LOCAL COHOMOLOGY IN EQUIVARIANT TOPOLOGY

J.P.C.GREENLEES

Abstract. The article describes the role of local homology and cohomology in understanding the equivariant cohomology and homology of universal spaces. This brings to light an interesting duality property related to the Gorenstein condition. The phenomena are studied and illustrated in several rather different families of examples. Both topology and commutative algebra benefit from the connection, and many interesting questions remain open.

Contents

1. Overview 2

Part I. Equivariant K-theory. 2 2. The complex representation ring 2 3. Free G-spaces. 4 4. The Atiyah-Segal completion theorem and its dual. 5 5. Duality for the representation ring. 7 6. A proof for p-groups. 8 7. Variations and extensions. 11

Part II. Ordinary cohomology and graded connected k-algebras. 13 8. Cohomology of some classes of groups. 13 8.1. Finite groups. 13 8.2. Compact Lie groups. 15 8.3. Arithmetic groups. 16 8.4. p-adic Lie groups. 16 9. The local cohomology theorem for group cohomology. 17 10. An algebraic proof of the local cohomology theorem for ordinary cohomology. 18 11. Structural implications of the local cohomology theorem. 20

Part III. Examples related to bordism. 22 12. The theorem for MU-modules. 22 13. The proof. 23 13.1. G-spectra. 23 13.2. Commutative algebra with G-spectra. 24 13.3. The map A is an equivalence. 25 13.4. The map B is an equivalence. 25 14. Homotopically Gorenstein rings. 26 15. The chromatic case. 27 16. Connective K theory. 27 References 29

1. Overview

The usefulness of local cohomology in equivariant topology is not just a superficial phenomenon. It arises because similar structures occur in both contexts. The aim of these lectures is to explain one particular connection that I am especially familiar with, showing the common structures in the process. This connection is useful in both directions. I hope to expose some interesting algebraic structures and recommend them for further study, and to display some topological phenomena that may be amenable to methods of commutative algebra.

The main ingredients on the topological side are a group G and a G-equivariant cohomology theory E_{G}^{∗}(·). On the commutative algebra side we begin with an augmented k-algebra S, and the augmentation ideal J = ker(S ⟶ k); in the topological examples S = E_{G}^{∗} is the equivariant coefficient ring and k = E^{∗} is the non-equivariant coefficient ring. Aside from the common structures, the reason for studying these phenomena is that the rings S have an interesting duality property generalizing the notion of Gorenstein rings.

The paper is organized around families of examples of rings S arising from equivariant cohomology theories. We start in Part I with a rather simple instance (K theory and the complex representation ring), where quite complete results are available in very concrete terms. This should give some life to the ideas. We then summarize the other examples we want to discuss, before considering each in more detail: these subsequent examples all correspond to complete rings. First in Part II we consider the case of ordinary cohomology, where S is a complete graded ring over a field k. In this simpler situation it is possible to give general geometric results of some substance. In Part III we turn to chromatic examples, where S is finite over k; the geometry is more complicated here, so we restrict attention to rather crude structural phenomena and to some very special cases.

During the discussion, there will always be a group G in the background. Most results are interesting even for the group of order 2, and the reader may want to concentrate on this case to begin with. A few sections treat groups which are not finite, and readers will lose little by ignoring them.

Part I. Equivariant K-theory.

In Part I, we concentrate on a particular example which is not only very close to algebra, but can also be made completely explicit. Most of the phenomena we are concerned with occur in this case in very concrete forms.

2. The complex representation ring

For further details of this section see [51, 52].

Let G be a compact Lie group and X a G-space (i.e. a topological space with continuous left G-action). A complex G-equivariant vector bundle over X is a continuously varying family of complex vector spaces, parametrized by X. More precisely, it is a G-map π : ξ ⟶ X so that for all x ∈ X, the fibre ξ_{x} := π^{−1}(X) is a complex vector space, and for each g ∈ G, the translation g : ξ_{x} ⟶ ξ_{gx} is linear. We also require that ξ is locally trivial: for each point x ∈ X there is an open neighbourhood U over which π is projection in the sense that there is a commutative diagram

\xymatrix{π^{−1}(U) \ar[rr]^{≅} \ar[dr]^{π} &  & U × ℂ^{n} \ar[dl]^{proj} \\  & U}

We let Vect_{G}(X) denote the set of isomorphism classes of vector bundles on X. Direct sum of vector spaces extends to vector bundles, making Vect_{G}(X) into a monoid. The equivariant K-theory of X is obtained by adjoining inverses to form an abelian group

K_{G}^{0}(X) = AbGp(Vect_{G}(X))/([ξ ⊕ η] = [ξ] + [η]).

Tensor product of vector spaces extends vector bundles, and this makes K_{G}^{0}(X) into a commutative ring.

Note that if X is a point, a vector bundle over X is just a vector space with linear G action, which is simply a complex representation:

K_{G}^{0}(pt) = R(G)

where R(G) is the complex representation ring. Indeed, tensor product with a representation makes K_{G}^{0}(X) into an R(G)-module, so from the algebraic point of view we are discussing the ring R(G) and modules over it. Example 2.1. (i) If G is cyclic of order n we may choose a faithful one dimensional representation α and R(G) = ℤ[α]/(α^{n} − 1). This is one dimensional, and using cyclotomic polynomials, we see that the irreducible components of spec(R(G)) correspond to the subgroups of G. (ii) If G is dihedral of order 8, character theory shows

R(D_{8}) = ℤ[a, b, σ]/(a^{2} = b^{2} = 1, aσ = bσ = σ, σ^{2} = 1 + a + b + ab). □

For an arbitary compact Lie group G, R(G) is Noetherian. Segal has shown [51] that its dimension is 1+rank(G) where rank(G) is the dimension of the maximal torus in G, and the irreducible components of spec(R(G)) correspond to the conjugacy classes of topologically cyclic subgroups of G.

Now in fact the functor K_{G}^{0} extends to an equivariant cohomology theory K_{G}^{∗} : G-spaces ⟶ AbGp. We will not explain in detail what this means, but it is a contravariant functor with good exactness properties (analogous to those of functors which take exact sequences of modules to long exact sequences of cohomology groups), and which takes sums to products. The extension can be given by stating that K_{G}^{∗} is 2-periodic in the sense that K_{G}^{n} = K_{G}^{n+2} and the odd part is given by K_{G}^{−1}(X) = ker(K_{G}^{0}(S^{1} × X) ⟶ K_{G}^{0}({1} × X)). The fact that this gives a suitably exact functor is Bott periodicity. For example

K_{G}^{∗}(pt) = R(G)[u, u^{−1}]

where u is a unit of degree 2; the value K_{G}^{∗}(X) on a G-space X is a module over K_{G}^{∗}(pt), and multiplication by u gives the periodicity. Henceforth we adopt the convenient abbreviation K_{G}^{∗} = K_{G}^{∗}(pt) for the coefficient ring.

The other consequence of having a cohomology theory is that there is an associated homology theory K_{∗}^{G}(X). This is more complicated to define, so we will be content to say that it is related to cohomology in such a way that a form of Poincaré duality holds when X is a manifold. The most trivial example of this is that K_{G}^{i}(pt) = K_{−i}^{G}(pt), so that, with the usual convention for relating upper and lower indexing, we have

K_{G}^{∗} = K_{∗}^{G} = R(G)[u, u^{−1}].

Where necessary we refer to lower indices as degrees and upper indices as codegrees; for example u ∈ K_{2}^{G} = K_{G}^{−2} is of degree 2 and codegree −2.

3. Free G-spaces.

For further details of this section see [52, 27].

The simplest sort of G-spaces are those with a free action (i.e. so that the identity element of the group is the only element fixing anything). In particular if X is a free Gspace then equivariant vector bundles ξ ⟶ X over X correspond to non-equivariant bundles η ⟶ X/G over the quotient: given ξ we take η = ξ/G and given η we take ξ to be the pullback of η along the quotient map X ⟶ X/G. This passes to K theory to say that if X is free

K_{G}^{∗}(X) = K^{∗}(X/G).

For finite groups there is a similar statement

K_{∗}^{G}(X) = K_{∗}(X/G)

for homology. However this is less elementary, and involves transfer arguments. Accordingly, if G is a general compact Lie group the statement must be modified by inserting a suitable kind of twist.

The other thing about free G-spaces is that there is a terminal free G space EG in the homotopy category. This means that for any free G-space X there is a G-map ν_{X} : X ⟶ EG, unique up to homotopy. In fact EG is characterized by two properties: it is free and non-equivariantly contractible. For example if G is cyclic we may view G as a subgroup of the unit complex numbers, and then EG = S(∞ℂ) is the unit sphere in the direct sum ∞ℂ of infinitely many copies of ℂ. This may be more familiar to some through the quotient BG = EG/G, called the classifying space of G: for instance BC_{2} = ℝP^{∞}.

Note in particular that for any free G-space X the universal map ν_{X} induces a diagram

\xymatrix{K_{G}^{0}(X) & K_{G}^{0}(EG) \ar[l] \\ K^{0}(X/G) \ar[u]^{≅} & K^{0}(BG). \ar[l] \ar[u]^{≅}}

Thus a knowledge of K_{G}^{0}(EG) gives canonical characteristic classes in K_{G}^{0}(X) = K^{0}(X/G). These can be useful invariants for distinguishing different G-spaces with the same quotient X/G. For example if G = C_{2} we may have X_{1} = G × ℝP^{3} whilst X_{2} = S(2ℂ) (i.e. the 3-sphere with the antipodal action). In both cases the quotient is projective space ℝP^{3}, but the different characteristic classes distinguish them.

A similar motivation for studying K_{∗}^{G}(EG) can be given, but a more convincing one can be given because quite powerful torsion invariants of free G-manifolds belong to this group

4. The Atiyah-Segal completion theorem and its dual.

For further details of this section see [3, 24].

In this section we describe the calculation of K_{G}^{∗}(EG) and the associated homology K_{∗}^{G}(EG). We present this in a quasi-historical way, as a series of theorems starting very concrete and proceeding to greater generality at the cost of some additional machinery.

Remembering the periodicity of K theory, we may work with the degree 0 part of the coefficient ring S = R(G), and it suffices to describe the 0th and 1st K groups. The ring S = R(G) is augmented over k = R(1) by the map S = R(G) ⟶ R(1) = ℤ taking the virtual dimension, and we need to consider the augmentation ideal J = ker(R(G) ⟶ R(1) = ℤ) of elements of virtual dimension 0. Theorem 4.1. (Atiyah (1961) [1]) The equivariant K theory of EG is

K_{G}^{0}(EG) = K^{0}(BG) = R(G)_{J}^{∧}

and

K_{G}^{1}(EG) = K^{1}(BG) = 0.

Example 4.2. (i) When G = C_{2}, we have seen R(C_{2}) = ℤ[α]/(α^{2} − 1), and since 1 and α are one dimensional representations, we see J = (1 − α). Now a short calculation allows us to identify the completion: (1 − α)^{2} = 1 − 2α + α^{2} = 2(1 − α), so that

(1 − α)^{n+1} = 2^{n}(1 − α),

and

R(C_{2})_{J}^{∧} = ℤ ⊕ ℤ_{2}^{∧}.

(ii) More generally, the Artin induction theorem easily shows that if G is a p-group

R(G)_{J}^{∧} = ℤ ⊕ J_{p}^{∧}.

(iii)It is not hard to see that for finite groups, J-completion is injective if and only if G is a p-group. □

There is a more conceptual way of expressing the phenomena. We consider the projection map EG ⟶ pt and the induced map

K_{G}^{∗}(X) ⟶ K_{G}^{∗}(EG × X) = K^{∗}(EG ×_{G} X)

in K-theory. Theorem 4.3. (Atiyah-Segal (1969) [3]) For any finite G-space X the natural map

K_{G}^{∗}(X) ⟶ K_{G}^{∗}(EG × X) = K^{∗}(EG ×_{G} X)

is completion at the augmentation ideal J of R(G).

We pause for some remarks. Firstly, Theorem 4.1 is the special case X = pt, since we have seen that K_{G}^{∗}(pt) = R(G)[u, u^{−1}]. Next, we comment that the codomain K_{G}^{∗}(EG × X) is a cohomology theory of X, so that for the statement to be plausible, it is necessary that K_{G}^{∗}(X)_{J}^{∧} is a cohomology theory on finite complexes X. However, for each i, the R(G)-module K_{G}^{i}(X) is Noetherian provided X is a finite G-complex. This is clear for K_{G}^{0}(G/H) = K_{H}^{0}(pt) = R(H), and the general case follows by induction on the number of cells, using the exactness properties of a cohomology theory. Accordingly, the Artin-Rees lemma implies that J-completion is exact, and so K_{G}^{∗}(X)_{J}^{∧} is a cohomology theory on finite complexes X. Since J-completion is definitely not exact for arbitrary R(G)-modules, the statement must be modified if it is to cover arbitrary complexes X. Atiyah and Segal used the device of pro-groups, but for our purposes we prefer an alternative solution. Theorem 4.4. (Greenlees-May [31]) For any G-space X there is a natural short exact sequence

0 ⟶ L_{1}^{J}(K_{G}^{∗}(X)) ⟶ K_{G}^{∗}(EG × X) ⟶ L_{0}^{J}(K_{G}^{∗}(X)) ⟶ 0,

where L_{i}^{J} is the ith left derived functor of J-completion.

The Artin-Rees lemma implies that for a Noetherian module M, we have L_{0}^{J}M = M_{J}^{∧} and L_{i}^{J}M = 0 for i > 0, so that that if we take X to be a finite complex, we obtain the previous versions of the theorem.

It is a general phenomenon in topology that statements in homology have better finiteness properties than in cohomology, so it is natural to seek statements about K homology. Finally, local cohomology is about to make an appearance. Theorem 4.5. (Greenlees (1993) [24]) For any finite group G and any G-complex X, there is a spectral sequence

H_{J}^{∗}(K_{∗}^{G}(X)) ⇒ K_{∗}^{G}(EG × X) = K_{∗}(EG ×_{G} X),

where J is the augmentation ideal of R(G).

We outline a proof for p-groups in Section 6, and a rather different proof for all groups in Part III (Section 13).

The next step is the major input from commutative algebra, and appears quite magical from the topological point of view. Grothendieck’s vanishing theorem states that local cohomology vanishes above the dimension; since R(G) is one dimensional, the spectral sequence collapses. Corollary 4.6. (Greenlees (1993) [24]) For any finite group G and any G-complex X, there is a short exact sequence

0 ⟶ H_{J}^{1}(K_{i+1}^{G}(X)) ⟶ K_{i}^{G}(EG × X) ⟶ H_{J}^{0}(K_{i}^{G}(X)) ⟶ 0.

In particular if X = pt

K_{0}^{G}(EG) = H_{J}^{0}(R(G)) = ℤ

and

K_{1}^{G}(EG) = H_{J}^{1}(R(G)). □

Example 4.7. If G = C_{2} we may make the calculation completely explicit. We work in R(C_{2}) = ℤ[α]/α^{2} − 1, and J = (1 − α). Since (1 − α)(1 + α) = 1 − α^{2} = 0, inverting (1 − α) kills 1 + α. Similarly, since (1 − α)^{2} = 2(1 − α), inverting 1 − α inverts 2. Thus the stable Koszul complex R(C_{2}) ⟶ R(C_{2})[1/(1 − α)] becomes

ℤ ⊕ ℤ ⟶ ℤ[1/2],

showing H_{J}^{0}(R(C_{2})) = ℤ and H_{J}^{1}(R(C_{2})) = ℤ/2^{∞}.

The calculation of K_{∗}^{G}(EG) is quite explicit in general. Indeed, H_{J}^{0}(R(G)) is easily calculated in terms of characters: J consists of representations whose characters vanish at the identity element e ∈ G. Since characters separate conjugacy classes, H_{J}^{0}(R(G)) consists of representations whose characters vanish except at e. This consists of the integer multiples of the regular representation.

It is not hard to calculate H_{J}^{1}(R(G)) either. Indeed, for any p-group, Artin’s induction theorem shows that the p-adic and J-adic topology on J coincide, and hence

H_{J}^{1}(R(G)) = R(G)/ℤ ⊗ ℤ/p^{∞}.

Remark 4.8. The theorem is also true for an arbitrary compact Lie group G. For this one must use the proof in Part III. The crucial ingredient is that the representing spectrum of K theory is a highly structured ring spectrum. Until recently this was only known for finite groups, but M.Joachim [43] tells me that he has proved the result for arbitrary compact Lie groups.

Since the dimension of R(G) is 1 + rank(G), the spectral sequence will not normally collapse. However it still collapses when K_{∗}^{G}(X) has depth ≥ rank(G). Remark 4.9. An alternative way of motivating this whole connection is that by definition J acts as 0 on K_{G}^{∗}(G) = K^{∗}(pt). By induction J acts nilpotently on K_{G}^{∗}(X) and K_{∗}^{G}(X) if X is finite and free. Since EG is a direct limit of finite free complexes, we expect K_{G}^{∗}(EG) to be the simplest possible “inverse limit” of J-power torsion complexes (hence it is calculated by the left derived functors of J-completion), and K_{∗}^{G}(EG) to be the simplest possible “direct limit” of J-power torsion complexes (hence it is calculated by the right derived functors of J-power torsion). Here ‘simplest possible’ has a rather complicated meaning, requiring homotopy invariance, functoriality and exactness.

5. Duality for the representation ring.

In this section we introduce one of the central properties of equivariant cohomology rings; in this case we are considering the ring S = R(G) augmented over k = R(1) = ℤ. For the representation ring, this property appears rather commonplace, but at least it is visible to inspection. The idea is to combine the local cohomology theorems with the universal coefficient theorem to obtain a duality statement.

We are going to use the form appropriate to complete rings, and the completion theorem in the non-equivariant form:

K^{0}(BG) = R(G)_{J}^{∧} and K^{1}(BG) = 0.

Similarly, we use the non-equivariant form of the local cohomology theorem; assuming G is finite, this takes the form of the identifications

K_{0}(BG) = H_{J}^{0}(R(G)) and K_{1}(BG) = H_{J}^{1}(R(G)).

We have used the non-equivariant forms since there is a Universal Coefficient Theorem [2] for calculating K^{∗}(X) from K_{∗}(X). This is a short exact sequence

0 ⟶ Ext_{ℤ}^{1}(K_{i+1}(X), ℤ) ⟶ K^{i}(X) ⟶ Hom_{ℤ}(K_{i}(X), ℤ) ⟶ 0.

We apply this to X = BG and substitute the algebraic expressions for its homology and cohomology to obtain a short exact sequence

0 ⟶ Ext_{ℤ}^{1}(H_{J}^{1}(R(G)), ℤ) ⟶ R(G)_{J}^{∧} ⟶ Hom_{ℤ}(H_{J}^{0}(R(G)), ℤ) ⟶ 0.

This no longer mentions K theory: it is just a statement about the augmented ℤ-algebra R(G). It is reminiscent of local duality for Gorenstein rings, but of course R(G) is not local or even Cohen-Macaulay. It is also notable that local cohomology has separated the ℤ-torsion free H_{J}^{0} from the torsion part H_{J}^{1}. The former gives the uncompleted part of R(G)_{J}^{∧} (namely multiples of the regular representation), and the latter gives the completed part (namely J_{(p)}^{∧} in the case of a p-group).

6. A proof for p-groups.

In this section I will outline a proof of the local cohomology theorem for p-groups. The reason for restricting to this case is that on the one hand it is a convincingly large family of examples, whilst on the other, it is possible to give a rather concrete version of the proof in which the correspondence between topological and commutative algebraic structures is highlighted.

In Part III we will see one method for going beyond p-groups using some quite sophisticated machinery. For the case of K-theory, and for finite groups, there is the more elementary approach used in [24]: the Burnside ring A(G) of finite G-sets is a good approximation to the representation ring. Indeed the permutation representation homomorphism A(G) ⟶ R(G) has the property that

\sqrt{J(A(G)) · R(G)} = J(R(G))

so that we may replace the representation ring by the Burnside ring in the proof, considering all R(G)-modules (such as R(G) itself) as modules over A(G). For topological reasons, manipulations with the Burnside ring are much more elementary. For infinite compact Lie groups this method fails since A(G) is still 1-dimensional whilst the dimension of R(G) is equal to the rank of G. For the present we return to a method applying to p-groups and complex orientable theories.

We introduce notation by summarizing standard constructions in the commutative algebra of a ring S. The unstable Koszul complex is the cochain complex K(x) = (S ⟶^{x} S) concentrated in codegrees 0 and 1. This is the desuspension of the mapping cone of x:

K(x) = Σ^{−1}M(x) ⟶ S ⟶^{x} S.

To form the stable Koszul complex, we may now do the analogous construction for powers of x, assemble them into a direct system and pass to limits:

K(x) = Σ^{−1}M(x) ⟶ S ⟶^{x} S
↓  ↓  ↓
K(x^{2}) = Σ^{−1}M(x^{2}) ⟶ S ⟶^{x^{2}} S
↓  ↓  ↓
K(x^{3}) = Σ^{−1}M(x^{3}) ⟶ S ⟶^{x^{3}} S
↓  ↓  ↓
⋮  ⋮  ⋮
↓  ↓  ↓
K(x^{∞}) = Σ^{−1}M(l) ⟶ S ⟶^{l} S[1/x]

Now given an ideal J = (x_{1}, x_{2}, …, x_{r}) in S we may form the stable Koszul complex

K(J^{∞}) = K(x_{1}^{∞}) ⊗ K(x_{2}^{∞}) ⊗ ⋯ ⊗ K(x_{r}^{∞}).

The notation is reasonable since the complex is independent of generators up to quasiisomorphism (exercise). This allows us to define local cohomology by

H_{J}^{∗}(M) := H^{∗}(K(J^{∞}) ⊗ M),

and its dual local homology by

H_{∗}^{J}(M) := H_{∗}(Hom(P K(J^{∞}), M)),

where P K(J^{∞}) is a complex of projectives quasi-isomorphic to K(J^{∞}). These definitions become interesting by Grothendieck’s theorem that local cohomology calculates right derived functors of J-power torsion

H_{J}^{∗}(M) = R^{∗}Γ_{J}(M)

where

Γ_{J}(M) = {x ∈ M | J^{N}x = 0 for N >> 0},

and the fact [32] that local homology calculates left derived functors of completion

H_{∗}^{J}(M) = L_{∗}Λ^{J}(M)

where

Λ^{J}(M) = lim_{←}_{k} M/J^{k}M.

Note also that Grothendieck vanishing and the evident universal coefficient theorem

E_{2}^{∗,∗} = Ext_{S}^{∗}(H_{J}^{∗}(S), M) ⇒ H_{∗}^{J}(M)

shows that the L_{i}Λ^{J}(M) = 0 for i > dim(S). This explains why only the zeroth and first derived functors entered into the Completion Theorem 4.4.

Now in the topological context, the analogue of the stable Koszul complex is the universal free space EG. Let us start with the case when G is a cyclic group, G ⊆ ℂ^{∗}. We note that for each k the group G acts freely on the unit sphere S(kℂ) in the k-fold direct sum kℂ. Since S(kℂ) is a (2k − 1)-sphere (and hence (2k − 2)-connected), the union S(∞ℂ) = ⋃_{k} S(kℂ) is contractible. Thus EG = S(∞ℂ) in this case.

Remark 6.1. For the best analogy with the above commutative algebra we need to work with G-spaces equipped with a G-fixed basepoint, so we need to describe the routine translation between the based and unbased context.

For cohomology, it is in terms of the reduced cohomology: if Y is a G-space with G-fixed basepoint y_{0} we have maps {y_{0}} ⟶ Y ⟶ {y_{0}} so that if we define the reduced cohomology by K̃_{G}^{∗}(Y) = ker(K_{G}^{∗}(Y) ⟶ K_{G}^{∗}(y_{0})), we have K_{G}^{∗}(Y) = K̃_{G}^{∗}(Y) ⊕ K_{G}^{∗}(y_{0}).

If X is an unbased G-space we form the based space X_{+}, where the subscript + denotes the addition of a disjoint basepoint fixed by G. Thus

K̃_{G}^{∗}(X_{+}) = K_{G}^{∗}(X). □

We need one more construction. In addition to the unit sphere S(V) of a finite dimensional inner product space V, we may form the one point compactification S^{V}. A useful feature of the based context is that S^{V} is the mapping cone of the map S(V)_{+} ⟶ pt_{+} taking S(V) to the non-base point. Consequently we may also say that S(V)_{+} is the desuspension of the mapping cone of the inclusion S^{0} ⟶ S^{V}.

Now we have the ingredients to construct a diagram exactly analogous to the one for the stable Koszul complex

EG_{+}^{(1)} = S(ℂ)_{+} ⟶ S^{0} ⟶^{[x]} S^{ℂ}
↓  ↓  ↓
EG_{+}^{(3)} = S(2ℂ)_{+} ⟶ S^{0} ⟶^{[x^{2}]} S^{2ℂ}
↓  ↓  ↓
EG_{+}^{(5)} = S(3ℂ) ⟶ S^{0} ⟶^{[x^{3}]} S^{3ℂ}
↓  ↓  ↓
⋮  ⋮  ⋮
↓  ↓  ↓
EG_{+} = S(∞ℂ)_{+} ⟶ S^{0} ⟶^{l} S^{∞ℂ}.

To complete the analogy, I should explain why it is reasonable to consider S^{0} and S^{kℂ} as analogues of the same object S, and why we have written [x^{k}] for the inclusion map. For this we may as well allow G to be arbitrary. The point is that equivariant Bott periodicity provides a specific isomorphism K̃_{G}^{0}(S^{V}) ≅ K̃_{G}^{0}(S^{0}) = R(G) for any complex representation V. We then have a diagram

\xymatrix{K̃_{0}^{G}(S^{0}) \ar[r]^{[x(V)]} \ar[d]_{=} & K̃_{0}^{G}(S^{V}) \ar[d]^{Bott}_{≅} \\ R(G) \ar[dr]_{·χ(V)} & K̃_{0}^{G}(S^{0}) \ar[d]^{=} \\  & R(G)}

This defines the element χ(V) ∈ R(G) called the Euler class of V, and by transitivity of Bott periodicity isomorphisms, χ(V ⊕ W) = χ(V)χ(W). Thus the map marked [x^{k}] above is χ(kℂ) = χ(ℂ)^{k}. For an arbitrary group G and representation V, the construction of the Bott map means that χ(V) is the alternating sum of exterior powers of V. It follows that when we apply K̃_{∗}^{G}(·) to

S(∞V)_{+} ⟶ S^{0} ⟶ S^{∞V}

we obtain

K̃_{G}^{∗}(S(∞V)_{+}) ⟶ R(G) ⟶ R(G)[1/χ(V)]

and it is in this sense that K̃_{∗}^{G}(S(∞V)_{+}) is the analogue of the stable Koszul complex K(χ(V)^{∞}).

Now, just as most ideals are not principal, so most groups are not cyclic. However, we can deal with this in the same way that we constructed a stable Koszul complex for an ideal with several generators. For example, if G = C_{1} ×C_{2} ×⋯×C_{r} is a product of cyclic groups, we have

EG = S(∞ℂ_{1}) × S(∞ℂ_{2}) × ⋯ × S(∞ℂ_{r})

where ℂ_{i} is the natural representation of the cyclic group C_{i} on ℂ. It is not hard to use character theory to shown that if G is a p-group (or more generally supersoluble) then it has complex representations V_{1}, V_{2}, …, V_{r} for some r so that G acts freely on S(V_{1}) × S(V_{2}) × ⋯ × S(V_{r}), and hence

EG = S(∞V_{1}) × S(∞V_{2}) × ⋯ × S(∞V_{r}).

(Equally it is not hard to show that most groups do not have such representations; the smallest example is A_{4}, but see [49] for much more detailed information). Supposing that we do have such a model for EG, we may immediately outline a proof of the local cohomology theorem.

Outline of proof of 4.5.

The theorem was stated with an unbased space X, but the proof works with based spaces, like X_{+}. The translation is routine as described in Remark 6.1. In particular EG_{+} ∧ X_{+} = (EG × X)_{+} and K̃_{∗}^{G}(X_{+}) = K_{∗}^{G}(X).

We apply K̃_{∗}^{G} to the filtered space EG_{+} ∧ X_{+}. Because the filtration precisely modelled that of the stable Koszul complex, we immediately find a spectral sequence

E_{1}^{∗,∗} = K(ℰ^{∞}) ⊗_{R(G)} K_{∗}^{G}(X) ⇒ K_{∗}^{G}(EG_{+}),

where

ℰ = (χ(V_{1}), χ(V_{2}), …, χ(V_{r}))

is the ideal generated by the Euler classes. Again, the correspondence between topology and commutative algebra makes clear that d_{1} is the Koszul differential, so that

E_{2}^{∗,∗} = H_{ℰ}^{∗}(K_{∗}^{G}(X)).

Finally we need to explain why ℰ may be replaced by J. First, the inclusion S^{0} ⟶ S^{V} is obviously null-homotopic non-equivariantly if V ≠ 0, so ℰ ⊆ J. In the cyclic case we had equality, and for a general p-group we have

\sqrt{ℰ} = J.

This can be observed explicitly for the representation ring, since the primes are known by [51], but in fact it follows from some finiteness assumptions [28].

Since local cohomology only depends on the radical it proves H_{J}^{∗}(M) = H_{ℰ}^{∗}(M) as required. □

As far as any applications we have made are concerned, there is no particular advantage to using the augmentation ideal J rather than the Euler class ideal ℰ. It is when comparing different groups that it comes into its own.

7. Variations and extensions.

There are two directions to develop the above ideas. The first replaces free G-spaces (i.e. spaces with all isotropy groups trivial) by spaces with isotropy in some other family ℱ of subgroups. On the ring theoretic side, the ideal J is replaced by the ideal J(ℱ) of elements restricting to zero in R(H) for all subgroups H in ℱ. For equivariant K theory this goes smoothly. However there is no comparable expression in terms of the non-equivariant case, and hence no duality statement. We will not pursue this variation any further here.

The second variation is to replace K theory by another cohomology theory; on the ring theoretic side this replaces the augmented ring R(G) by another. If K theory is replaced by E-theory then R(G) is replaced by the coefficient ring S = E_{G}^{∗} (or its degree zero part E_{G}^{0}), R(1) is replaced by k = E^{∗} (or its degree zero part E^{0}) and J is replaced by the kernel J = ker(E_{G}^{∗} ⟶ E^{∗}) of the map forgetting equivariance. Context 7.1. The correspondence between commutative algebra and topology is via

• S = E_{G}^{∗}

• k = E^{∗}

• J = ker(E_{G}^{∗} ⟶ E^{∗}). □

The remainder of this article is structured round a number of these. In this section we give an overview.

In the following table the first row summarizes the properties of K theory discussed above, and each subsequent row is an analogous example. To avoid discussion of special cases, the information refers only to finite groups G. The final two columns refer to the question of whether the homology and cohomology of EG can be calculated using local cohomology and completions as was the case for K theory.

Cohomology theory Coefficients Dimension Augmented over Homology

Cohom.

K-theory  R(G)  1  R(1) = ℤ  [24]  [3]
E_{n}-theory  E_{n}^{0}(BG)  n  ℤ_{p}^{∧}[[u_{1}, …, u_{n−1}]] [24, 20, 22]  T
Stable cohomotopy A(G) + nilp  1  A(1) = ℤ

FALSE!

[15]

Ordinary  H^{∗}(BG) rank_{p}(G)  H^{∗} = 𝔽_{p}  [26]  T
Connective K-theory ku^{∗}(BG)  2  ku^{∗} = ℤ[u]

[24, 20, 22]

T

Remark 7.2. (i) There are various ways of grouping these examples. First, K-theory and E_{n}-theory are periodic, so we may work with ungraded rings. In the cohomotopy example the Burnside ring A(G) is in degree 0, and all other elements are nilpotent (Nishida’s nilpotence theorem); accordingly we may work over the degree zero part A(G) in this case too. The other examples are graded. (ii) More significant is the fact that apart from K-theory and cohomotopy, all the examples are complete for the augmentation ideal. This means that the completion theorem is a tautology for finite complexes (hence the entries T for ‘true’ and ‘tautology’) and concentrates attention on local cohomology and the duality statement.

This is no disadvantage for present purposes, but in topology it is a major problem to identify natural rings of which these are completions. To see why it is valuable, one may imagine we only knew the completion of R(G). We may then seek to refine it to R(G) itself, hoping thereby to invent representation theory. (iii) In the first three examples, the ring is essentially finite over its nonequivariant counterpart: in the other two we have finitely generated algebras over the non-equivariant coefficients. (iv) All the examples for which the local cohomology theorem is known to be true can be proved for p-groups using the method of Part I. They can be proved for finite groups by the method described in [24, Appendix], using [28] to establish finiteness properties. However, the technology necessary for implementing the strategy was not introduced until [20]; it was shown to apply to these cases in [22].

(v) Stable cohomotopy is quite exceptional. The fact that local cohomology of π_{G}^{∗} does not calculate π_{∗}(BG) means that the type of proof for the completion theorem we will give in the complex orientable case (by formally deducing it from the local cohomology theorem) cannot apply. Carlsson’s proof of the completion theorem (“the Segal conjecture”) is not a simple formality like the proofs we give here, and it seems that the substantial calculational input is necessary. Other examples of this type occur in algebraic K theory, but we shall not be concerned with them further.

In Part II (Sections 8 to 11) we discuss ordinary cohomology: here the topological technology is at a minimum, but the commutative algebra is correspondingly increased. In Section 12 we discuss a family of examples arising from the topological theory of manifolds: this has the advantage of being a rather extensive family from the ring-theoretic point of view. Finally, we look at two special cases of this. In Section 15 we discuss the E_{n}-theory example because an interesting interaction of Cousin filtrations occurs. In Section 16 we turn to connective K theory; the interest here is that although it is very close to examples where rather complete results are available, some rather intricate commutative algebra appears.

Part II. Ordinary cohomology and graded connected k-algebras.

We spend Sections 8 to 11 considering ordinary cohomology. This has several special attractions, especially to algebraists. First, the cohomology of a point can be described in purely algebraic terms as an Ext algebra relevant to representation theory. Second, the ring S is a connected commutative graded algebra over a field k, so it is especially easy to apply the methods of commutative algebra.

8. Cohomology of some classes of groups.

For the duration of Part II, S is an ℕ-cograded algebra over a field k, and S^{0} = k. Properly speaking, S is graded commutative in the sense that ab = (−1)^{deg(a) deg(b)}ba, and all the results we need can be proved in this context. Those not wishing to worry about graded commutativity can restrict attention to the case that char(k) = 2. For most of the discussion it suffices to replace S by its even-graded commutative subring.

In this section the augmentation ideal J = ker(S ⟶ k) is the unique maximal ideal of the graded local ring S. To emphasize this we write 𝔪 for it.

8.1. Finite groups. The first class of examples we discuss are the rings

S = H^{∗}(G; k) := Ext_{kG}^{∗}(k, k)

for a finite group G, where kG denotes the group ring. General references for this subsection are [5, 11]. One of the principal reasons for studying this is representation theory. For example the rate of growth of a minimal resolution of k is exactly captured by the Hilbert series

h(G; k) := ∑_{i≥0} dim_{k}(H^{i}(G; k))t^{i};

for Noetherian rings this power series is easily seen to be the expansion of a rational function about t = 1. The cohomology ring H^{∗}(G; k) is known explicitly for a large number of groups, and we tabulate some examples to bring the discussion down to earth. All the examples are 2-groups, and k is a field of characteristic 2. We use the convention that a subscript on a generator indicates cohomological degree. The Cohen-Macaulay defect for a local ring S is defined by CM-defect(S) = dim(S) − depth(S). The groups are elementary abelian, quaternion of order 8, dihedral of order 8, semidihedral of order 16, a certain group of order 32, and the extra-special 2-groups.

Group G

H^{∗}(G; k)

dim depth defect

Hilbert series

(C_{2})^{r}  k[x(1)_{1}, x(2)_{1}, ⋯, x(r)_{1}]  r  r  0  1/(1 − t)^{r}
Q_{8}  k[x_{1}, y_{1}, z_{4}]/(x^{2} + xy + y^{2}, x^{3}, y^{3})  1  1  0 (1 + t + t^{2})/(1 − t)(1 + t^{2})
D_{8}  k[x_{1}, y_{1}, u_{2}]/(xy)  2  2  0  1/(1 − t)^{2}
SD_{16}  See below  2  1  1  1/(1 − t)^{2}(1 + t^{2})
Γ_{7}a_{2}  3  1  2
2^{1+n}  k[x(1)_{1}, x(2)_{1}, …, x(n)_{1}]/I ⊗ k[ζ_{2^{n−r}}] r  r  0

Remark 8.1. (i) There is a copious supply of further examples on J.F.Carlson’s webpages

[14] (the cohomology of all but 5 of the 267 groups of order 64 are there, together with all 2-groups of smaller order). (ii) Quillen [46] has shown that

dim(H^{∗}(G; k)) = rank_{p}(G),

where the p-rank rank_{p}(G) of G is the rank of the largest elementary abelian p-subroup (C_{p})^{r} in G. (iii) For p-groups of small order, the Cohen-Macaulay defect tends to be quite small. Indeed, Duflot [16] has shown that

depth(H^{∗}(G; k)) ≥ rank_{p}(Z(G))

where char(k) = p, and that in any case the depth is at least 1 [17]. (iv) The cohomology ring of the semidihedral group is

k[x_{1}, y_{1}, z_{3}, t_{4}]/(xy + y^{2}, y^{3}, yz, x^{3}z + x^{3}y + y^{2}t + z^{2}).

The cohomology ring of Γ_{7}a_{2} has 8 generators and 18 relations, but the relevant information is well summarized in [6].

(v) The case of the extra special groups of order 2^{1+2m} is included because Quillen’s calculation [47] is so elegant (see also [5, 5.5] for a brief account). The ideal I is generated by a regular sequence.

To be more precise, consider an extension 1 ⟶ C_{2} ⟶ G ⟶ E ⟶ 1 where E is elementary abelian of rank n. If C_{2} is the commutator subgroup and the centre of G then G is called extraspecial, n = 2m is even and there are only two isomorphism types of extraspecial groups of this order, but the cohomology calculation applies to any extension of the given form. As usual, r is the dimension of the cohomology ring, or equivalently the rank of the largest elementary abelian subgroup.

To describe the ideal, we use the fact that any such extension is classified by the map q : E ⟶ C_{2} given by squaring the preimage in G of an element of E. This q is a quadratic form with associated bilineary form b. Now if e_{1}, …, e_{n} is a basis of E as an 𝔽_{2}-vector space and v = Σ_{i}e_{i}x_{i} then I is generated by the sequence

q(v), q(v, F (v)), …, q(v, F^{n−r}(v)),

where F is the Frobenius map, and the sequence turns out to be regular. The codegrees are 2, 4, 8, …, 2^{n−r+1}, so it is easy to write down the Hilbert series.

8.2. Compact Lie groups. If G is a compact Lie group the ring

S = H^{∗}(BG; k)

is the cohomology ring of the classifying space of G that we met in Section 3. One reason for studying this is exactly parallel to the motivation we gave for K theory: it tells us about characteristic classes. One often thinks of compact Lie groups such as the orthogonal and unitary groups, but of course finite groups are perfectly good examples, and using the bar construction for BG one sees that it can be constructed with cells corresponding to copies of kG in the bar resolution: thus H^{∗}(BG; k) = Ext_{kG}^{∗}(k, k), and our notation is consistent. We may give a good supply of examples here too.

Group G

H^{∗}(BG; k)

dim depth defect

Hilbert series

(U(1))^{r}  k[x(1)_{2}, x(2)_{2}, ⋯, x(r)_{2}]  r  r  0  1/(1 − t^{2})^{r}
SU(2)  k[x_{4}]  1  1  0  1/(1 − t^{4})
U(n)  k[c_{1}, c_{2}, …, c_{n}]  n  n  0 1/(1 − t^{2})(1 − t^{4}) ⋯ (1 − t^{2n})
O(2n)  k[p_{1}, p_{2}, …, p_{n}]  n  n  0 1/(1 − t^{4})(1 − t^{8}) ⋯ (1 − t^{4n})
Spin(n) k[w_{2}, w_{3} …, w_{n}]/I ⊗ k[ζ_{2^{n−r}}] r  r  0

Remark 8.2. (i) By force of precedent, the Chern class c_{i} is of cohomological degree 2i and the Pontrjagin class p_{i} is of cohomological degree 4i. Otherwise we have retained the convention that subscripts refer to codegrees. (ii) The field k can be of any characteristic except that char(k) ≠ 2 for O(2n) and char(k) = 2 for Spin(n). (iii) The case O(2n) is included because its behaviour is slightly more complicated than its polynomial cohomology ring suggests (see Example 9.5(iii)). (iv) The cohomology of Spin(n) is also calculated in [47], and was Quillen’s motivation for considering the extraspecial 2-groups. The ideal I is generated by the regular sequence

w_{2}, Sq^{1}w_{2}, …, Sq^{2^{n−r}}Sq^{2^{n−r−1}} ⋯ Sq^{2}Sq^{1}w_{2},

where Sq^{i} is the ith Steenrod square. Quillen deduces this calculation by comparison with the calculation for the extraspecial subgroup of Spin(n).

One other feature will be important. The group G acts on itself smoothly by conjugation. Since it preserves the identity element e ∈ G, a group element gives a self-map of the tangent space T_{e}G, and we obtain the adjoint representation

ad : G ⟶ GL(T_{e}G),

and hence the k-orientation representation

G ⟶^{ad} GL(T_{e}G) ⟶^{det} ℝ^{∗} ⟶^{sign} {+1, −1} ⟶ k^{∗}.

If the image is trivial we say that ad is orientable over k.

Evidently if char(k) = 2, the adjoint representation is orientable for every group G. By continuity, each connected component maps to the same point, so that if G is connected, or if it has an odd number of components then ad is orientable over any field. However if char(k) ≠ 2 then ad(O(2n)) is not orientable over k (this is easy to check for n = 1, and follows in general).

8.3. Arithmetic groups. For an arbitrary discrete group G we may consider the cohomology ring

S = H^{∗}(G; k) = Ext_{kG}^{∗}(k, k).

This can be arbitrarily unpleasant unless we place a restriction on G. We are interested in arithmetic groups, such as SL_{n}(ℤ) and O_{n}(ℤ). However the appropriate level of generality is a little wider: virtual duality groups. A general reference for this subsection is [11].

Before reaching virtual duality groups we should discuss duality groups. The reader may like to construct an analogy with Gorenstein local rings. A Gorenstein ring can be characterized as one having finite injective dimension over itself, but perhaps the reason Gorenstein rings are so important is their duality. Similarly, if we restrict attention to groups G with the finiteness condition that k admits a resolution by finitely generated projective kG modules, we may characterize duality groups as torsion free groups G so that

H^{i}(G; kG) := Ext_{kG}^{i}(k, kG) = 0 unless i = n

for some n. The number n is called the dimension of G and the module

I = H^{n}(G; kG)

is called the dualizing module. Such groups then have a duality isomorphism

H^{i}(G; M) ≅ H_{n−i}(G; M ⊗ I).

We say that G is a D^{n}-group; if I is one dimensional over k we say G is a Poincaré duality group (P D^{n}-group), and if in addition G acts trivially on I we say it is orientable.

A group G is said to be a virtual duality group if it has a subgroup G′ of finite index which is a duality group. The virtual dimension of G is the dimension of G′ (and is well defined) and the dualizing module for G is

I = H^{n}(G′; kG′) ≅ H^{n}(G; kG)

where the isomorphism is Shapiro’s lemma and this shows I is a G-module. Example 8.3. (i) Borel and Serre [13] show that any torsion free arithmetic group is a duality group, and a general arithmetic group is a virtual duality group. (ii) Fundamental groups of a knot complement are D^{3}-groups. (iii) Mapping class groups are virtual duality groups [41, 40]. (iv) Automorphism groups of free groups are virtual duality groups [10].

Thus the class of virtual duality groups is very extensive. However there are very few non-trivial examples where the cohomology ring is known explicitly.

8.4. p-adic Lie groups. It is possible to transpose the entire previous section into the category of profinite groups. If G is a profinite group we may consider modules for its (discrete) finite quotients. However it is important to be able to discuss inverse limits of these (compact G-modules) and direct limits (discrete G-modules). It is possible to put both classes of modules into a reasonably well behaved category [54], and make most of the usual constructions of homological algebra in this category. In particular the complete group ring k[[G]] behaves like a free module and the cohomology ring may be defined by

S := H^{∗}(G; k) = Ext_{k[[G]]}^{∗}(k, k).

Just as in the discrete case we may define duality groups and virtual duality groups. Virtual duality groups have a virtual dimension and a (compact) dualizing module I = H^{n}(G; k[[G]]). Example 8.4. Lazard shows that any p-adic Lie group is a profinite virtual duality group [44]. As in the discrete case, the interesting examples that come up are automorphism groups of suitable structures. Familiar examples are the groups SL_{n}(ℤ_{p}^{∧}), and another class of examples of considerable interest to topologists are the Morava stabilizer groups (automorphism groups of certain formal groups).

For all these more general classes of groups, the ring S is known to share with the case of finite groups the properties that it is a finitely generated k-algebra and of dimension rank_{p}(G) [46].

9. The local cohomology theorem for group cohomology.

We return to the local cohomology theorem, treated in Part I for the representation ring. Now we consider what it says for ordinary cohomology, which is to say for the augmented rings k-algebras S discussed in Section 8. This case is much more interesting because S can be of higher dimension, and exhibits a wide variety of behaviours as the group varies.

We are writing 𝔪 for the augmentation idea: it is the maximal ideal of positive codegree elements. We write

DM = Hom_{S}(M, I(k)) = Hom_{k}(M, k)

for the Matlis dual, where I(k) is the injective envelope of k. Theorem 9.1. [26] If S = H^{∗}(G; k) for a finite group G there is a spectral sequence

E_{2}^{∗,∗} = H_{𝔪}^{∗}(S) ⇒ DS,

with differentials d_{r} : E_{r}^{s,t} ⟶ E_{r}^{s+r,t−r+1}. This is a spectral sequence of S-modules in the sense that d_{r} : E_{r}^{s,∗} ⟶ E_{r}^{s+r,∗} is a map of S-modules for all r and s.

We will outline the proof of the result in Section 10 below, but first we discuss some special cases. Example 9.2. (i) If G is an elementary abelian 2-group of rank r and k is of characteristic 2 then S = k[x_{1}, …, x_{r}]. It is easy to calculate the local cohomology from the stable Koszul complex, and we deduce

H_{𝔪}^{∗}(S) = H_{𝔪}^{r}(S) = Σ^{−r}DS.

More generally, whenever S is Cohen-Macaulay of dimension r, we find

H_{𝔪}^{∗}(S) = H_{𝔪}^{r}(S)

and hence the spectral sequence collapses to give

H_{𝔪}^{r}(S) = Σ^{−r}DS,

and hence S is Gorenstein. This recovers a theorem of Benson and Carlson [7]. (ii) The simplest non Cohen-Macaulay example is the semidihedral group of order 16, of dimension 2 and depth 1. The spectral sequence again collapses since the E_{2}-term is concentrated in two adjacent columns. (iii) The simplest example with Cohen-Macaulay defect 2 is the group Γ_{7}a_{2}. The cohomology is calculated by Rusin [50], and the relevant features are highlighted by Benson and Carlson [7]. One may show that the spectral sequence does not collapse, and one may identify the differential.

Turning next to compact Lie groups, recall the conjugation of G induces the adjoint representation G ⟶ k^{∗}, and let k(ad) denote k with this action. Theorem 9.3. [9] If S = H^{∗}(BG; k) for a compact Lie group G of dimension d, then there is a spectral sequence

H_{𝔪}^{∗}(S) ⇒ Σ^{−d}H_{∗}(BG; k(ad)),

of S-modules where k(ad) is the coefficient system in which each element g of the group acts on k by +1 or −1 according to whether conjugation by g preserves or reverses the orientation in a neighbourhood of the identity. Remark 9.4. When G is orientable the spectral sequence is just like the finite case with a shift

H_{𝔪}^{∗}(S) ⇒ Σ^{−d}H_{∗}(BG; k),

Example 9.5. (i) All the finite group examples discussed before are also examples here. (ii) If G is the circle group (of dimension d = 1) then S = k[x_{2}] and if G = SU(2) (of dimension 3) then S = k[x_{4}]. Both groups are connected, so that k(ad) = k. These examples show how the dimension shift Σ^{−d} comes into play. (iii) We have already commented that the adjoint representation of O(2n) is non-trivial if char(k) ≠ 2. This accounts for the fact that the local cohomology of H^{∗}(BO(2n)) = k[p_{1}, …, p_{n}] is a copy of its dual shifted by −2n(n + 1), whereas the dimension shift for an orientable group of the same dimension (namely n(2n − 1)) and rank (namely n) would be −n − n(2n − 1). The difference of 2n is explained by the non-trivial adjoint action.

Finally, we turn to virtual duality groups. Theorem 9.6. [8] If S = H^{∗}(BG; k) for a (discrete or profinite) virtual duality group G of dimension n there is a spectral sequence

H_{𝔪}^{∗}(S) ⇒ Σ^{n}H_{∗}(G; I),

of S-modules where I is the dualizing module. Remark 9.7. When G is an orientable Poincaré duality group the spectral sequence is just like the finite case with a shift

H^{∗}(S) ⇒ Σ^{n}H_{∗}(BG; k),

We will discuss the implications of these theorems for the structure of the ring S in the Section 11, and in Section 10 we sketch a proof.

10. An algebraic proof of the local cohomology theorem for ordinary cohomology.

In this section we outline the proof of the local cohomology theorem in the case of finite groups. This is the proof given in [26]; it is possible to give topological proofs as in Part I or Part III and another proof is given in [19].

In fact we will prove a more general result for cohomology of a kG-module M. This states that there is a spectral sequence

H_{𝔪}^{∗}(H^{∗}(G; M)) ⇒ H_{∗}(G; M).

The case M = k recovers the theorem. Similarly there are versions of the results for other classes of groups with modules of coefficients.

The simplest case is when G has periodic cohomology. This is known to be a very restrictive assumption, but once this case is clear we should be able to describe the proof in general. The assumption means that there is a codegree n and an element x ∈ S of codegree n so that multiplication by x gives an isomorphism S^{m} ≅ S^{m+n} whenever m ≥ 0. Thus S ⊇ k[x]. Now we may view x as an extension class x ∈ Ext_{kG}^{n}(k, k) and represent it by an exact sequence

0 ⟶ k ⟶ C_{n−1} ⟶ C_{n−2} ⟶ ⋯ ⟶ C_{1} ⟶ C_{0} ⟶ k ⟶ 0.

If we begin a minimal resolution the C_{0}, C_{1}, …, C_{n−2} are projective by construction, and C_{n−1} is projective since H^{n}(G) ≅ H^{0}(G) = k. We thus consider the complex

C(x) = C_{n−1} ⟶ C_{n−2} ⟶ ⋯ ⟶ C_{1} ⟶ C_{0}

and form a projective resolution P of k by concatenating copies of C(x), as we may do because H_{n−1}(C(x)) = k = H_{0}(C(x)). We thus let

P = [… ⟶ C_{n−1} ⟶ C_{n−2} ⟶ ⋯ ⟶ C_{1} ⟶ C_{0} ⟶

C_{n−1} ⟶ C_{n−2} ⟶ ⋯ ⟶ C_{1} ⟶ C_{0} ⟶ C_{n−1} ⟶ C_{n−2} ⟶ ⋯ ⟶ C_{1} ⟶ C_{0} ] .

Now we may write the stable Koszul complex

S ⟶ lim_{→}_{k} 1/x^{k}S = S[1/x].

It may help to visualize the term 1/x^{k}S as a copy of S, displayed vertically, so that the bottom is in degree −kn, and the maps in the direct system just include each column in the next. We model this at the level of resolutions, at least after reversing arrows, to obtain

P = L_{0} ⟵ L_{1} = lim_{←}_{k} L[−kn, ∞) = lim_{←}_{k} Σ^{−kn}P.

Again, one may view L[−kn, ∞) as the chain complex L, displayed vertically, with the bottom is in degree −kn, and the maps in the inverse system just project each column onto the previous one. Thus L_{•} = (L_{0} ⟵ L_{1}) is a double complex, and the proof proceeds by considering the double complex

X = Hom_{kG}(L_{•}, M)

There are two spectral sequences for calculating the cohomology of the double complex. The one in which we take Koszul cohomology first collapses to show

H_{∗}(X) = H_{∗}(H_{Koszul}^{∗}(Hom_{kG}(L_{•}, M))) = H_{∗}(Hom_{kG}(H_{∗}^{Koszul}(L_{•}), M))

since

H_{∗}^{Koszul}(L_{•}) = H_{1}(L_{•}) = P^{!},

where P^{!} = L_{1}/P is the part of L_{1} in negative codegrees. Now

Hom_{k}(P^{!}, M) = P ⊗_{k} M

so that

H_{∗}(X) = H_{∗}(G; M).

Taking the spectral sequence arising from the other filtration we obtain

H_{Koszul}^{∗}(H^{∗}(Hom_{kG}(L_{•}, M))) ⇒ H_{∗}(X) = H_{∗}(G; M).

This is the required spectral sequence. Indeed, by definition we have H^{∗}(Hom_{kG}(L_{0}, M)) = H^{∗}(G; M), and for L_{1} we calculate

H^{∗}(Hom_{kG}(L_{1}, M)) = H^{∗}(Hom_{kG}(lim_{←}_{n} L[−kn, ∞), M))
= lim_{→}_{n} H^{∗}(Hom_{kG}(L[−kn, ∞), M))
= lim_{→}_{n} 1/x^{n}H^{∗}(G, M))
= H^{∗}(G, M))[1/x],

where the second equality used the fact that the limit is achieved in each degree. Thus we obtain the stable Koszul complex,

(H^{∗}(Hom_{kG}(L_{•}, M))) = (H^{∗}(G; M) ⟶ H^{∗}(G; M)[1/x]).

The cohomology of this complex is the local cohomology of H^{∗}(G; M).

When G does not have periodic cohomology, Noether normalization shows that S is finite over a polynomial subring k[x_{1}, …, x_{r}]. We replace the single complex C(x) by C = C(x_{1})⊗ ⋯ ⊗ C(x_{r}), which we can view as an r-dimensional box. This time the top modules in each complex C(x_{i}) are not projective, but by the theory of support varieties [5, Chapter 5], because S is finite over the polynomial subring, their tensor product is projective. Now the proof is directly analogous. We mimic the construction of the multigraded Koszul complex by stacking boxes. For example with r = 2 we have

(L_{0} ⟵ L_{1} ⟵ L_{2}) = (P ([0, ∞)×[0, ∞)) ⟵ P ((−∞, ∞)×[0, ∞))⊕P ([0, ∞)×(−∞, ∞)) ⟵ P ((−∞, ∞)×(−∞, ∞))),

where P ([0, ∞) × [0, ∞)), for instance, is the result of stacking boxes in the first quadrant.

11. Structural implications of the local cohomology theorem.

This section summarizes the contents of [30], outlining the implications for a finitely generated k-algebra S of the existence of a spectral sequence

H_{𝔪}^{∗}(S) ⇒ Σ^{a}DS.

Duality is exact in the context so it is equivalent to say there is a spectral sequence

DH_{𝔪}^{∗}(S) ⇒ Σ^{−a}S.

We say that S has a local cohomology theorem with shift a, or that it is an LCT^{a} ring.

First we repeat the immediate observation that if S is Cohen-Macaulay of dimension r then the spectral sequence collapses to give an isomorphism

DH_{𝔪}^{r}(S) = Σ^{r−a}S,

so that S is Gorenstein.

Next, consider the case that S has Cohen-Macaulay defect 1 (in the sense that its depth is one less than its dimension). We write Ω = DH_{𝔪}^{r}(S) for the canonical module, and Λ = DH_{𝔪}^{r−1}(S) for the subcanonical module. Thus the spectral sequence collapses to the short exact sequence

0 ⟶ Σ^{a−r+1}Λ ⟶ S ⟶ Σ^{a−r}Ω ⟶ 0.

Proposition 11.1. If S is an LCT^{a} ring with Cohen-Macaulay defect 1 then • Ω has depth r and H_{𝔪}^{r}(Ω) = Σ^{r−a}DΩ, and

• Λ has depth r − 1 and H_{𝔪}^{r−1}(Λ) = Σ^{r−1−a}DΛ. It is natural to say that a ring with Cohen-Macaulay defect 1 is almost Cohen-Macaulay

and to say that it is almost Gorenstein if it satisfies the conclusion of the proposition. Proof: Note first that the depth statements are equivalent to the vanishing of local cohomology up to the top degree. To give the idea of proof, we will just explain the vanishing of lower local cohomology groups. The isomorphisms for the top local cohomology groups follow by extending the analysis one more step.

Grothendieck’s spectral sequence

E_{2}^{p,q} = H_{𝔪}^{p}(DH_{𝔪}^{r−q}(S)) ⇒ DS

(which applies to an arbitrary commutative ring) gives H_{𝔪}^{i}(Λ) = H_{𝔪}^{i+2}(Ω) for i ≤ r − 3. Applying the local cohomology to the short exact sequence from the local cohomology theorem gives H_{𝔪}^{i−1}(Ω) ≅ ΣH_{𝔪}^{i}(Λ) for i ≤ r − 2. Combining these gives the statements about depth. □

Remark 11.2. It is worth recording some consequences of these results for an arbitrary LCT^{a} ring. First, we recall that Grothendieck defined the dual localization functor L_{℘}M = D(D(M)_{℘}), which is useful for Artinian modules like local cohomology modules. It is exact, and an easy consequence of local duality is the fact that if ℘ is a prime with S/℘ of dimension d then L_{℘}H_{𝔪}^{i}(M) = H_{𝔪}^{i−d}(M_{℘}). Accordingly, we may apply L_{℘} to see that if S is an LCT^{a} ring then S_{℘} is an LCT^{a−d} ring. Hence we conclude that for any minimal prime ℘, the ring S_{℘} is Gorenstein (one says S is Gorenstein in codimension 0), and that if ℘ is of height 1, the ring S_{℘} is almost Gorenstein (one says S is almost Gorenstein in codimension 1).

On a more concrete level, if S is a finitely generated k-algebra in codegrees ≥ 0, we may obtain conditions on its Hilbert series [S](t) = Σ_{i} dim_{k}(S_{i})t^{i}. Corollary 11.3. (i) (Stanley [53]) If S is a Cohen-Macaulay LCT^{a} ring then

[S](1/t) = t^{−a}(−t)^{r}[S](t).

(ii) If S is an almost Cohen-Macaulay LCT^{a} ring then

[S](1/t) − t^{−a}(−t)^{r}[S](t) = (−1)^{r−1}(1 + t)[Λ](t)

and

[Λ](1/t) = t^{a}(−t)^{1−r}[Λ](t).

Remark 11.4. In [30] the functional equation for Λ is misstated: two signs are negated, one in the statement and one in the proof. Proof: To give the idea, we present the proof of Part (i) in a way that suggests that of Part (ii), referring the reader to [30] for further details. We first observe that the result is elementary for a polynomial ring. Now by Noether normalization we may find a polynomial subring S̃ ⊆ S over which S is a finitely generated module. We may work entirely with S̃-modules in the rest of the proof. By the Auslander-Buchsbaum formula, since S is Cohen-Macaulay, S = F_{0} ⊗_{k} S̃ as S̃-modules, where F_{0} is a finite dimensional graded vector space. Thus for Hilbert series, [S] = [S̃][F_{0}]. Now calculate

[S](1/t) = [F_{0}^{∨}][S̃](1/t) = (−1)^{r}[F_{0}^{∨}][DH_{𝔪}^{r}(S̃)] = (−1)^{r}[DH_{𝔪}^{r}(S)] = (−1)^{r}t^{r−a}[S],

where the last equality is the local cohomology theorem. The proof in the almost Cohen-Macaulay case uses exactly the same ingredients, now working with a short exact sequence 0 ⟶ F_{1} ⊗_{k} S̃ ⟶ F_{0} ⊗_{k} S̃ ⟶ S ⟶ 0. □

Example 11.5. We consider the semidihedral group SD_{16}. It is of dimension 2 and has Hilbert series f(t) = 1/(1 − t)^{2}(1 + t^{2}). We calculate

f(1/t) − (−t)^{2}f(t) = \frac{t^{4}}{(1 − t)^{2}(1 + t^{2})} − \frac{t^{2}}{(1 − t)^{2}(1 + t^{2})} = −(1 + t)\frac{t^{2}}{(1 − t)(1 + t^{2})}.

It is then easy to check that with δ(t) = t^{2}/(1 − t)(1 + t^{2}) we have

δ(1/t) = (−t)^{−1}δ(t)

as required.

It is interesting to note that f(t) also satisfies the single functional equation

f(1/t) = t^{2}(−t)^{2}f(t)

just as if it was the the Hilbert series of a Cohen-Macaulay LCT ring with shift a = −2.

Part III. Examples related to bordism.

There is a rather general class of cohomology theories for which one can give a uniform treatment very close to ideas from commutative algebra. Curiously, these examples arise from the study of a certain class of manifolds, but these geometric antecedents are not relevant here. Even in topology, Quillen’s theorem [48] that there is an intimate relation between bordism and the algebraic theory of formal groups means that much discussion of bordism is conducted in purely algebraic terms.

12. The theorem for MU-modules.

There is an equivariant version of bordism constructed from certain manifolds with group action. The cohomology theory MU_{G}^{∗}(·) is called equivariant (homotopical) complex cobordism. Its value MU_{G}^{∗}(X) on a G-space X is a module over the coefficient ring S = MU_{G}^{∗}. Theorem 12.1. (Greenlees-May [37]) If M_{G}^{∗}(·) is module valued over MU_{G}^{∗} then for any finite group there are spectral sequences

H_{J}^{∗}(M_{∗}^{G}(X)) ⇒ M_{∗}^{G}(EG × X) = M_{∗}(EG ×_{G} X)

and

H_{∗}^{J}(M_{G}^{∗}(X)) ⇒ M_{G}^{∗}(EG × X) = M^{∗}(EG ×_{G} X)

where J = ker(MU_{G}^{∗} ⟶ MU^{∗}) is the augmentation ideal. Remark 12.2. (i) In fact MU_{G}^{∗} is not Noetherian, and J is not known to be of finite arithmetic rank. Accordingly there is work involved in showing the initial terms of the spectral sequences make sense: one shows that H_{J′}^{∗}(·) is independent of J′ for all sufficiently large finitely generated ideals J′ ⊆ J. (ii) We shall only be applying this in the case that M_{G}^{∗} is a Noetherian ring in its own right, and one may replace J with J_{M} = ker(M_{G}^{∗} ⟶ M^{∗}). We view this as an example in the form of Context 7.1 by replacing S with S_{M} = M_{G}^{∗}, and k with k_{M} = M^{∗}.

(iii) The sense in which M_{G}^{∗}(·) is required to be module valued will be explained in Section 13. Example 12.3. The easiest class of examples to describe are those which are J-complete in a suitable sense. Fortunately these are the ones we intend to discuss later.

For this class we first define the non-equivariant theory. Indeed the coefficient ring of complex cobordism is a polynomial ring

MU^{∗} = ℤ[x_{1}, x_{2}, x_{3}, …]

on infinitely many variables, where x_{i} has degree 2i. For any flat module M_{∗} over MU_{∗} the definition

M_{∗}(X) = MU_{∗}(X) ⊗_{MU_{∗}} M_{∗}

gives an exact functor of X, and is therefore a homology theory, and hence represented by a spectrum M. For topological reasons the flatness condition can be weakened to something much more easily verified (Landweber exact functor theorem). There are also many module valued theories for which M_{∗} is not flat, but other means must be used to construct them. From any such non-equivariant theory one may form the complete theory

M_{G}^{∗}(X) = M^{∗}(EG ×_{G} X).

One family of well known examples are the complete Johnson-Wilson theories M = E_{n} with

E_{n}^{∗} = ℤ_{p}^{∧}[[u_{1}, …, u_{n−1}]][u, u^{−1}],

where the u_{i} are of degree 0 and u is of degree 2. This may be constructed using the Landweber exact functor theorem. One may show that for any finite group G, E_{n}^{∗}(BG) is finite over E_{n}^{∗}.

Another example has ku^{∗} = ℤ[x_{1}], and here ku^{∗}(BG) is only finite over a completion of ku^{∗}. The coefficient ring does not satisfy the hypotheses of the Landweber exact functor theorem, so other methods must be used to construct it.

We will return to an algebraic investigation of these two last examples in Sections 15 and 16, but first we spend Section 13 explaining the idea of the proof of the the local cohomology theorem for Noetherian MU-algebras. This is so closely analogous to commutative algebra in the derived category that it inspires a number of definitions of interest in algebra as well as topology, and we briefly introduce them in Section 14.

13. The proof.

The main thing is that there is a good category to work in [20]. Working there, the proof is essentially formal and just like working in an algebraic derived category. This is the category of G-spectra, briefly introduced in Subsection 13.1. In Subsection 13.2 we transpose some commutative algebra into the category of G-spectra and in the final two subsections we complete the proof.

13.1. G-spectra. A G-spectrum may be thought of as a generalized based G-space. The purpose of the generalization is to ensure that any equivariant cohomology theory E_{G}^{∗}(·) is represented in the category of G-spectra. This means that there is a G-spectrum E so that for any unbased G-space X,

E_{G}^{∗}(X) = [X_{+}, E]_{G}^{∗}

where the expression on the right denotes G-homotopy classes of G-maps of G-spectra. Sufficiently well behaved cohomology theories (such as MU) are represented by ring objects R in the category of G-spectra, and there is a category of module spectra over R. These rings R are analogous to differential graded algebras, and the homotopy category of modules over R is directly analogous to the derived category of differential graded modules over the differential graded ring. This homotopy category is where we work. Thus for a module M over R, and an unbased G-space X,

M_{G}^{∗}(X) = [X_{+}, M]_{G}^{∗} = [R ∧ X_{+}, M]_{R,G}^{∗}

and

M_{∗}^{G}(X) = [S^{0}, X_{+} ∧ M]_{∗}^{G} = [R, X_{+} ∧ M]_{∗}^{R,G},

where the decoration R, G refers to equivariant homotopy classes of R-module maps. Warning 13.1. Even if R_{G}^{∗}(·) is represented by a ring R, it is not automatic that the representing G-spectrum M of a module valued cohomology theory M_{G}^{∗}(·) is an R-module. However this is true in many cases, and specifically for the complete theories formed from E_{n} and ku discussed above.

The proof of the the local cohomology theorem described in this section does not in fact apply to R = MU. It requires us to work with a cohomology theory R_{G}^{∗}(·) represented by a ring G-spectrum R, with two properties. Firstly, the coefficient ring R_{G}^{∗} must be Noetherian (and for all subgroups H the modules R_{H}^{∗} must be finitely generated), and secondly, it must be complex oriented (this is equivalent to saying that R is an MU-algebra up to homotopy, but it has a more concrete meaning that will be described at the appropriate point). The coefficient ring of MU is not Noetherian, so the main obstacle is the construction of enough elements of the ideal J: this is interesting but not relevant to our applications. In fact the complete theories of both E_{n} and ku are both represented by ring spectra R to which the argument of this section does apply.

13.2. Commutative algebra with G-spectra. In the category of modules over R one can mimic most constructions in the derived category. For example we can construct the homotopy I-power torsion functor. If x ∈ R_{∗}^{G} = [R, R]_{∗}^{G,R} we may form

Γ_{(x)}R := fibre(R ⟶ R[1/x])

and then if I = (x_{1}, …, x_{n}) we take

Γ_{I}R := Γ_{(x_{1})}R ∧_{R} Γ_{(x_{2})}R ∧_{R} … ∧_{R} Γ_{(x_{n})}R.

Up to equivalence, Γ_{I}R does not depend on the generators used, and it only depends on the radical of the ideal I: this is an easy exercise exactly as in the algebraic derived category of modules over a commutative ring (see Section 6). The case of a principal ideal is constructed as a fibre and so comes with a filtration of length 1, and hence Γ_{I}R has a filtration of length n. Because the construction is modelled on the stable Koszul comples, it is clear that the homotopy spectral sequence of the filtration takes the form

E_{2}^{∗,∗} = H_{I}^{∗}(M_{G}^{∗}) ⇒ [R, Γ_{I}M]_{∗}^{R,G}.

To prove the local cohomology theorem it therefore suffices to establish the two equivalences

R ∧ EG_{+} ⟵^{A} Γ_{J}R ∧ EG_{+} ⟶^{B} Γ_{J}R

of equivariant R-modules.

We may deduce the local cohomology theorem by applying (M ∧ X_{+}) ∧_{R} (·) to obtain the equivalence

M ∧ EG_{+} ∧ X_{+} ≃ Γ_{J}R ∧_{R} M ∧ X_{+}.

The homotopy of the left hand side is M_{∗}^{G}(EG × X) and that of the right hand side is calculated by a spectral sequence with E_{2} term H_{J}^{∗}(M_{∗}^{G}(X)).

We may deduce the completion theorem by applying Hom_{R}(· ∧ X_{+}, M) to obtain the equivalence

Hom_{R}(R ∧ EG_{+} ∧ X_{+}, M) ≃ Hom_{R}(Γ_{J}R ∧ X_{+}, M).

The homotopy of the left hand side is M_{G}^{∗}(EG × X), and the homotopy of the right hand side may be calculated by a spectral sequence with E_{2}-term H_{∗}^{J}(M_{G}^{∗}(X)).

It remains to prove that the maps A and B are equivalences of equivariant R-modules.

13.3. The map A is an equivalence. Since EG_{+} is free, it is built from free G-cells S^{n} ∧ G_{+}, so it suffices to show that R ∧ G_{+} ⟵ Γ_{J}R ∧ G_{+} is an equivalence, which is to say that R ⟵ Γ_{J}R is a non-equivariant equivalence. But res_{1}^{G}J = 0 by definition, so

res_{1}^{G}Γ_{J}R = Γ_{res_{1}^{G}J}R = Γ_{0}R = R

as required.

13.4. The map B is an equivalence. We show the mapping cone of B is contractible. In fact we may assume by induction on the group order that it is H-contractible for all proper subgroups H. This uses the fact that res_{H}^{G}J has the same radical as J(H) = ker(R_{H}^{∗} ⟶ R^{∗}) as follows in the Noetherian setting from [28]. From this we deduce

res_{H}^{G}Γ_{J}R = Γ_{res_{H}^{G}J}R = Γ_{J(H)}R.

Now the mapping cone of B is Γ_{J}R ∧ ẼG where ẼG is the mapping cone of EG_{+} ⟶ S^{0}. Because (ẼG)^{G} = S^{0}, there is unique map ẼG ⟶ S^{∞V} which is the identity in G-fixed points, where V is the reduced regular representation. It suffices to show that Γ_{J}R ∧ S^{∞V} is contractible. Indeed, if we define Q by the cofibre sequence

ẼG ⟶ S^{∞V} ⟶ Q,

we see that Q is built from cells G/H_{+} for proper subgroups H. It follows from the inductive hypothesis Γ_{J}R ∧ Q ≃ ∗.

Since Γ_{J}R∧S^{∞V} is obviously H-contractible for all proper subgroups H it suffices to show its G-homotopy is zero. Now for any R-module M we may define Euler classes χ(V) exactly as was done for K-theory in Section 6. Indeed, it is immediate from the construction of MU that MU ∧S^{V} ≃ MU ∧S^{|V |}, (where |V | denotes V with the trivial G-action) so that we may take χ(V) to be the pullback of 1 ∈ MU_{G}^{|V |} = [S^{|V |}, MU]_{G}^{∗} under the inclusion S^{0} ⟶ S^{V}. From the definition we see that

π_{∗}^{G}(M ∧ S^{∞V}) = lim_{→}_{n} π_{∗}^{G}(M ∧ S^{nV}) = π_{∗}^{G}(M)[1/χ(V)].

On the other hand, since the inclusion S^{0} ⟶ S^{V} is non-equivariantly nullhomotopic, χ(V) ∈ J so that inverting it kills J-local cohomology and therefore π_{∗}^{G}(Γ_{J}R ∧ S^{∞V}) = 0.

This completes the proof. □

14. Homotopically Gorenstein rings.

For further details see [29, 19].

To smooth the transition between topology and commutative algebra we now write k^{∗}(X) = E^{∗}(X) for the non-equivariant cohomology theory and S^{∗}(X) = E_{G}^{∗}(X) for the equivariant one. In view of the completion theorem we may also write Ŝ^{∗}(X) = E^{∗}(EG ×_{G} X) for the complete theory. We restrict attention to finite groups G, and therefore to the case with shift a = 0 (in the notation of Section 11).

We have obtained duality statements by combining the local cohomology theorem with the universal coefficient theorem. In the first instance these were spectral sequences

H_{J}^{∗}(k^{∗}(BG)) ⇒ k_{∗}(BG)

and

Ext_{k_{∗}}(k_{∗}(BG), k_{∗}) ⇒ k^{∗}(BG),

but we have already reformulated the first as

Γ_{J}F (EG_{+}, k) ≃ k ∧ EG_{+},

and the second can be written

Hom_{k}(k ∧ BG_{+}, k) ≃ F (BG_{+}, k),

which we would like to think of as the fixed points of an equivariant equivalence

Hom_{k}(k ∧ EG_{+}, k) ≃ F (EG_{+}, k).

The analogy is with the algebraic derived category, so Hom corresponds to the total right derived functor of ordinary homomorphisms. Substituting in the local cohomology theorem expression for k ∧ EG_{+} ≃ F (EG_{+}, k) ∧ EG_{+}, and writing Ŝ = F (EG_{+}, k) we find

Hom_{k}(Γ_{J} Ŝ, k) ≃ Ŝ.

More generally we can consider this condition on any ring Ŝ up to homotopy equipped with ring maps k ⟶ Ŝ ⟶ k making it into a supplemented k-algebra. We may say that a supplemented ring Ŝ is homotopically Gorenstein if there is such an equivalence. If k is a field we may take homotopy and deduce that Ŝ_{∗} is an LCT ring. If k is not a field the left hand side is the composite of two functors. Each of these functors could be calculated with a spectral sequence, but it is harder to extract information about their composite. However when k is of small injective dimension we can still get striking duality properties for Ŝ_{∗}.

For complete rings it is quite often equivalent to consider the statement

Γ_{J} Ŝ ≃ Hom_{k}(Ŝ, k),

and this is better behaved for non-complete rings, so we may say that a supplemented kalgebra S is homotopically Gorenstein if

Γ_{J}S ≃ Hom_{k}(S, k).

It turns out [19] that under quite weak hypotheses this is equivalent to requiring

Hom_{S}(k, S) ≃ k

as modules over Hom_{S}(k, k), which may be a more familiar form of the Gorenstein condition. This is related to the notion of Gorenstein differential graded commutative algebras considered by Avramov and Foxby [4], and to the ideas of Félix, Halperin and Thomas [23] but they only require the isomorphism as S-modules. This circle of ideas is investigated further by Dwyer, Greenlees and Iyengar [18, 19].

15. The chromatic case.

For further details see [39].

We have explained that it is of interest to calculate the local cohomology modules H_{J}^{∗}(S) where S = R_{G}^{∗} = E_{n}^{∗}(BG), because they give the E_{2}-term of a spectral sequence for calculating the more subtle invariant (E_{n})_{∗}(BG). Playing this spectral sequence off against the universal coefficient theorem implies that the ring R_{G}^{∗} has very special duality properties.

Now here we have k = R^{∗} = ℤ_{p}^{∧}[[u_{1}, …, u_{n−1}]][u, u^{−1}], and we may consider the Landweber sequence of prime ideals

0 ⊆ (p) ⊆ (p, u_{1}) ⊆ (p, u_{1}, u_{2}) ⊆ ⋯ ⊆ (p, u_{1}, …, u_{n−1}).

It turns out that a natural ring of endomorphisms acts on R^{∗} and that the Landweber sequence is the unique maximal sequence of invariant prime ideals. For any module M we may form the Cousin complex

C(M) = (M[1/p] ⟶ M/(p^{∞})[1/u_{1}] ⟶ M/(p^{∞}, u_{1}^{∞})[1/u_{2}] ⟶ ⋯ ⟶ M/(p^{∞}, u_{1}^{∞}, …, u_{n−1}^{∞}).

If (p, u_{1}, …, u_{n−1}) is M-regular we say M is good; in this case M ≃ C(M). It is then natural to use this filtration to approach the calculation of local cohomology. Theorem 15.1. [39] If E_{n}^{∗}(BG) is good, then local cohomology is trivial on pure chromatic strata in the sense that

H_{J}^{∗}(E_{n}^{∗}(BG)) = H^{∗}(Γ_{J}C(E_{n}^{∗}(BG)). □

Remark 15.2. (i) The module E_{n}^{∗}(BG) is known to be good in many cases, including all abelian groups G, and all symmetric groups [42]. (ii) The complex Γ_{J}C(E_{n}^{∗}(BG)) is highly non-trivial. Indeed, Γ_{J}C(E_{n}^{∗}(BG)) is usually nonzero up to degree n: for example if G is abelian then in degree i it is |G|^{i}C_{i}(E_{n}^{∗}). On the other hand if G is abelian of rank r, then J has arithmetic rank ≤ r, and hence the local cohomology is trivial above degree r. If n > r the exactness of the complex above degree r must involve interesting differentials. (iii) The case of periodic K-theory gives something we have already seen. It states

H_{J}^{∗}(R(G)_{p}^{∧}) = H^{∗}(Γ_{J}R(G)_{p}^{∧}[1/p] ⟶ Γ_{J}R(G)/p^{∞}) = H^{∗}(ℤ_{p}^{∧}[1/p] ⟶ R(G)/p^{∞}).

This is very effective, and recovers the local cohomology calculations of Section 4.

16. Connective K theory.

For further information see [12, Chapter 4].

In this section we present a very concrete example where the commutative algebra is very intricate and very striking. The local cohomology theorem states there is a spectral sequence

H_{J}^{∗}(ku^{∗}(BG)) ⇒ ku_{∗}(BG),

for any finite group G, but we want to discuss a case where the entire behaviour is understood. We take G = V an elementary abelian 2-group of rank r. In fact there is a short exact sequence

0 ⟶ T ⟶ ku^{∗}(BV) ⟶ Q ⟶ 0.

Here Q is the extended Rees ring for the J-completion R̂(V) of the representation ring

R(V) = ℤ[α_{1}, …, α_{r}]/(α_{1}^{2} = ⋯ = α_{r}^{2} = 1).

This completion has the effect of 2-adic completion on J, and leaving alone the ℤ summand generated by the regular representation. Thus

Q = R̂(V)[v, y_{1}, y_{2}, …, y_{r}] ⊆ R̂(V)[u, u^{−1}]

where y_{i} = (1−α_{i})/u, and we may take J to be generated by elements mapping to y_{1}, …, y_{r}. One may calculate H_{J}^{∗}(Q) without too much difficulty: H_{J}^{0}(Q) = ℤ[u] and H_{J}^{1}(Q) is zero below degree −2r and the order of its ℤ-torsion increases with degree.

On the other hand T is a bit more complicated. It is a P -submodule of 𝔽_{2}[x_{1}, …, x_{r}] where P = 𝔽_{2}[y_{1}, …, y_{r}] and x_{i}^{2} = y_{i}. It turns out that it is generated by elements the q_{S} = ∏_{i∈S} x_{i}(Σ_{j,k∈S}x_{j}x_{k}^{3} as S ranges over subsets of {1, 2, …, r} with at least 2 elements.

Now we can decompose T = T_{2} ⊕ T_{3} ⊕ ⋯ ⊕ T_{r}, where T_{i} is generated by the q_{S} with |S| = i. These P -modules T_{i} show some remarkable behaviour.

Proposition 16.1. (i) T_{i} is of dimension r and depth i. (ii) H_{𝔪}^{j}(T_{i}) = 0 unless j = i or r. (iii) H_{𝔪}^{i}(T_{i})^{∨} is 1-dimensional if i < r. (iv) H_{𝔪}^{r}(T_{i})^{∨} = T_{r−i+2}(−r + 4) if i < r.

(v) H_{𝔪}^{r}(T_{r})^{∨} = P (−r + 4). □

Thus in codimension r − 2, the module T is CM and H^{r}(T)^{∨} = T (−r + 4): it is very nearly Gorenstein.

However it turns out the structure of ku^{∗}(BV) is bound even more tightly.

Lemma 16.2. The modules H_{𝔪}^{i}(T_{i})^{∨} are the subquotients in the 2-adic filtration of H_{J}^{1}(Q):

H_{𝔪}^{i}(T_{i})^{∨}(i − 2) = (2^{i−2}H_{J}^{1}(Q)/2^{i−1}H_{J}^{1}(Q))^{∨}

for i = 2, 3, … r − 1. □

It turns out this is just what is required for ku_{∗}(BV) to be well-behaved. There is a short exact sequence

0 ⟶ Σ^{−4}T^{∨} ⟶ k̃u_{∗}(BV) ⟶ Σ^{−1}(2^{r−1}H_{J}^{1}(Q)) ⟶ 0

and furthermore this corresponds under the universal coefficient theorem to the exact sequence above for ku^{∗}(BV) in the sense that

Ext_{ku_{∗}}^{2}(Σ^{−2}T^{∨}, ku_{∗}) = T

and

Ext_{ku_{∗}}^{1}(2^{r−1}H_{J}^{1}(Q), ku_{∗}) = Q/(ℤ[u]).

This shows that even when we are not working over a field, the implications of the homotopy Gorenstein condition are very striking.

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[18] W.G.Dwyer and J.P.C.Greenlees “The equivalence of categories of complete and torsion modules.” Preprint (1999) 20pp

[19] W.G.Dwyer, J.P.C.Greenlees and S.B.Iyengar “Duality in algebra and topology.” (In preparation) 14pp [20] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May. Rings, Modules and Algebras in Stable Homotopy

Theory, Volume 47 of Amer. Math. Soc. Surveys and Monographs. American Mathematical Society, 1996.

[21] A.D.Elmendorf, J.P.C.Greenlees, I.Kriz and J.P.May “Commutative algebra in stable homotopy theory and a completion theorem.” Mathematical Research Letters 1 (1994) 225-239.

[22] A. D. Elmendorf, and J. P. May “Algebras over equivariant sphere spectra” J. Pure and Applied Algebra 116 (1997) 139-149

[23] Y. Félix, S. Halperin and J.-C. Thomas “Gorenstein spaces” Advances in Mathematics, 71, (1988) 92-112

[24] J.P.C.Greenlees “K-homology of universal spaces and local cohomology of the representation ring” Topology 32 (1993) 295-308.

[25] J.P.C.Greenlees “Tate cohomology in commutative algebra.” J. Pure and Applied Algebra 94 (1994) 59-83

[26] J.P.C.Greenlees “Commutative algebra in group cohomology.” J.Pure and Applied Algebra 98 (1995) 151-162

[27] J.P.C.Greenlees “An introduction to equivariant K-theory.” CBMS Regional Conference Series 91 American Math. Soc. (1996) 143-152.

[28] J.P.C.Greenlees “Augmentation ideals of equivariant cohomology rings.” Topology 37 (1998) 1313-1323 [29] J.P.C.Greenlees “Tate cohomology in axiomatic stable homotopy theory” Preprint 16pp.

[30] J.P.C.Greenlees and G.Lyubeznik “Rings with a local cohomology theorem and applications to cohomology rings of groups.” J. Pure and Applied Algebra 149 (2000) 267-285.

[31] J.P.C.Greenlees and J.P. May “Completions of G-spectra at ideals of the Burnside ring” Proc. Adams Memorial Conference II, Cambridge University Press (1992) 145-178.

[32] J.P.C.Greenlees and J.P. May “Derived functors of I-adic completion and local homology” Journal of Algebra 149 (1992) 438-453.

[33] J.P.C.Greenlees and J.P.May “Completions in algebra and topology” Handbook of Topology (ed. I.M.James) North Holland (1995) 255-276.

[34] J.P.C.Greenlees and J.P.May “Equivariant stable homotopy theory.” Handbook of Topology (ed. I.M.James) North Holland (1995) 277-323.

[35] J.P.C.Greenlees and J.P.May “Brave new equivariant algebra.” CBMS Regional Conference Series 91 American Math. Soc. (1996) 299-314.

[36] J.P.C.Greenlees and J.P.May “Localization and completion in complex bordism.” CBMS Regional Conference Series 91 American Math. Soc. (1996) 315-326.

[37] J.P.C.Greenlees and J.P.May “Localization and completion theorems for MU-module spectra.” Annals of Maths. 146 (1997) 509-544

[38] J.P.C.Greenlees and H. Sadofsky “The Tate spectrum of v_{n}-periodic complex oriented theories.” Math. Zeits. 222 (1996) 391-405.

[39] J.P.C.Greenlees and N.P.Strickland “Varieties and local cohomology for chromatic group cohomology rings.” Topology 38 (1999) 1093-1139.

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Department of Pure Mathematics, Hicks Building, Sheffield S3 7RH. UK.

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