⟶^{j′} (∐_{A} P_{A} × L_{A})/G,

and that these maps are mutually inverse.

Theorem 9.2. There is a natural map of formal schemes

(∐_{A} π_{0}(Z^{A}) × Level_{k}(A^{∗}, 𝔾)) /G ⟶X_{k}(Z, G),

which induces a V -isomorphism

(∐_{A} π_{0}(Z^{A}) × Level′_{k}(A^{∗}, 𝔾)) /G ⟶X′_{k}(Z, G).

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

Level_{k}(B^{∗}, 𝔾) ⟶Hom_{k}(B^{∗}, 𝔾) ⟶Hom_{k}(A^{∗}, 𝔾).

These can be assembled to give a map

M_{A} = ∐_{B≤A} Level_{k}(B^{∗}, 𝔾) ⟶Hom_{k}(A^{∗}, 𝔾),

and one can check that this is natural for maps A ⟶A′ in 𝒜. We therefore get an induced map of coends

f: (∐_{A} π_{0}(Z^{A}) × Level_{k}(A^{∗}, 𝔾)) /G = ∫ ^{A} π_{0}(Z^{A}) × M_{A} ⟶∫ ^{A} π_{0}(Z^{A}) × Hom_{k}(A^{∗}, 𝔾).

We compose this with the map

g: ∫ ^{A} π_{0}(Z^{A}) × Hom_{k}(A^{∗}, 𝔾) ⟶X_{k}(Z, G)

provided by Theorem 2.4 to get the map claimed in the Proposition.

General nonsense provides a map

(∐_{A} π_{0}(Z^{A}) × Level′_{k}(A^{∗}, 𝔾)) /G ⟶X′ ×_{X} (∐_{A} π_{0}(Z^{A}) × Level_{k}(A^{∗}, 𝔾)) /G,

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

f′: (∐_{A} π_{0}(Z^{A}) × Level′_{k}(A^{∗}, 𝔾)) /G ⟶∫ ^{A} π_{0}(Z^{A}) × Hom′_{k}(A^{∗}, 𝔾).

We know by Theorem 6.4 that the maps X′_{k} ×_{X} M_{A} ⟶Hom′_{k}(A^{∗}, 𝔾) are V -isomorphisms, and it follows using Proposition 3.13 that f′ is a V -isomorphism. We also know from Theorem 2.4 that g′ = 1_{X′_{k}} ×_{X} g is a V -isomorphism, so g′f′ is a V -isomorphism, as required.

Remark 9.3. One way to think about this result is that it guarantees that the intersection of any two irreducible components of X_{k}(Z, G) lies over an infinitesimal thickening of X_{k+1}. There is another way that one might hope to prove this, at least in the case where E is Morava E-theory. As the components are distinct and have Krull dimension n − k, the image of the intersection in X will have dimension strictly less than n − k. Because of the way that the components arise from Abelian subgroups, this image will also be invariant under the action of the Morava stabiliser group. It is widely believed that the only prime ideals of E^{0} that are invariant under this action are the ideals I_{k}, and that this is a simple consequence of the classification of invariant primes in MU_{∗}. We suspect that the first of these beliefs is true, but the second seems to be false. We also suspect that the I_{k} are the only invariant radical ideals, or equivalently that they are the only primes that are invariant 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 X_{j} 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 π_{0}(Z^{A}). The pair (A, z) defines a point of the finite G-set ∐_{A} π_{0}(Z^{A}). We write N_{G}(A, z) for its stabiliser and [A, z] for its orbit, and put W_{G}(A, z) = N_{G}(A, z)/A.

There is a closed subscheme Ŷ′_{k}(Z, G, A, z) ≤ X̂′_{k}(Z, G) depending only on the orbit of (A, z), such that

X̂′_{k}(Z, G) = ∐_{[A,z]} Ŷ′_{k}(Z, G, A, z)

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

Level′_{k}(A^{∗}, 𝔾)/W_{G}(A, z) ⟶X_{k} ×_{X} Ŷ′_{k}(Z, G, A, z),

which is an isomorphism on the underlying varieties.

Proof. It is not hard to see that

(∐_{A} π_{0}(Z^{A}) × Level′_{k}(A^{∗}, 𝔾)) /G = ( ∐_{(A,z)} Level′_{k}(A^{∗}, 𝔾)) /G

= ∐_{[A,z]} Level′_{k}(A^{∗}, 𝔾)/W_{G}(A, z).

This scheme maps by a V -isomorphism to X′_{k}(Z, G) = spec(u_{k}^{−1}E_{G}^{0}(Z)/I_{k}), which can be infinitesimally thickened to get the scheme spec(u_{k}^{−1}E_{G}^{0}(Z)/I_{k}^{m}) for any m > 0. Applying Proposition 3.15, we get an induced splitting of the ring u_{k}^{−1}E_{G}^{0}(Z)/I_{k}^{m} into pieces indexed by the orbits in ∐_{A} π_{0}(Z^{A}). 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 Ŷ′_{k}(A, B) for Ŷ′_{k}(∗, A, B, ∗).

Theorem 9.5. If A is an Abelian p-group and B ≤ A then 𝒪_{Ŷ′(A,B)} is a free module of rank |A|^{k}| Mon(B^{∗}, Λ_{k})| over 𝒪X̂′_{k}.

Proof. The decomposition X̂′_{k}(A) = ∐_{B} Ŷ′_{k}(A, B) gives a decomposition

Hom′_{k}(A^{∗}, 𝔾) = X′_{k}(A) = X̂′_{k}(A) ×_{X} X_{k} = ∐_{B} Ŷ′_{k}(A, B) ×_{X} X_{k}.

It is not hard to see that this must coincide with the splitting in Theorem 8.3, so that 𝒪_{Ŷ′_{k}(A,B)}/I_{k} = 𝒪_{Y′(A,B)}. We also know that 𝒪_{Y′(A,B)} can be generated over u_{k}^{−1}E^{0}/I_{k} = 𝒪X̂′_{k}/I_{k} by d_{B} elements, where d_{B} = |A|^{k}| Mon(B^{∗}, Λ_{k})|. It follows easily that we can choose an epimorphism t_{B}: 𝒪d_{B} ; X̂′_{k} ↠ 𝒪_{Ŷ′_{k}(A,B)}. The direct sum of all these is an epimorphism

𝒪d ; X̂′_{k} ↠ 𝒪_{X̂′_{k}(A)} where d = ∑_{B} d_{B} = |A|^{n}. In other words, it is an epimorphism between free modules of the same finite rank over the Noetherian ring 𝒪X̂′_{k}, hence an isomorphism. This implies that t_{B} 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 equivariant means. We have been much concerned with the cohomology theory E_{G}^{∗}(Z) = E^{∗}(EG×_{G}Z), and now we consider the calculation of E_{∗}^{G}(EG×Z) = E_{∗}(EG×_{G}Z) for finite G-complexes Z, particularly if Z = ∗ when we obtain E_{∗}^{G}(EG) = E_{∗}(BG). It is necessary to remind the reader that E_{∗}^{G}(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 E_{G}^{0}, so J = ker(E_{G}^{0} ↠ E^{0}). We will apply Theorem 2.7 to the study of the local cohomology H_{J}^{∗}(E_{G}^{0}(Z)) and the local Tate cohomology Ĥ_{J}^{∗}(E_{G}^{0}(Z)), which is the same as the Čech cohomology Ȟ_{J}^{∗}(E_{G}^{0}(Z)) because E_{G}^{0}(Z) is complete at J. In good cases, these will be relevant to the calculation of E_{∗}^{G}(EG × Z) = E_{∗}(EG ×_{G} Z) and the E-Tate cohomology groups t(E)_{G}^{∗}(Z) of [8]; indeed, we expect to have spectral sequences

E_{2}^{∗∗} = H_{J}^{∗}(E_{G}^{∗}(Z)) ⟹ E_{∗}^{G}(EG × Z) = E_{∗}(EG ×_{G} Z)

and

E_{2}^{∗∗} = Ȟ_{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 E_{G}^{0} is discussed in detail in [11].

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

Recall that for a finitely generated ideal J = (α_{1}, …, α_{d}) of E_{G}^{0} we may define the flat stable Koszul complex

K^{•}(J) = (A ⟶A[1/α_{1}]) ⊗ ⋯ ⊗ (A ⟶A[1/α_{d}]);

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

H_{J}^{∗}(M) = H^{∗}(K^{•}(J) ⊗ M).

Note that

H_{J}^{0}(M) = Γ_{J}(M) := {x ∈ M | J^{N}x = 0 for N sufficiently large};

geometrically, Γ_{J}(M) is the module of sections of the sheaf M̃ with support in V (J). Since E_{G}^{0} is a Noetherian ring, a result of Grothendieck states that local cohomology calculates the right derived functors of the left exact functor Γ_{J}. Note that in general the local cohomology of a ring depends on the nilpotents, so that it is not a functor of the variety var(R). However the J-local cohomological dimension only depends on the variety, and in particular H_{J}^{i}(R) = 0 for i > 0 if and only if J is nilpotent, or equivalently var(R) = V (J) = var(R/J).

For any module M over E_{G}^{0} we may form the Cousin complex corresponding to the chromatic filtration

C^{•}(M) = (p^{−1}M ⟶u_{1}^{−1}M/p^{∞} ⟶u_{2}^{−1}M/p^{∞}, u_{1}^{∞} ⟶⋯ ⟶u_{n−1}^{−1}M/I_{n−1}^{∞} ⟶M/I_{n}^{∞}) ,

where the terms are defined recursively by the exact sequences

M/I_{k}^{∞} ↣ u_{k}^{−1}M/I_{k}^{∞} ↠ M/I_{k+1}^{∞}.

If p, u_{1}, …, u_{n−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 over E^{∗} concentrated in even degrees, there is a natural isomorphism

H_{J}^{∗}(E_{G}^{0}(Z)) = H^{∗}Γ_{J}C^{•}(E_{G}^{0}(Z)).

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

Lemma 10.2. Let M be a module over E_{G}^{0}. If p, u_{1}, …, u_{n−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, u_{1}, …, u_{n−1} is a regular sequence for M then if we take Cousin cohomology of T^{•,•} we obtain M ⊗ K^{•}(J) whose Koszul cohomology is H_{J}^{∗}(M); this is therefore the cohomology of the total complex. The spectral sequence is the one for calculating the cohomology of the total complex obtained by taking Koszul homology first.

For any module M over E_{G}^{0}, it is easy to see that C^{k}(M) = u_{k}^{−1}M/I_{k}^{∞} admits a unique structure as a module over (u_{k}^{−1}E_{G}^{0}/I_{k})_{I_{k}}^{∧} = 𝒪_{X̂′_{k}(G)} = ∏_{(A)} 𝒪_{Ŷ′_{k}(G,A)} extending its structure as an E_{G}^{0}-module. It follows that C^{k}(M) has a canonical splitting as a direct sum of pieces C_{A}^{k}(M), where C_{A}^{k}(M) is a module over 𝒪_{Ŷ′_{k}(G,A)}. In particular, we have a piece C_{1}^{k}(M) corresponding to the trivial subgroup A = 1.

Proposition 10.3. For any module M over E_{G}^{0} we have

Γ_{J}C^{k}(M) = H_{J}^{0}C^{k}(M) = C_{1}^{k}(M)

and H_{J}^{i}(C^{k}(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)/W_{G}(A) ⟶ Y′_{k}(G, A). In particular, we have V′_{k} ⊆ Y′_{k}(G, 1).

The splitting gives an idempotent e ∈ u_{k}^{−1}E_{G}^{0}/I_{k} such that e ↦ 0 in 𝒪_{Y′_{k}(G,1)} and e ↦ 1 in 𝒪_{Y′_{k}(G,A)} for all A ≠ 1. For large enough N we have a = u_{k}^{N}e ∈ E_{G}^{0}/I_{k}. As V′_{k} ⊆ Y′_{k}(G, 1), we see that a becomes zero in u_{k}^{−1}E^{0}/I_{k}, but E^{0}/I_{k} has no u_{k}-torsion so a becomes zero in E^{0}/I_{k}. We can thus choose a lift b ∈ E_{G}^{0} of a such that b becomes zero in E^{0}, in other words b ∈ J. When A ≠ 1, it is clear that b becomes a unit in 𝒪_{Y′_{k}(G,A)} and thus also in 𝒪_{Ŷ′_{k}(G,A)}.

Now let M be an E_{G}^{0}-module. If A ≠ 1 then b acts as an isomorphism on C_{A}^{k}(M) and b ∈ J; by standard properties of local cohomology, we see that H_{J}^{∗}(C_{A}^{k}(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 𝒪_{Y′_{k}(G,A)} = 𝒪_{Ŷ′_{k}(G,A)}/I_{k} is nilpotent. It follows that for each m ≥ 1 the image of J in 𝒪_{Ŷ′_{k}(G,A)}/I_{k}^{m} is nilpotent. As every element of C_{1}^{k}(M) is annihilated by I_{k}^{m} for some m, we see that it is also annihilated by a power of J. The definition of local cohomology now tell us that H_{J}^{0}(C_{1}^{k}(M)) = C_{1}^{k}(M), and since Γ_{J} preserves injectives, H_{J}^{i}(C_{1}^{k}(M)) = 0 for i > 0.

The geometric content of the proof is that the components Ŷ′_{k}(G, A) do not meet V (J) unless A = 1 and that Ŷ′_{k}(G, 1) is essentially a colimit of infinitesimal thickenings of V (J).

We may argue rather similarly for Čech cohomology, and obtain a chromatic route to calculating it, and hence to calculating the Tate cohomology t(E)_{G}^{∗}(Z) via the local Tate spectral sequence. Note that there is a fibre sequence K^{•}(J) ⟶A ⟶Č^{•}(J), where Č^{•}(J) is the usual flat complex for calculating Čech cohomology; therefore we may argue exactly as in 10.2 to obtain a spectral sequence for calculating the Čech cohomology. Furthermore in case H_{J}^{1}(M) = 0 we find Ȟ_{J}^{0}(M) = M/Γ_{J}M, so that by 10.3 we obtain the following in which Ψ_{J}(M) := M/Γ_{J}M.

Corollary 10.4. Provided that p, u_{1}, …, u_{n−1} is a regular sequence for E_{G}^{0}(Z), the Čech cohomology groups Ȟ_{J}^{∗}(E_{G}^{0}(Z)) are the cohomology groups of the complex Ψ_{J}C^{•}(E_{G}^{0}(Z)).

11. Some examples.

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

Γ_{J}(p^{−1}R(G)_{p}^{∧}) ⟶Γ_{J}(R(G)_{p}^{∧}/p^{∞})

By character theory we see that J-power torsion in the zero’th term consists of multiples of the regular representation, and if G is a p-group J^{n} ⊆ (p) for some n, so all of the first term 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] H_{J}^{0}(R(G)) = ℤ and H_{J}^{1}(R(G)) = (R(G)/ℤ) ⊗ ℤ/p^{∞}. Geometrically, there are components Y_{0}(1), and Y_{0}(A) when A is a cyclic p-subgroup. The statement that the local cohomology of p^{−1}R(G)_{p}^{∧} is concentrated in degree 0, corresponds to the fact that Y′_{0}(1) does not meet Y′_{0}(A) if A ≠ 1, or that Y_{0}(1) only meets other components over the point (p) ∈ Spec(ℤ_{p}^{∧}). The statement that there is a single component Y_{1}(1) mod p suggests that the components Y_{0}(A) meet Y_{0}(1) ‘tangentially’ in an infinitesimal neighbourhood of p, and the rank of the answer suggests that Y_{0}(A) contributes with multiplicity equal to the number of generators of the group A. In this case one may make a more precise integral analysis using Segal’s description of Spec(R(G)).

The results of [11] concern the admissible theory

E_{n,i} = E_{n}/(p, u_{1}, …, u_{i−1}, u_{i+1}, …, u_{n−1}),

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

Γ_{J}(v_{i}^{−1}E_{n,i}^{0}(BG)) ⟶Γ_{J}(E_{n,i}^{0}(BG)/v_{i}^{∞}).

If i = n−1 these terms correspond to the nonzero terms of the complex of Γ_{J}C^{•}(E_{n,i}^{0}(BG)) in degrees n − 1 and n. If G is elementary Abelian of order p^{r}, the (n − 1)’st term is a free v_{i}^{−1}E^{0}-module of rank p^{ir}, and the nth term is a direct sum of p^{nr} copies of E^{0}/v_{i}^{∞}. This again suggests that X_{0}(G) has components Y_{0}(1) and Y_{0}(A) for A a cyclic p-subgroup, and that Y_{0}(1) only meets the other components over (v_{i}). The difference is that it now has multiplicity p^{ir}. Again, in an infinitesimal neighbourhood of (v_{i}), there is a single component Y_{i}(1), but now with multiplicity p^{nr}. This suggests that the intersection of Y_{0}(A) with Y_{0}(1) is ‘tangential’, but now the multiplicity contributed by each group element of order p is (p^{nr} − p^{ir})/(p^{n} − 1).

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

u_{k}^{−1}E^{0}(BA)/I_{k}^{m} ≅ ∏_{B} D′_{k}(A, B, m).

Furthermore, D′_{k}(A, B, m) is a free u_{k}^{−1}E^{0}/I_{k}^{m}-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 ‘chromatic complex with multiplicities’

|A|^{0}[p^{−1}E^{∗}] ⟶|A|^{1}[u_{1}^{−1}E^{∗}/p^{∞}] ⟶|A|^{2}[u_{2}^{−1}E^{∗}/p^{∞}, u_{1}^{∞}] ⟶⋯

⋯ ⟶|A|^{n−1}[u_{n−1}^{−1}M/I_{n−1}^{∞}] ⟶|A|^{n}[E^{∗}/I_{n}^{∞}].

This then suggests the general picture that each Γ_{J}(u_{k}^{−1}E^{0}(BG)/I_{k}^{∞}) is a direct sum of l_{k}(G) copies of u_{k}^{−1}E^{0}/I_{k}^{∞}. However, the differential

Γ_{J}(u_{k}^{−1}E^{0}(BG)/I_{k}^{∞}) ⟶Γ_{J}(u_{k+1}^{−1}E^{0}(BG)/I_{k+1}^{∞})

is not given by just projecting onto l_{k+1}(G) factors and then composing with the natural map

u_{k}^{−1}E^{0}/I_{k}^{∞} ⟶u_{k+1}^{−1}E^{0}/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 E^{0} in degree 0, and a direct sum of p^{kr} − p^{(k−1)r} copies of E^{0}/I_{k}^{∞} in degree k ≥ 1. This is non-zero for 0 ≤ k ≤ n. However, the local cohomology vanishes above dimension r if J can be generated as a radical ideal by r elements (for example if G is Abelian of rank r); this gives a 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 of Hopkins-Kuhn-Ravenel; we record it partly because of notational differences, and partly because [14] is not easily accessible.

Proof of Theorem 2.6. First we construct a natural map

ν: p^{−1}E_{G}^{∗}(Z) ⟶(∏_{A} p^{−1}E^{∗}(Z^{A}) ⊗_{E^{0}} D(A))^{G} .

Indeed, it suffices to construct the components

ν_{A} : E_{G}^{∗}(Z) ⟶E^{∗}(Z^{A}) ⊗_{E^{0}} D(A)

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

E_{G}^{∗}(Z) ⟶E_{A}^{∗}(Z) ⟶E_{A}^{∗}(Z^{A}) = E^{∗}(Z^{A} × BA) = E^{∗}(Z^{A}) ⊗_{E^{0}} E^{0}(BA)

together with the natural map E^{0}(BA) ⟶D(A). It remains to show that ν is an isomorphism for all finite G-complexes Z.

The domain is evidently a cohomology theory, and the codomain is too, because D(A) is flat and the group order is inverted before we take invariants. Thus, if ν is an isomorphism for Z = G/B whenever B is an Abelian subgroup, it is an isomorphism whenever Z has finite Abelian isotropy groups. The general case follows since both sides give equaliser diagrams 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 diagram Z^{A}×T^{A}×T^{A} ⇉ Z^{A}×T^{A} ⟶Z^{A} we obtain an equaliser. Indeed, the subsequent operations preserve equalisers: D(A) is flat, localisation and products are exact, and passage to Ginvariants is exact since the group order is invertible. However T^{A} is the space of Ainvariant complete flags in V, and this is a disjoint union of products of the flag spaces of the A-isotypical parts of V. The result therefore again follows from the nonequivariant instance of Corollary 5.3.

It remains to show ν is an isomorphism if Z = G/B, which we do by direct calculation and the theory of level structures. A simple argument reduces us to the case where B is a p-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^{−1}D(C)) = p^{−1} ∏_{C≤B} D(C).

By Theorem 6.4, the natural map

p^{−1}E_{B} = p^{−1}𝒪_{Hom(B^{∗},𝔾)} ⟶p^{−1} ∏_{C≤B} D(C)

is surjective (with nilpotent kernel). Both sides are free modules over p^{−1}E. The domain has rank | Hom(B^{∗}, Λ_{0})|, and the codomain has rank ∑_{C≤B} | Mon(C^{∗}, Λ_{0})|, which is the same. An epimorphism of free modules of the same rank over a Noetherian ring is an isomorphism, so p^{−1}E_{B} = p^{−1} ∏_{C≤B} D(C), as required.

Appendix B. The Evens norm map

In this section we outline an alternative approach to Theorem 2.5, parallel to that of Evens [4] for the case of ordinary cohomology. This has the advantage that it constructs some useful elements of E_{G}^{0} explicitly. However, we need to assume that E has the structure of an H_{∞} ring spectrum [1]. There are essentially only two admissible theories for which there is a published construction of an H_{∞} structure (rational cohomology and p-adic Ktheory). 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 the foundations 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 Paul Goerss and Mike Hopkins.

For this appendix we assume that E has an H_{∞} structure. For simplicity, we also assume that k > 0. The H_{∞} structure allows us to construct an Evens norm map E_{H}^{0} ⟶ E_{G}^{0} 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 = {x_{1}H, …, x_{d}H} gives rise to a homomorphism G ⟶ Σ_{d} ≀ H, which is canonical up to conjugacy. If u ∈ E_{H}^{0} = [BH_{+}, E] then we define

norm_{H}^{G}(u) = (BG_{+} ⟶B(Σ_{d} ≀ H) = D_{d}(BH_{+}) ⟶^{D_{d}(u)} D_{d}E ⟶^{θ_{d}} E),

where D_{d} is the d’th extended power construction, and θ_{d} 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 under conjugation.

2. It is multiplicative: norm_{H}^{G}(uv) = norm_{H}^{G}(u) norm_{H}^{G}(v).

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

res_{K}^{G} norm_{H}^{G}(u) = ∏_{KgH∈K\G/H} norm_{K∩^{g}H}^{K} res_{K∩^{g}H}^{^{g}H}c_{g}(u).

Definition B.1. For any Abelian p-subgroup A of G we write χ(A) ∈ E_{A}^{0} for the Euler 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 by ϕ ⟼ ∏_{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 |W_{G}A| = p^{a}b where p does not divide b, and we note that g(x) = (1 + x)^{1/b} − 1 is a power series over ℤ_{p}. We define

z_{A} = g(norm_{A}^{G}(1 + χ(A)) − 1).

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

res_{B}^{G}(z_{A}) = \begin{cases}0 & if B contains no conjugate of A \\ χ(A)^{p^{a}} & if B = A. \end{cases}

If A and B are non-conjugate Abelian p-subgroups, with |B| ≤ |A| say, then B cannot contain a conjugate of A. Thus, z_{A} is zero over the image of Level′_{k}(B^{∗}, 𝔾) in X′_{k}(G), and invertible over the image of Level′_{k}(A^{∗}, 𝔾), so these images are disjoint.

Consider the natural map

f: u_{k}^{−1}E_{G}^{0}/I_{k} = 𝒪_{X′_{k}(G)} ⟶D′_{k}(A)^{W_{G}A} = 𝒪_{Level′_{k}(A^{∗},𝔾)/W_{G}A}.

We know that χ(A)^{p^{a}} lies in the image of f and that it is invertible. The inverse is integral over the image of f (because everything is finite over X′_{k}). After multiplying a monic equation of integral dependence by a suitable power of χ(A)^{p^{a}}, we find that χ(A)^{−p^{a}} also lies in the image of f, say χ(A)^{−p^{a}} = f(ξ(A)). Now suppose that y ∈ D_{k}(A)^{W_{G}A}, and choose an element ỹ ∈ E_{A}^{0} lifting y. One can check that

f(g(norm_{A}^{G}(1 + χ(A)ỹ))ξ(A)) = y^{p^{a}}.

Thus, f is F -surjective in the evident sense. A little more work in this direction recovers Theorem 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_{G}^{0} = E_{G}^{0}(Z).

Appendix C. Varieties and reduction mod I_{k}.

In this appendix we give the elementary argument that E^{∗}(BG)/I_{k} and (E/I_{k})^{∗}(BG) have isomorphic varieties, assuming that Bockstein spectral sequences with appropriate multiplicative 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)/I_{k} ⟶(E/I_{k})^{∗}(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 to X = BG.

By a Bockstein spectral sequence argument given in [11] or by 5.4 above, since E is complete, \overline{E}^{∗}(X) is finitely generated over \overline{E}^{∗} for \overline{E} = E/I_{k} for any k ≤ n. We argue by induction on k, supposing that we have constructed an F-isomorphism E^{∗}(X)/I_{k} ⟶ (E/I_{k})^{∗}(X), using the identity if k = 0. Now let \overline{E} = E/I_{k} and v = u_{k}, and consider the cofibre sequence \overline{E} ⟶^{v} \overline{E} ⟶\overline{E}/v; this gives a ring map \overline{E}^{∗}(X)/v ⟶(\overline{E}/v)^{∗}(X). Combining this with the F-isomorphism already constructed, we obtain

E^{∗}(X)/I_{k+1} = (E^{∗}(X)/I_{k})/v ⟶{E/I_{k}}^{∗}(X)/v ⟶{E/(I_{k} + (v))}^{∗}(X) = {E/I_{k+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 \overline{θ}: A/a ⟶B/θ(a) is also an F-isomorphism.

It therefore suffices to deal with the principal case.

Lemma C.3. If \overline{E}^{∗}(X) is finitely generated over the characteristic p ring \overline{E}^{∗} then for any v the map \overline{E}^{∗}(X)/v ⟶(\overline{E}/v)^{∗}(X) is an F-isomorphism.

Proof. Of course the map is injective by construction, so it suffices to show that there is an integer t such that for any y ∈ (\overline{E}/v)^{∗}(X) the power y^{p^{t}} is the reduction of some class x ∈ \overline{E}^{∗}(X). This is a job for the Bockstein spectral sequence arising from the diagram

\overline{E}  \overline{E}  \overline{E}  \overline{E}  ⋯ \overline{E}/v  \overline{E}/v  \overline{E}/v^{v} w  ^{v} w_{u_{ρ}}  ^{v} w_{u_{ρ}}  ^{v} w_{u_{ρ}} ^{[ [ [[^}_{β}  ^{[ [ [[^}_{β}  ^{[ [ [[^}_{β}

by applying [X, ·]^{∗}. Here β is the connecting homomorphism (\overline{E}/v)^{∗}(X) ⟶ \overline{E}^{∗}(Σ^{−1}X). Since \overline{E}^{∗}(X) is Noetherian it has bounded v-torsion in the sense that there is an integer t so that if v^{t+1}y = 0 then also v^{t}y = 0. Now by definition, if d_{t}z = 0 we have βz = v^{t}z_{t} for some z_{t} ∈ \overline{E}^{∗}(X), and since 0 = vβz = v^{t+1}z_{t} it follows that 0 = v^{t}z_{t} = βz, and hence z is the reduction of an element of \overline{E}^{∗}(X). It thus suffices to show that any p^{t}th power y^{p^{t}} ∈ (\overline{E}/v)^{∗}(X) is a t-cycle. But the spectral sequence is multiplicative by assumption, and so for any s-cycle z, d_{s+1}z^{p} = 0, thus y^{p^{t}} is a t-cycle by induction.

If we assume E is an S-algebra in the sense of [3] then the tower can be constructed in 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 \overline{E}/v is an E-algebra up to homotopy, and that the diagram

\overline{E}/v ∧_{E} \overline{E}/v  \overline{E}/v Σ\overline{E} ∧_{E} \overline{E}/v ∨ \overline{E}/v ∧_{E} Σ\overline{E} Σ\overline{E} Σ\overline{E}/v ∨ Σ\overline{E}/v  Σ\overline{E}/v_{_{( \begin{matrix}β∧1 \\ 1∧β \end{matrix} )} u}  ^{μ} w _{u_{β}}_{_{≃} u}  _{u_{Σρ}} w_{∇}

commutes. This follows by elementary obstruction theory as in Chapter V of [3] provided E_{∗} is concentrated in degrees divisible by 4, and I_{k} is generated by a regular sequence (and also under slightly weaker hypotheses). Indeed, \overline{E} is the E-smash product of the spectra E/u for the elements u in the regular sequence, so it suffices to deal with the case \overline{E} = E. Here it is easy to be explicit since E/v ∧_{E} E/v = E/v ∨ ΣE/v, and the product μ is the identity on the first summand and zero on the second. The clockwise composite is zero on the second factor, and E/v ⟶^{β} ΣE ⟶ ΣE/v on the first. Consider the anticlockwise contribution from the second factor: the components into each of the factors at the bottom left are ±1 and ∓1, and thus cancel. On the first factor, one of the terms is zero and the other is E/v ⟶^{β} ΣE ⟶ΣE/v, depending on the isomorphisms used.

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

  E-mail address: j.greenlees@sheffield.ac.uk

Trinity College, Cambridge CB2 1TQ, UK.

  E-mail address: n.strickland@pmms.cam.ac.uk