MODULES WITH INJECTIVE COHOMOLOGY, AND LOCAL DUALITY FOR A FINITE GROUP

DAVID BENSON

1. Introduction

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et G be a finite group and k be a field of characteristic p. By a theorem of Evens and Venkov [10, 23], the cohomology ring H^{∗}(G, k) is a finitely generated graded commutative k-algebra. Further restrictions on the structure of H^{∗}(G, k) can be expressed in terms of the existence of certain spectral sequences. The first of these was constructed by Benson and Carlson [2] using multiple complexes and related

finite Poincaré duality complexes of projective kG-modules. It follows from the existence of this spectral sequence that if H^{∗}(G, k) is Cohen–Macaulay, then it is Gorenstein. Even if H^{∗}(G, k) is not Cohen–Macaulay, the existence of the spectral sequence gives severe restrictions on the possibilities for the ring structure. An example of an application of this spectral sequence is the statement that if H^{∗}(G, k) is a polynomial ring, then it follows that p = 2, the generators are in degree one, and G/O(G) is an elementary abelian 2-group.

Greenlees [11] found a way of using the same techniques coming from multiple complexes to construct a cleaner spectral sequence expressing much of the same information. This spectral sequence can be written as

H_{𝔪}^{∗} H^{∗}(G, k) ⇒ H_{∗}(G, k).

Here, 𝔪 denotes the maximal ideal generated by the elements of positive degree in H^{∗}(G, k), and H_{𝔪}^{∗} denotes local cohomology in the sense of Grothendieck [13, 14]. Greenlees and Lyubeznik [12] showed how to apply dual localization to this spectral sequence, with respect to a homogeneous prime ideal 𝔭 of H^{∗}(G, k). This leads to a spectral sequence whose E_{2} page is

H_{𝔭}^{∗} H^{∗}(G, k)_{𝔭}

and which converges to the injective hull, in the graded sense, of H^{∗}(G, k)/𝔭 as a H^{∗}(G, k)-module, suitably shifted in degree.

The first purpose of this paper is to provide a construction of the Greenlees spectral sequence using Rickard’s idempotent modules [22]. This construction is essentially a stabilized version of a construction which Carlson and Wheeler [8] introduced in the context of complexity quotient categories. The construction is given in Section 3, where the following theorem is made more precise.

Theorem 1.1. There is a double complex consisting of homomorphisms from projective kGmodules to Rickard idempotent modules, whose spectral sequence is isomorphic to the Greenlees spectral sequence, from the E_{1} page onwards.

Date: March 3, 2001.

The author is partly supported by a grant from the NSF.

We give an interpretation in terms of this spectral sequence of the “last survivor” method of section 7 of [2]. The result is a slightly stronger theorem, the proof of which can be found in Section 4.

Theorem 1.2. Let ζ_{1}, …, ζ_{r} be a homogeneous set of parameters for H^{∗}(G, k), and let a_{i} = deg(ζ_{i}) (1 ≤ i ≤ r). Then there is an element of degree ∑_{i=1}^{r}(a_{i} − 1) in H^{∗}(G, k) which is not in the ideal (ζ_{1}, …, ζ_{r}) and is not annihilated by any nonzero element of k[ζ_{1}, …, ζ_{r}].

In Section 5, we provide an analagous construction of a spectral sequence with the same E_{2} page as the Greenlees–Lyubeznik dual localized spectral sequence. We have been unable to prove that these two spectral sequences are isomorphic, although this seems very likely. Our spectral sequence converges to the Tate cohomology Ĥ^{∗}(G, κ_{𝔭}) with coefficients in the module κ_{𝔭} introduced in introduced by Benson, Carlson and Rickard [3] (in that paper, the module is called κ(V), where V is the closed homogeneous irreducible subvariety of the cohomology variety V_{G} corresponding to 𝔭).

The modules κ_{𝔭} are interesting because they are idempotent, in the sense that κ_{𝔭} ⊗ κ_{𝔭} ≅ κ_{𝔭} in the stable module category, and tensoring with κ_{𝔭} picks out the “layer” in the stable module category corresponding to the prime 𝔭 ⊆ H^{∗}(G, k).

If it could be proved that the two spectral sequences are isomorphic, then it would follow that Ĥ^{∗}(G, κ_{𝔭}) is an injective H^{∗}(G, k)-module. This would allow the modules κ_{𝔭} to be compared with the modules T (I_{𝔭}) introduced by Benson and Krause [5]. The second purpose of this paper is to investigate some special cases in which we are able to make this comparison. We give evidence for the following conjecture.

Conjecture 1.3. Let 𝔭 ≠ 𝔪 be a homogeneous prime ideal of dimension d in H^{∗}(G, k). Then T (I_{𝔭}) and Ω^{−d}κ_{𝔭} are stably isomorphic.

The modules T (I_{𝔭}) are the representing objects for the contravariant exact functor which takes a module to the Matlis dual of its Tate cohomology, in the sense that there are functorial isomorphisms

D_{𝔭}Ĥ^{∗}(G, M) ≅ Êxt_{kG}^{∗}(M, T (I_{𝔭})).

In particular, the T (I_{𝔭}) are pure injective modules, and their cohomology is given by Ĥ^{∗}(G, T (I_{𝔭})) ≅ I_{𝔭}, the injective hull of H^{∗}(G, k)/𝔭.

The reason for the interest in the above conjecture is that what we know about the properties of T (I_{𝔭}) and of Ω^{−d}κ_{𝔭} have little overlap. That a single module should have all these properties seems to have be significant. It seems likely that there are many consequences for the interactions between commutative algebra and modular representation theory.

So for example, it would follow from this conjecture that κ_{𝔭} is pure injective, so that there are no phantom maps into κ_{𝔭}. It also follows that Ĥ^{∗}(G, κ_{p}) ≅ I_{𝔭}[d], and that End_{k}(κ_{𝔭}) is a kG-module whose Tate cohomology is the completion H^{∗}(G, k)𝔭^{∧} of the localized cohomology ring with respect to the powers of its maximal ideal. Another consequence of the conjecture is the formula

T (I_{𝔭}) ⊗_{k} T (I_{𝔭}) ≅ Ω^{−d}T (I_{𝔭})

in the stable module category.

We prove the above conjecture in the case where H^{∗}(G, k)_{𝔭} is Cohen–Macaulay of Krull dimension r−d, and also in the case where the depth is r−d−1 and the Krull dimension is r−d. In particular, since any localization of a Cohen–Macaulay ring is Cohen–Macaulay, the conjecture holds whenever H^{∗}(G, k) is Cohen–Macaulay. The proof of the following theorem can be found in Section 7.

Theorem 1.4. Let G be a finite group of p-rank r, and let k be a field of characteristic p. Suppose that 𝔭 ≠ 𝔪 is a homogeneous prime ideal in H^{∗}(G, k) of Krull dimension d, such that the localization H^{∗}(G, k)_{𝔭} satisfies one of the following conditions:

(*) H^{∗}(G, k)_{𝔭} is Cohen–Macaulay of Krull dimension r − d (i.e., its depth is also equal to r − d), or

(**) H^{∗}(G, k)_{𝔭} has depth r − d − 1 and Krull dimension r − d.

Then the following hold:

(i) The cohomology Ĥ^{∗}(G, κ_{𝔭}) is injective in the category of graded H^{∗}(G, k)-modules; in particular, it is isomorphic to I_{𝔭}[d].

(ii) The modules T (I_{𝔭}) and Ω^{−d}κ_{𝔭} are stably isomorphic.

(iii) The modules κ_{𝔭} are pure injective.

(iv) There are no phantom maps into κ_{𝔭}.

(v) The ring Êxt_{kG}^{∗}(κ_{𝔭}, κ_{𝔭}) ≅ Ĥ^{∗}(G, End_{k}(κ_{𝔭})) is isomorphic to the completion H^{∗}(G, k)𝔭^{∧} of the localized cohomology H^{∗}(G, k)_{𝔭} with respect to the maximal ideal 𝔭_{𝔭}.

(vi) The formula T (I_{𝔭}) ⊗ T (I_{𝔭}) ≅ Ω^{−d}T (I_{𝔭}) holds in the stable module category.

If H^{∗}(G, k) is Cohen–Macaulay, then (*) holds for H^{∗}(G, k)_{𝔭}. Furthermore, H^{∗}(G, k) is Cohen– Macaulay in (at least) the following cases:

(a) G has abelian Sylow p-subgroups,

(b) G = GL_{n}(𝔽_{q}) with q a power of a prime different from p,

(c) p = 2 and G has extraspecial Sylow 2-subgroups,

(d) p = 2 and G is a finite simple group of 2-rank at most three, or isomorphic to U_{3}(2^{n}) or Sz(2^{2n+1}),

(e) p = 3 and G has extraspecial Sylow 3-subgroups of order 3^{1+2} and exponent 3.

Finally, if G has p-rank at most two, then either (*) or (**) is satisfied for every prime ideal 𝔭.

Background material on commutative algebra can be found in Matsumura [16]. References on local cohomology include Brodmann and Sharp [6], Bruns and Herzog [7], and Grothendieck [13, 14].

Conventions and notations. We assume throughout that G is a finite group of order divisible by p, and that k is a field of characteristic p. When we talk about modules and maps over a ℤ-graded ring, we always mean graded modules unless otherwise specified. So for example when we talk about injective hulls, these are taken in the category of graded modules. This is not the same as injective hulls in the category of all modules. Also, when we use the term “ideal”, we mean graded ideal unless otherwise specified. If M is a graded module, we write M[n] for the graded module with M[n]_{i} = M_{i+n}.

Acknowledgements. I would like to take this opportunity to thank Jon Carlson, Henning Krause and Jeremy Rickard for the interest they have taken in this work.

2. Depth

Much of this paper is concerned with the depth and the Krull dimension of the cohomology ring, and various localizations of it. We begin with a statement that must be well known, but for which it does not seem easy to find a reference in the literature.

Theorem 2.1. Let P be a Sylow p-subgroup of G, and k a field of characteristic p. Then the depth of H^{∗}(G, k) is at least as big as the depth of H^{∗}(P, k).

Proof. There are several ways to prove this theorem. One way is to notice that because of the existence of the transfer map, H^{∗}(G, k) is a direct summand of H^{∗}(P, k) as an H^{∗}(G, k)-module. Since H^{∗}(P, k) is finitely generates as an H^{∗}(G, k)-module, the depth of H^{∗}(P, k) as a ring is equal to its depth as an H^{∗}(G, k)-module. Its depth is therefore at most as big as the depth of the summand H^{∗}(G, k).

A localized version of the same theorem is also true.

Theorem 2.2. Let 𝔭 be a prime ideal of H^{∗}(P, k), and let 𝔮 = 𝔭 ∩ H^{∗}(G, k) be the prime ideal of H^{∗}(G, k) over which it lies. Then the depth of the localization H^{∗}(G, k)_{𝔮} is at least as big as the depth of H^{∗}(P, k)_{𝔭}.

Proof. The proof is essentially the same as the proof of Theorem 2.1, but this time, we need to observe that H^{∗}(G, k)_{𝔮} is a direct summand of H^{∗}(P, k)_{𝔭} as an H^{∗}(G, k)_{𝔮}-module.

3. The Greenlees spectral sequence

We begin by recalling what we need from Rickard’s theory of idempotent modules [22]. If 𝒞 is a thick subcategory of the stable category stmod(kG) of finitely generated kG-modules, then there is a triangle in StMod(kG) of the form

E_{𝒞} → k → F_{𝒞} → Ω^{−1}E_{𝒞}

where E_{𝒞} is a direct limit of modules in 𝒞, and F_{𝒞} is 𝒞-local in the sense that \underline{Hom}_{kG}(X, F_{𝒞}) = 0 for all modules X in 𝒞. The modules E_{𝒞} and F_{𝒞} are idempotent modules in the sense that each is stably isomorphic to its tensor square.

If 𝒱 is a collection of closed homogeneous irreducible subvarieties of the cohomology variety V_{G}, closed under specialization (i.e., if V ∈ 𝒱 and W ⊆ V then W ∈ 𝒱), let 𝒞_{𝒱} be the thick subcategory consisting of the modules M such that each irreducible component of V_{G}(M) is in 𝒱. In this case, we write E_{𝒱} and F_{𝒱} for the corresponding idempotent modules. Finally, if ζ ∈ H^{n}(G, k), write E_{ζ} and F_{ζ} for the idempotent modules corresponding to the hypersurface in V_{G} determined by ζ. The module F_{ζ} can be described as the colimit of

k ⟶^{ζ̂} Ω^{−n}k ⟶^{ζ̂} Ω^{−2n}k → …

where the maps ζ̂ are arbitrary representatives of ζ. In particular, Ĥ^{∗}(G, F_{ζ}) is the localization of H^{∗}(G, k) (or equivalently of Ĥ^{∗}(G, k)) obtained by inverting ζ. More generally, if 𝒱 is any nonempty collection generated by hypersurfaces, in the sense that each element of 𝒱 is contained in a hypersurface in 𝒱, then Ĥ^{∗}(G, F_{𝒱}) is the localization of H^{∗}(G, k) (or of Ĥ^{∗}(G, k), it doesn’t matter) obtained by inverting the elements corresponding to the hypersurfaces. Furthermore, for any kG-module M, Ĥ^{∗}(G, F_{𝒱} ⊗_{k} M) is the localization of Ĥ^{∗}(G, M) obtained by inverting the same elements.

Choose a homogeneous set of parameters ζ_{1}, …, ζ_{r} for H^{∗}(G, k). For each ζ_{i}, we have a cochain complex

0 → k → F_{ζ_{i}} → 0

where k is taken to be in degree zero and F_{ζ_{i}} in degree one. The cohomology of this complex is Ω^{−1}E_{ζ_{i}}, concentrated in degree one.

Tensoring these complexes together (over k, with diagonal G-action), we obtain a complex

Λ^{∗} = Λ^{∗}(ζ_{1}, …, ζ_{r}) : 0 → k → ⨁_{1≤i≤r} F_{ζ_{i}} → ⨁_{1≤i<j≤r} F_{ζ_{i}} ⊗_{k} F_{ζ_{j}} → … → ⨂_{1≤i≤r} F_{ζ_{i}} → 0.

By the Künneth theorem, this is exact except in degree r, where its cohomology is

Ω^{−1}E_{ζ_{1}} ⊗_{k} ⋯ ⊗_{k} Ω^{−1}E_{ζ_{r}}

which is a projective kG-module.

Now let P̂_{∗} be a Tate resolution of the trivial kG-module. In other words, P̂_{∗} is a complex of the form

\xymatrix{… \ar[r] & P_{2} \ar[r] & P_{1} \ar[r] & P_{0} \ar[rr] \ar[dr] &  & P_{−1} \ar[r] & P_{−2} \ar[r] & … \\  &  &  &  & k \ar[ur] \ar[dr] \\  &  &  & 0 \ar[ur] &  & 0}

obtained by splicing together a projective resolution and an injective resolution of the trivial module.

Consider the double complex Ê_{0}^{∗,∗} = Hom_{kG}(P̂_{∗}, Λ^{∗}). This gives rise to two spectral sequences. The spectral sequence where we take cohomology with respect to the differential coming from Λ^{∗} first has its E_{1} page concentrated on a single line, where it consists of

Hom_{kG}(P̂_{∗}, Ω^{−1}E_{ζ_{1}} ⊗_{k} ⋯ ⊗_{k} Ω^{−1}E_{ζ_{r}}).

The E_{2} page is then the Tate cohomology of this projective module, which is zero. It follows that the cohomology of the total complex Tot Ê_{0}^{∗,∗} is zero.

On the other hand, if we take cohomology with respect to the differential coming from P̂_{∗} first, then the E_{1} page is

Ê_{1}^{s,t} = Ĥ^{t}(G, Λ^{s}).

Now each Λ^{s} is a direct sum of modules of the form F_{ζ}, where ζ is a product of a subset of size s of ζ_{1}, …, ζ_{r}. The cohomology of this module is the localization Ĥ^{∗}(G, k)_{ζ}. The maps in the complex are the signed inclusions. So this is the stable Koszul complex

Ê_{1}^{∗,∗} = C^{∗}(Ĥ^{∗}(G, k); ζ_{1}, …, ζ_{r}).

See for example section 3.5 of Bruns and Herzog [7] for a description of the stable Koszul complex and the fact that it calculates local cohomology. Thus we have

Ê_{2}^{s,t} = H_{𝔪}^{s,t} Ĥ^{∗}(G, k) ⇒ 0.

Here, we have given local cohomology a double grading. The first grading is the local cohomological grading, and the second is the internal grading coming from the fact that Ĥ^{∗}(G, k) is a graded ring. The differentials in this spectral sequence are

d_{j}: Ê_{j}^{s,t} → Ê_{j}^{s+j,t−j+1}.

Now, this is not quite the spectral sequence which Greenlees [11] constructed, so we make the following modification. Let P̂_{∗}^{−} be the subcomplex

0 → P_{−1} → P_{−2} → …

of P̂_{∗}, and let P_{∗} be the quotient complex

… → P_{1} → P_{0} → 0.

Then we can consider Hom_{kG}(P̂_{∗}^{−}, k) as a quotient complex of Ê_{0}^{∗,∗} by a subcomplex which we call E_{0}^{∗,∗} of the following form.

\xymatrix{ & ⋮ \ar[d] & ⋮ \ar[d] &  & ⋮ \ar[d] \\ 0 \ar[r] & 0 \ar[r] \ar[d] & Hom_{kG}(P_{−2}, Λ^{1}) \ar[r] \ar[d] & ⋯ \ar[r] & Hom_{kG}(P_{−2}, Λ^{r}) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & 0 \ar[r] \ar[d] & Hom_{kG}(P_{−1}, Λ^{1}) \ar[r] \ar[d] & ⋯ \ar[r] & Hom_{kG}(P_{−1}, Λ^{r}) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Hom_{kG}(P_{0}, k) \ar[r] \ar[d] & Hom_{kG}(P_{0}, Λ^{1}) \ar[r] \ar[d] & ⋯ \ar[r] & Hom_{kG}(P_{0}, Λ^{r}) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Hom_{kG}(P_{1}, k) \ar[r] \ar[d] & Hom_{kG}(P_{1}, Λ^{1}) \ar[r] \ar[d] & ⋯ \ar[r] & Hom_{kG}(P_{1}, Λ^{r}) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Hom_{kG}(P_{2}, k) \ar[r] \ar[d] & Hom_{kG}(P_{2}, Λ^{1}) \ar[r] \ar[d] & ⋯ \ar[r] & Hom_{kG}(P_{2}, Λ^{r}) \ar[r] \ar[d] & 0 \\  & ⋮ & ⋮ &  & ⋮}

We have a short exact sequence of complexes

0 → Tot E_{0}^{∗,∗} → Tot Ê_{0}^{∗,∗} → Hom_{kG}(P̂_{∗}^{−}, k) → 0.

Since Tot Ê_{0}^{∗,∗} is exact, this gives

H^{n}(Tot E_{0}^{∗,∗}) ≅ H^{n+1}(Hom_{kG}(P̂_{∗}^{−}, k)).

The cohomology of Hom_{kG}(P̂_{∗}^{−}, k) is the negative part of Tate cohomology, shifted in degree by one, Ĥ^{−}(G, k)[−1], which is the same as ordinary homology with degrees negated. So we have

H^{n}(Tot E_{0}^{∗,∗}) ≅ \begin{cases}H_{−n}(G, k) & n ≤ 0 \\ 0 & n > 0. \end{cases}

Now, localizing Tate cohomology at any nonzero element kills all negative degree elements, and so the result is the same as the localization of ordinary cohomology. It follows that if we take cohomology with respect to the vertical differential first, the E_{1} page of the spectral sequence is the stable Koszul complex for ordinary cohomology,

E_{1}^{∗,∗} = C^{∗}(H^{∗}(G, k); ζ_{1}, …, ζ_{r}).

Thus we have

E_{2}^{s,t} = H_{𝔪}^{s,t} H^{∗}(G, k) ⇒ H_{−s−t}(G, k).

This is the required module theoretic construction of the Greenlees spectral sequence.

Note that as a module over H^{∗}(G, k), the homology H_{∗}(G, k) is just the injective hull I_{𝔪} of k = H^{∗}(G, k)/𝔪. So another way of writing this spectral sequence is

E_{2}^{∗,∗} = H_{𝔪}^{∗} H^{∗}(G, k) ⇒ I_{𝔪}.  (3.1)

4. The last survivor

In this section, we give an interpretation of the last survivor of section 7 of [2] in terms of the Greenlees spectral sequence (3.1).

Since H^{∗}(G, k) and H_{∗}(G, k) are vector space duals, the identity element of H^{0}(G, k) corresponds to an isomorphism α_{G} : H_{0}(G, k) → k with the property that if i_{H,G} : H → G is the inclusion of a subgroup H of G then α_{G} ∘ (i_{H,G})_{∗} = α_{H}.

Now suppose that G has p-rank r, so that by a theorem of Quillen [19, 20], H^{∗}(G, k) has Krull dimension r. Then H_{𝔪}^{i} H^{∗}(G, k) = 0 for i > r, and so there is a right-hand edge homomorphism for the Greenlees spectral sequence (3.1)

H_{𝔪}^{r,t} H^{∗}(G, k) → H_{−r−t}(G, k).

This is an isomorphism precisely when H^{∗}(G, k) is Cohen–Macaulay. Composing with α_{G}, we obtain a homomorphism

γ_{G} : H_{𝔪}^{r,−r} H^{∗}(G, k) → k.

If H is a subgroup of G then transfer in negative Tate cohomology Ĥ^{−n}(H, k) → Ĥ^{−n}(G, k) is the same as the map (i_{H,G})_{∗} : H_{n−1}(H, k) → H_{n−1}(G, k) induced by the inclusion i_{H,G} : H → G. It follows that transfer induces a homomorphism of spectral sequences

\xymatrix{H_{𝔪}^{s,t} H^{∗}(H, k) \ar@{=>}[r] \ar[d]^{(Tr_{H,G})_{∗}} & H_{−s−t}(H, k) \ar[d]^{(i_{H,G})_{∗}} \\ H_{𝔪}^{s,t} H^{∗}(G, k) \ar@{=>}[r] & H_{−s−t}(G, k)}  (4.1)

Now let E be an elementary abelian p-subgroup of G of maximal rank r. Then H^{∗}(E, k) is Cohen–Macaulay of Krull dimension r, and so the edge homomorphism is an isomorphism in this case. So the edge homomorphisms fit into a commutative diagram

[figure: H_{𝔪}^{r,−r}H^{∗}(E, k); ≅; (Tr_{E,G})_{∗}; H_{0}(E, k); (i_{E,G})_{∗}; α_{E}; k; H_{𝔪}^{r,−r}H^{∗}(G, k); H_{0}(G, k); α_{G}; k]

It follows that

γ_{G} ∘ (Tr_{E,G})_{∗} = γ_{E}.  (4.2)

Since the top horizontal map in the above commutative square is an isomorphism, it follows that γ_{E} is nonzero, and hence also γ_{G} is nonzero.

Theorem 4.1. There is a nonzero canonical homomorphism

γ_{G}: H_{𝔪}^{r,−r}H^{∗}(G, k) → k.

A good way of interpreting this theorem is in terms of Grothendieck duality [13, 14]. Let R be a polynomial subring k[ζ_{1}, …, ζ_{r}] of H^{∗}(G, k) over which the latter is finitely generated as a module (i.e., a Noether normalization of H^{∗}(G, k)). Set a_{i} = deg(ζ_{i}), and a = ∑_{i=1}^{r} a_{i}. Then the dualizing module for R is R[−a]. So Grothendieck duality for R states that for any finitely generated Rmodule M, the graded vector space dual of H_{𝔪}^{i,∗}(M) is isomorphic to Ext_{R}^{r−i,−∗}(M, R[−a]). So for example, dualizing the spectral sequence (3.1) gives a spectral sequence

Ext_{R}^{r−s,−t}(H^{∗}(G, k), R[−a]) ⇒ H^{−s−t}(G, k).

Reindexing gives

Ext_{R}^{s,t}(H^{∗}(G, k), R[−a + r]) ⇒ H^{s+t}(G, k).

Similarly, dualizing the transfer diagram of spectral sequences (4.1) gives

\xymatrix{Ext_{R}^{s,t}(H^{∗}(G, k), R[−a + r]) \ar@{=>}[r] \ar[d]^{_{Tr{∗ \atop H,G}}} & H^{s+t}(G, k) \ar[d]^{_{res_{H,G}}} \\ Ext_{R}^{s,t}(H^{∗}(H, k), R[−a + r]) \ar@{=>}[r] & H^{s+t}(H, k),}

provided that R is also a Noether normalization of H^{∗}(H, k) via restriction, which happens precisely when H has p-rank r. Since Ext^{0} = Hom, this means that the element γ_{G} can be interpreted as an element of

Hom_{R}^{∗}(H^{∗}(G, k), R[−a])

of degree r (dualizing negates the degree), or in other words as a degree preserving R-module homomorphism

γ̃_{G}: H^{∗}(G, k) → R[−a + r].

In this guise, the transfer formula (4.2) now reads

γ̃_{G} ∘ Tr_{E,G} = γ̃_{E}.  (4.3)

The homomorphism γ̃_{E} is easy to interpret. As an R-module, H^{∗}(E, k) is free on a finite set of homogeneous generators. These generators lie in degrees zero through a − r = ∑_{i=1}^{r}(a_{i} − 1), and exactly one generator, say u, lies in the largest degree a − r. It follows that γ̃_{E} must vanish on all the generators except u, and must take u to some scalar in k times the identity element of R[−a+r]. In particular, γ̃_{E} is surjective. It follows from the transfer formula that γ̃_{G} is also surjective.

Theorem 4.2. There is a surjective R-module homomorphism γ̃_{G} : H^{∗}(G, k) → R[−a + r].

In particular, choosing an inverse image under γ̃_{G} for the identity element of R, we see that H^{∗}(G, k) has a nonzero element in degree a − r which is not in the ideal (ζ_{1}, …, ζ_{r}), and is not annihilated by multiplication by any nonzero element of R. Such an element can be chosen to be a transfer from E. It is not unique, but it is called the “last survivor”.

In fact, this interpretation of the last survivor gives more information than was given in section 7 of [2], because in that context it was not at all clear that the last survivor is not annihilated by any nonzero element of R.

5. The kappa modules

Let 𝔭 ≠ 𝔪 be a prime ideal in H^{∗}(G, k), corresponding to an irreducible subvariety V of the cohomology variety V_{G}. Let 𝒱 denote the set of subvarieties W satisfying W ⊉ V. Then the kappa module κ_{𝔭} is defined to be E_{V} ⊗ F_{𝒱}. Actually, this is not quite the definition given in [3] (where the notation κ(V) is used), but is easily seen to be equivalent. This definition is better suited to our purposes, because 𝒱 is determined by the hypersurfaces in it, and this makes Ĥ^{∗}(G, F_{𝒱}) easier to calculate. Namely, we have

Ĥ^{∗}(G, F_{𝒱}) = H^{∗}(G, k)_{𝔭},

the localization of H^{∗}(G, k) obtained by formally inverting all the elements not in 𝔭.

Now let h be the height of 𝔭, which is equal to the Krull dimension of the localization H^{∗}(G, k)_{𝔭}. Using the form of Noether normalization proved in Nagata [17], we can choose a homogeneous system of parameters ζ_{1}, …, ζ_{r} for H^{∗}(G, k) in such a way that ζ_{1}, …, ζ_{h} lie in 𝔭. This means that the images of ζ_{1}, …, ζ_{h} in H^{∗}(G, k)_{𝔭} form a system of parameters there. Consider the complex

Λ^{∗}(ζ_{1}, …, ζ_{h}) : 0 → k → ⨁_{1≤i≤h} F_{ζ_{i}} → ⨁_{1≤i<j≤h} F_{ζ_{i}} ⊗_{k} F_{ζ_{j}} → … → ⨂_{1≤i≤h} F_{ζ_{i}} → 0

obtained by tensoring together the complexes

0 → k → F_{ζ_{i}} → 0

(1 ≤ i ≤ h) as in Section 3. The cohomology of this complex is

Ω^{−1}E_{ζ_{1}} ⊗_{k} ⋯ ⊗_{k} Ω^{−1}E_{ζ_{h}} ≅ Ω^{−h}(E_{ζ_{1}} ⊗_{k} ⋯ ⊗_{k} E_{ζ_{h}}) ⊕ (projective) ≅ Ω^{−h}E_{V} ⊕ (projective),

concentrated in degree h. So defining

Λ_{𝔭}^{∗} = Λ^{∗}(ζ_{1}, …, ζ_{h}) ⊗ F_{𝒱} ,

the cohomology of Λ_{𝔭}^{∗} is

Ω^{−h}E_{V} ⊗_{k} F_{𝒱} ≅ Ω^{−h}κ_{𝔭} ⊕ (projective)

concentrated in degree h.

Now consider the double complex obtained by taking homomorphisms from a Tate resolution of k to Λ_{𝔭}^{∗},

E_{0}^{∗,∗}(𝔭) = Hom_{kG}(P̂_{∗}, Λ_{𝔭}^{∗}).

If we take cohomology with respect to the differential coming from Λ_{𝔭}^{∗} first, the E_{1} page is concentrated in horizontal degree h, and is equal to Hom_{kG}(P̂_{∗}, Ω^{−h}κ_{𝔭}). So the E_{2} page is also concentrated in horizontal degree h, and is equal to Ĥ^{∗}(G, Ω^{−h}κ_{𝔭}) = Ĥ^{∗}(G, κ_{𝔭})[h]. So we have

H^{∗}(Tot E_{0}^{∗,∗}(𝔭)) ≅ Ĥ^{∗}(G, κ_{𝔭}).

On the other hand, if we take cohomology with respect to the differential coming from P̂_{∗} first, then we have

E_{1}^{s,t}(𝔭) = Ĥ^{t}(G, Λ_{𝔭}^{s}).

This is the stable Koszul complex

E_{1}^{∗,∗}(𝔭) = C^{∗}(H^{∗}(G, k)_{𝔭}; ζ_{1}, …, ζ_{h}).

So doing the differential coming from Λ_{𝔭}^{∗} gives us the E_{2} page as local cohomology with respect to 𝔭 of the localized cohomology ring. So we have

E_{2}^{∗,∗}(𝔭) = H_{𝔭}^{∗} H^{∗}(G, k)_{𝔭} ⇒ Ĥ^{∗}(G, κ_{𝔭}).  (5.1)

6. Dual localization

There is another way to obtain a spectral sequence with E_{2} page isomorphic to H_{𝔭}^{∗} H^{∗}(G, k)_{𝔭}, due to Greenlees and Lyubeznik [12]. Namely, they use Matlis duality twice, in a process which they call “dual localization”. If 𝔭 is a prime ideal in H^{∗}(G, k), and X is a module over H^{∗}(G, k)_{𝔭}, then we write D_{𝔭}(X) for the Matlis dual of X (Matlis [15]),

D_{𝔭}(X) = Hom_{H^{∗}(G,k)_{𝔭}}^{∗}(X, I_{𝔭}).

Here, I_{𝔭} denotes the injective hull of H^{∗}(G, k)/𝔭 as an H^{∗}(G, k)-module. The module I_{𝔭} is in a natural way a module for the completion H^{∗}(G, k)𝔭^{∧} of H^{∗}(G, k)_{𝔭} with respect to the powers of the maximal ideal 𝔭_{𝔭}. So D_{𝔭} takes H^{∗}(G, k)_{𝔭}-modules to H^{∗}(G, k)𝔭^{∧}-modules. It takes Artinian modules to Noetherian modules and vice-versa. Applying D_{𝔭} twice to an Artinian module returns the same module, whereas applying D_{𝔭} twice to a Noetherian module returns its 𝔭-completion.

The functor T from injective H^{∗}(G, k)-modules to the stable category of kG-modules, defined in Benson and Krause [5], is designed to deal with Matlis duality in this setting. If M is a kG-module, it gives an isomorphism (lemma 3.2 of [5])

D_{𝔭}Ĥ^{∗}(G, M) ≅ Êxt_{kG}^{∗}(M, T (I_{𝔭}))  (6.1)

which may be viewed as a generalization of Tate duality. Namely, in case 𝔭 = 𝔪, the ideal generated by the homogeneous elements of positive degree, D_{𝔪} is the same as (graded) vector space duality over k, and T (I_{𝔪}) is just Ω(k) (lemma 3.1 of [5]).

Grothendieck duality [13, 14] implies that applying D_{𝔪} to the Greenlees spectral sequence (3.1) gives a spectral sequence

Ext_{R}^{r−∗,−∗}(H^{∗}(G, k), R[−a]) ⇒ H^{∗}(G, k).

Localization with respect to 𝔭 is exact, and so we obtain a spectral sequence

Ext_{R_{𝔮}}^{r−∗,−∗}(H^{∗}(G, k)_{𝔭}, R_{𝔮}[−a]) ⇒ H^{∗}(G, k)_{𝔭},

where 𝔮 = 𝔭 ∩ R, which is a prime ideal in R. Now if d is the Krull dimension of H^{∗}(G, k)/𝔭, then H^{∗}(G, k)_{𝔭} has Krull dimension r − d. So Grothendieck duality implies that applying D_{𝔭} to the above spectral sequence gives

H_{𝔭}^{∗−d,∗} H^{∗}(G, k)_{𝔭} ⇒ I_{𝔭}.

Regrading, we obtain

H_{𝔭}^{∗,∗} H^{∗}(G, k)_{𝔭} ⇒ I_{𝔭}[d].  (6.2)

This is the Greenlees–Lyubeznik spectral sequence.

In the Cohen–Macaulay case, this immediately calculates Ĥ^{∗}(G, κ_{𝔭}) for us, by comparing the spectral sequences (5.1) and (6.2). In the case where H^{∗}(G, k)_{𝔭} is not Cohen–Macaulay, it would be desirable to have a more direct way to compare these spectral sequences.

Theorem 6.1. Suppose that H^{∗}(G, k)_{𝔭} is Cohen–Macaulay. Then Ĥ^{∗}(G, κ_{𝔭}) ≅ I_{𝔭}[d].

Proof. If H^{∗}(G, k)_{𝔭} is Cohen–Macaulay then local cohomology is concentrated in a single degree. Namely, H_{𝔭}^{s,t} H^{∗}(G, k)_{𝔭} = 0 for s ≠ d. So the spectral sequences (5.1) and (6.2) are each concentrated in a single column, and there are no differentials or ungrading problems. Thus the two spectral sequences give isomorphisms

Ĥ^{t}(G, κ_{𝔭}) ≅ H_{𝔭}^{d,t−d} H^{∗}(G, k)_{𝔭} ≅ I_{𝔭}[d]_{t}.

7. Comparison

We would like to compare the spectral sequences (5.1) and (6.2) and prove that they are isomorphic, so that Ĥ^{∗}(G, κ_{𝔭}) ≅ I_{𝔭}[d]. Once this is achieved, the next stage would be to prove that if M is any kG-module satisfying Ĥ^{∗}(G, M) = 0, then \underline{Hom}_{kG}(M, κ_{𝔭}) = 0. This would enable us to apply proposition 4.2 of Benson and Krause [5] to deduce that T (I_{𝔭}) ≅ Ω^{−d}κ_{𝔭}, and Conjecture 1.3 would be proved.

We have been unable to carry out this program in general. Instead, we explain an alternative route which proves Conjecture 1.3 under some extra hypotheses.

Applying D_{𝔭} to the spectral sequence (5.1) and using (6.1), we obtain a spectral sequence

Ext_{R_{𝔮}^{∧}}^{r−d−s,−t}(H^{∗}(G, k)𝔭^{∧}, R𝔮^{∧}[−a]) ⇒ Êxt_{kG}^{−s−t}(κ_{𝔭}, T (I_{𝔭})).

Reindexing gives

Ext_{R_{𝔮}^{∧}}^{s,t}(H^{∗}(G, k)𝔭^{∧}, R𝔮^{∧}[−a + r]) ⇒ Êxt_{kG}^{s+t}(Ω^{−d}κ_{𝔭}, T (I_{𝔭})).  (7.1)

The desired isomorphism between Ω^{−d}κ_{𝔭} and T (I_{𝔭}) should be an element of degree zero in the right hand side of (7.1). There is an obvious candidate in the E_{2} page, namely the image γ̃_{G,𝔭} of the element γ̃_{G} of Section 4 under 𝔭-completion. The problem is that it is not at all obvious that this element should be a universal cycle, without a more direct comparison of the spectral sequences.

Theorem 7.1. Let 𝔭 be a prime ideal in H^{∗}(G, k) corresponding to a homogeneous irreducible subvariety V which is contained in an irreducible component of V_{G} of maximal dimension (equal to the p-rank of G).

Suppose that γ̃_{G,𝔭} is a universal cycle in the spectral sequence (7.1). Then any representative in

Êxt_{kG}^{0}(Ω^{−d}κ_{𝔭}, T (I_{𝔭})) = \underline{Hom}_{kG}(Ω^{−d}κ_{𝔭}, T (I_{𝔭}))

of γ̃_{G,𝔭} is a stable isomorphism.

Proof. Write ρ : Ω^{−d}κ_{𝔭} → T (I_{𝔭}) for a representative of γ̃_{G,𝔭}, and complete ρ to a triangle

Ω^{−d}κ_{𝔭} → T (I_{𝔭}) → U → Ω^{−d−1}κ_{𝔭}

in StMod(kG). By theorem 7.3 of [5], the variety of T (I_{𝔭}) is

𝒱_{G}(T (I_{𝔭})) = {V }.

By lemma 10.4 of [3], we also have

𝒱_{G}(Ω^{−d}κ_{𝔭}) = {V }.

So it follows that 𝒱_{G}(U) ⊆ {V }.

Let E be a maximal elementary abelian subgroup of G such that the image V_{G,E} of V_{E} → V_{G} is an irreducible component of maximal dimension (equal to the p-rank of G) containing V. Then we have a diagram of spectral sequences

[figure: Ext_{R_{𝔮}^{∧}}^{s,t}(H^{∗}(G, k)𝔭^{∧}, R𝔮^{∧}[−a + r]); _{Tr{∗ \atop E,G}}; Êxt_{kG}^{s+t}(Ω^{−d}κ_{𝔭}, T (I_{𝔭})); _{res_{G,E}}; Ext_{R_{𝔮}^{∧}}^{s,t}(H^{∗}(E, k)𝔭^{∧}, R𝔮^{∧}[−a + r]); Êxt_{kE}^{s+t}(Ω^{−d}κ_{𝔭}↓_{E}, T (I_{𝔭})↓_{E}).]

Now by lemma 8.2 of [3], we have

κ_{𝔭}↓_{E} ≅ ⨁_{res_{G,E}^{−1}(𝔓)=𝔭} κ_{𝔓}.

Here, 𝔓 runs over the (finite) set of prime ideals in H^{∗}(E, k) whose inverse image under restriction are equal to 𝔭. This set corresponds to the set of irreducible components in V_{E} of the inverse image of V under res_{G,E}^{∗}. Similarly, by proposition 7.1 of [5], we have

T (I_{𝔭})↓_{E} ≅ T (r_{G,E}(I_{𝔭})) ≅ ⨁_{res_{G,E}^{−1}(𝔓)=𝔭} T (I_{𝔓}).

Now H^{∗}(E, k) is Cohen–Macaulay, so the spectral sequence

Ext_{R_{𝔮}^{∧}}^{s,t}(H^{∗}(E, k)𝔭^{∧}, R𝔮^{∧}[−a + r]) ⇒ Êxt_{kE}^{s+t}(Ω^{−d}κ_{𝔭}↓_{E}, T (I_{𝔭})↓_{E})

degenerates to an isomorphism

Hom_{R_{𝔮}^{∧}}^{t}(H^{∗}(E, k)𝔭^{∧}, R𝔮^{∧}[−a + r]) ≅ Êxt_{kE}^{t}(Ω^{−d}κ_{𝔭}↓_{E}, T (I_{𝔭})↓_{E}) ≅ ⨁_{res_{G,E}^{−1}(𝔓)=𝔭} Êxt_{kE}^{t}(Ω^{−d}κ_{𝔓}, T (I_{𝔓})).

By equation (4.3) we have γ̃_{G} ∘ Tr_{E,G} = γ̃_{E}, or Tr_{E,G}^{∗}(γ̃_{G}) = γ̃_{E}. So the element Tr_{G,E}^{∗}(γ̃_{G,𝔭}) is the sum of the elements γ̃_{E,𝔓}, each of which, regarded as an element of \underline{Hom}_{kE}(Ω^{−d}κ_{𝔓}, T (I_{𝔓})), is a stable isomorphism. So the restriction of ρ, which represents Tr_{E,G}^{∗}(γ̃_{G}), is a stable isomorphism Ω^{−d}κ_{𝔭}↓_{E}→ T (I_{𝔭})↓_{E}. It follows that U ↓_{E} is projective. Since 𝒱_{G}(U) ⊆ {V }, this forces 𝒱_{G}(U) = ∅ by theorem 10.6 of [3]. It then follows, using section 10 of [3], that U is projective as a kG-module, so that ρ is a stable isomorphism.

We now turn to the proof of Theorem 1.4. If H^{∗}(G, k)_{𝔭} is Cohen–Macaulay, then it is equidimensional, and the spectral sequence (7.1) is concentrated on a single column. In this case, the conditions of the above theorem are satisfied: every irreducible subvariety is contained in a component of maximal dimension, and γ̃_{G,𝔭} is always a universal cycle.

If the depth and Krull dimension of H^{∗}(G, k)_{𝔭} differ by one, and if 𝔭 also contains a minimal prime of dimension r, then it still follows that γ̃_{G,𝔭} is a universal cycle, since the spectral sequence is concentrated in two adjacent columns and there is no room for nonzero differentials. So the conditions of the theorem are satisfied in case (**) of Theorem 1.4.

We have now shown that if condition (*) or (**) of Theorem 1.4 hold then any representative ρ : Ω^{−d}κ_{𝔭} → T (I_{𝔭}) of γ̃_{G,𝔭} is a stable isomorphism, thereby proving that part (ii) of the theorem holds. Since Ĥ^{∗}(G, T (I_{𝔭})) is injective in the category of stable H^{∗}(G, k)-modules, a degree shift shows that part (i) of the theorem holds. By theorem 5.1 of [5], the modules T (I_{𝔭}) are pure injective. They are even Σ-pure injective. So part (iii) of the theorem holds. By theorem 1.1.4 of [4], this implies that part (iv) holds. Finally, part (v) of the theorem follows from corollary 3.7 of [5] and part (vi) follows from part (ii) and the idempotent property for κ_{𝔭}.

The fact that H^{∗}(G, k) is Cohen–Macaulay in the cases (a)–(e) follows from the work of a number of people [1, 9, 18, 21]. Finally, if G has p-rank two, then every maximal elementary abelian psubgroup has p-rank two, and so H^{∗}(G, k) is equidimensional with depth one or two. It follows that (**) holds in this case.

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Department of Mathematics, University of Georgia, Athens GA 30602, USA

E-mail address: djb@byrd.math.uga.edu