CALCULATING GROUP COHOMOLOGY: TESTS FOR COMPLETION

JON F. CARLSON

1. Introduction

Suppose that G is a finite p-group and that k is a field of characteristic p. In the past few years I have been writing computer programs for doing homological algebra in the category of kG-modules. I am particularly interested in computing projective resolutions of modules, chain maps between resolutions and cup products. A major focus has been the development of programs to calculate generators and relations for the cohomology ring H^{∗}(G, k) of a p-group. This project, which is described in Section 2 is similar to that outlined in [8]. All of my computer programs are written in the MAGMA language and supported by that system [4]. However stand-alone versions of a part of the system have also been written by David Green [12].

The main purpose of this paper is to address another problem which arises in the calculation. It is that the cohomology ring H^{∗}(G, k) is an infinite object but any calculation of it is necessarily finite. So the question is: when do we know that we are done? One method would be to check the calculated answer against various spectral sequences that might be available. For groups which are not too large, this method is not impractical. Indeed, Rusin’s [14] impressive calculations of the mod-2 chomology rings of the 51 groups of 32 were done using the Eilenberg-Moore spectral sequence. However many of the cohomology rings of the 267 groups of order 64 are much more complicated. Indeed the method is not very appealing for the reason that there are choice to be made in the spectral sequences that seem always ad hoc and the calculations must be done by hand. It would be much better to have a method which could be implemented on a computer.

The scheme which we propose in this paper is actually based on a couple of conjectures about the nature of the cohomology rings. Several justifications for this approach can be made. First of all, in the course of the calculation the computer checks that the cohomology rings really do satisfy the conjectures. Hence there is no ambiguity in the final result. The conjectures hold in all of the cases that we have looked at to this point. Moreover the conjectures have been proved for some classes of groups. However the most important justification is that checking conjectures is one of the aims of the project. In fact, some of the technicalities of the conjectures came from looking at the data of these calculations. We hope that the system will be useful for testing and refining other ideas.

The organization of the paper is roughly as follows. In section 2, we describe the calculation of the cohomology ring. The primary interest here is the explicit description of the output in terms of the actual structure of the cohomology ring. In Sections 3 and 4 we present the conditions, one for generators and one for relations, which are conjectured to hold for cohomology rings of p-groups. Finally in Section 5 we show how the two conditions are related through the hypercohomology spectral sequence. Roughly, the main theorem says that if both conditions are verified to hold out to a certain degree in the cohomology then they must hold in general and the calculation is complete.

∗ Partially supported by a grant from NSF.

For general references for group cohomology we refer the reader to [2], [7] or [11]. Mostly the reader will only need to know some standard homological algebra as in [13].

The functions for computing cohomology rings have also been used to calculate the cohomology of some sporadic simple groups [1] [9]. We hope that these functions will be included in the MAGMA package in the near future. We are trying also to run the mod-2 cohomology rings of all groups of order 64. Most of this has been accomplished though some few cohomology rings have been particularly difficult to get. The outputs from the calculations, that have been made up to this time, have been posted on the author’s web page (http://www.math.uga.edu/∼jfc).

I want to thank John Cannon and Allan Steele of the MAGMA project for many instances of help in providing support for the cohomology calculations. I also thank David Green for general help with the manuscript and for pointing out some minor errors.

2. Computing Cohomology

In this section we present a brief outline of the steps in computer calculations of a cohomology ring H^{∗}(G, k). Throughout the section and the rest of the paper we assume that G is a p-group and k is a field of characteristic p for p a prime number. As stated earlier, the methods we use are similar to the ones outlined in [8]. One of the main differences is that in [8] we replaced the group algebra with its basic algebra in order to reduce the size of the calculation. However, for a pgroup the group algebra is isomorphic to its basic algebra and there is nothing to be gained by making the replacement. Another difference is that we have chosen to do the computations numerically to the extent possible. That is, the elements of the group algebra are represented by vectors in a k-vector space rather than as classes in a polynomial ring. Hence products are achieved by matrix multiplication and the noncommutative Gröbner basis algorithms used in [8] are not required. This should have the advantage of giving faster running times, particularly for small fields. However more memory may be required for some calculations. It will be interesting to make comparisons once the new implementation of Ed Green’s Gröbner basis material is available.

The first step in the computation is the creation of a “standard” free module for the group algebra kG. That is, we construct a representation of kG as a kG-module which is standard in the sense that it is fixed and to be used for all homological calculations for G. Calculations performed with different standard free modules are not comparable.

Basically we create the module by first choosing for the group a set of PCgenerators x_{1}, …, x_{n}. This is a set of generators with the property that for each i between 1 and n the elements x_{i+1}, …, x_{n} generate a normal subgroup of order p^{n−i}. Next we create a vector space V of dimension |G|, the order of G, and let the j^{th} vector in the standard basis for V represent the element

Π(x_{i} − 1)^{j_{i}} ∈ kG

where j − 1 = j_{1} + j_{2}p + ⋯ + j_{n}p^{n−1}, for 0 ≤ j_{i} < p. That is, we are choosing a specific representation for the group algebra in such a way that each element of kG is represented by a vector of length |G| and determined by the PC-generators. The advantage to this representation is that if (as often happens) x_{i}^{p} = 1 then (x_{i} − 1)^{p} = 0. Therefore elements which are contained in a high power of the radical of kG are represented by vectors having a large number of zeros as entries.

Of course, the representation depends on the choice of the PC-generators. In the case that G is not given in PC-form it is important that these data be computed only once. If G is given by generators y_{1}, …, y_{r} then the ResolutionInput function, which creates the standard free module, returns the matrices of the actions of the elements y_{1}, …, y_{r} as well as the matrices of the computed PC-generators. To save time in other parts of the calculation it is useful to have both a minimal set and a PC-set of generators for G.

For the next stage, let M be a kG-module given by the action of matrices for y_{1}, …, y_{r} on the vector space of the module. Our aim is to get a minimal projective resolution of M. First find a basis for the radical

RadM = ∑_{i=1}^{r} M(y_{i} − 1)

This means finding a basis for the sum of the row spaces of the matrices of y_{1} − 1, …, y_{r} − 1. A minimal generating set for M as a kG-module is a basis for a subspace of M complementary to RadM.

Let m_{1}, …, m_{s} be a minimal set of generators for M. Let P_{0} = ⊕ ∑_{j=1}^{s} kGu_{j} be

a free kG-module with free kG-basis u_{1}, …, u_{s}. For practical purposes P_{0} is just a direct sum of s copies of the standard free module. A projective cover of M is the homomorphism

θ : P_{0} ⟶ M

given by θ(u_{i}) = m_{i} for i = 1, …, s. The matrix for θ is easily constructed, since the j^{th} vector in the standard basis for kG · u_{i} has image

θ(u_{i} · ∑(x_{i} − 1)^{j_{i}}) = m_{i} · ∑(x_{i} − 1)^{j_{i}}

for j − 1 = j_{1} + j_{2}p + ⋯ + j_{n}p^{n−1} as before.

Next we compute the kernel of θ, Ω(M), which as vector space is the null space of θ. Notice here that Ω(M) ≤ P_{0} so that it is not necessary to actually realize the null space as a kG-module. That is, the action of G is known from the standard free module, and we can avoid the costly process of making matrices for the generators of G on Ω(M).

Now that we have the null space we can get its radical and minimal set of generators by the same process as before. Also as before we construct a projective cover for Ω(M). From the composition we get the first step in the projective resolution

∂ : P_{1} ⟶ P_{0}

where P_{1} = ∑_{i=1}^{t} kGv_{i} is a projective cover of Ω(M). Notice that for each i, ∂(v_{i}) =

∑_{j=1}^{s} α_{ij}u_{j} for α_{ij} ∈ kG. The only thing which we record of ∂ is the list of vectors α_{ij}.

That is, the output of the ProjectiveResolution function consists of list of ranks, (s, t, etc) of the projective modules and the list of vectors α_{ij} which are stored as the rows of a large matrix.

If the process just described is repeated N times then the output represents a portion

P_{N} ⟶^{∂_{N}} P_{N−1} ⟶ ⋯ ⟶^{∂_{2}} P_{1} ⟶^{∂_{1}} P_{0} ⟶ M ⟶ 0

of the minimal projective resolution of M. The matrices of the boundary maps can be easily recreated from the stored data.

The next stage is the computation of chain maps for the purposes of determining cup products. At present these routines are only implemented for case that M = k is the trivial kG-module. In this case, because of the minimality of the resolution and the irreducibility of the trivial module, we have that

H^{n}(G, k) ≅ Ext_{kG}^{n}(k, k) ≅ Hom_{kG}(P_{n}, k)

for all n. That is, a cohomology element ζ ∈ H^{n}(G, k) can be considered to be a homomorphism ζ : P_{n} → k. Such an element can always be lifted to a chain map (ζ̃_{0}, ζ̃_{1}, …) as in the diagram:

… ⟶ P_{n+1} ⟶ P_{n} ⟶ … ⟶ P_{0} ⟶ k ⟶ 0
ζ̃_{1}↓ ζ̃_{0}↓ ↘ζ
… ⟶ P_{1} ⟶ P_{0} ⟶ k

The process, which is adequately described in [8], is similar to the construction of the maps in the projective resolution. The main difference is that constructing ζ̃_{i} so that ζ̃_{i}∂ = ∂ζ̃_{i−1} means solving a system of linear equations. Again only the images of the identity element in the copies of kG are saved.

Of course, the point of this exercise is that the chain map of the cup product of two cohomology elements is chain homotopic to the composition of the chain maps. The chain homotopy does not present a problem because its image is always in the radical and this part will be discarded when it comes to computing generators and relations.

Now notice that the dimension of the cohomology group H^{n}(G, k) = Hom_{kG}(P_{n}, k) is equal to the number of copies in kG in P_{n}. In practice we compute the chain maps only for a minimal set of generators of the ring H^{∗}(G, k). That is, by calculating the chain maps for generators of degrees less than n we can see which elements in degree n are cup products of elements in smaller degrees and how many new generators are needed.

At this point we have a set of generators for cohomology and their chain maps. To get the relations we should first note that if γ : P_{n} → k is a cohomology element then γ(RadP_{n}) = 0. So it is really only necessary to look at the chain map as a sequence of homomorphisms (γ̂_{0}, γ̂_{1}, …) where

P_{n+r}/RadP_{n+r} ⟶^{γ̂_{r}} P_{r}/RadP_{r}.

The actual calculation of the relations is obtained as follows. Let x_{1}, …, x_{m} denote the generators of the cohomology. In degree n, form all monomials x_{1}^{r_{1}} … x_{m}^{r_{m}} whose degree ∑ r_{i}deg(x_{i}) = n. Each of these is represented uniquely by a homomorphism from P_{n}/RadP_{n} to P_{o}/RadP_{0} ≅ k. That is each is an element of the dual space (P_{n}/RadP_{n})^{∗}. Hence the relations in degree n are the k-dependence relations among these vectors in the dual space. In general the method produces many more relations than is necessary. Gröbner basis techniques can be used to reduce the set of relations to a managable collection.

So what do we have? First we have a graded-commutative “polynomial ring” Q = k[x_{1}, …, x_{m}]. Also we have an ideal I of relations which have been computed. We need to be careful at this point because cohomology rings satisfy the relations that xy = −yx if x and y are elements of odd degree. So if p > 2 then the ring Q actually has the form Q = k[y_{1}, …, y_{r}] ⊗ Λ(z_{1}, …, z_{s}) where y_{1}, …, y_{r} are the generators in even degrees and Λ is an exterior algebra in the generators z_{1}, …, z_{s} in odd degrees. Here {x_{1}, …, x_{m}} = {y_{1}, …, y_{r}} ∪ {z_{1}, …, z_{s}}. For convenience of notation we continue to write Q as k[x_{1}, …, x_{m}] as if it were a polynomial ring. It is still a free object in the category of graded-commutative polynomial rings.

In practice we can avoid some problems by always computing the monomials which are used to get the relations with a specific order on the generators. Then a person reading the data must be aware that there is an extra set of relations (yx = (−1)^{deg(x)deg(y)}xy) arising from the graded commutativity. Of course, if p = 2 then none of this matters.

The following should now be clear from the discussion.

Theorem 2.1. Suppose that R is the result of a computation to degree N of a cohomology ring H^{∗}(G, k). Then R satisfies the Standard Hypothesis given below for R = R_{N}.

Standard Hypothesis 2.2. Let N be a fixed integer and let P = k[x_{1}, …, x_{n}] be a graded commutative polynomial ring (see the above discussion). Assume that the generators are numbered so that deg(x_{i}) ≤ deg(x_{j}) if i < j. Let I be the ideal in P so that

H^{∗}(G, k) = P/I.

Let R_{N} = Q/J where the following hold.

1) Q = k[x_{1}, …, x_{m}] ⊆ P is the subring generated by all x_{j} with deg(x_{j}) ≤ N. So deg(x_{j}) > N if j > m.

2) J is the ideal of Q generated by all homogeneous elements of I of degree at most N.

Note that R = R_{N} in the Standard Hypothesis is a graded ring and has property that

R^{n} = H^{n}(G, k)

for all n ≤ N.

3. Generators

Let R = ∑_{i≥0} R^{i} be a graded-commutative finitely generated k-algebra. Let X = X_{R}

be its maximal ideal spectrum and for an ideal I ⊆ R, let X (I) ≤ X be the closed set of all maximal ideals in R which contain I.

Definition 3.1. Let r = DimX_{R}. A homogeneous set of parameters for R is a sequence ζ_{1}, …, ζ_{r} of homogeneous elements of positive degree with the property that X (I) = {0} for I = (ζ_{1}, …, ζ_{r}).

Here the zero point of X is the maximal ideal of all elements of positive degree. An equivalent definition of a homogeneous set of parameters is sequence ζ_{1}, …, ζ_{r} with the property that R is finitely generated module over k[ζ_{1}, …, ζ_{r}]. Notice that because r = Dim(X_{R}), the subring generated by a homogeneous set of parameters is a polynomial subring.

A sequence ζ_{1}, …, ζ_{t} ∈ R is a regular sequence provided ζ_{1} is a regular element of R (i.e. multiplication by ζ_{1} is an injective map from R to R) and for each i = 2, …, t, ζ_{i} is a regular element on R/(ζ_{1}, …, ζ_{i−1}). Any regular sequence can be filled out to get a homogeneous set of parameters. The depth of R is the length of the longest regular sequence in R. If the depth of R is equal to r = Dim(X_{R}) the R is said to be Cohen-Macaulay.

In the case that R = H^{∗}(G, k) we let V_{G}(k) denote the maximal ideal spectrum. Also we fix the notation τ = τ(G) for the dimension of V_{G}(k). The following facts are well known.

Proposition 3.2. (i) (See [3]). A sequence of homogeneous elements ζ_{1}, …, ζ_{τ} ∈ H^{∗}(G, k) is a homogeneous set of parameters if and only if for every elementary abelian p-subgroup E ⊆ G the quotient

H^{∗}(E, k)/(res_{G,E}(ζ_{1}), …, res_{G,E}(ζ_{τ}))

is a finite dimensional algebra. (Here res_{G,E} is the restriction map). The latter happens if and only if the dimension of the ideal (res_{G,E}(ζ_{1}), …, res_{G,E}(ζ_{τ})) is zero.

(ii) (See [5], [10]). A sequence ζ_{1}, …, ζ_{d} ∈ H^{∗}(G, k) is a regular sequence if the sequence res_{G,E}(ζ_{1}), …, res_{G,E}(ζ_{d}) is a sequence of parameters for the cohomology ring H^{∗}(E, k) where E is the unique maximal elementary abelian p-subgroup of Z(G). In particular, the depth of H^{∗}(G, k) is at least equal to the p-rank of the center of G

Remark: One importance of the above conditions is that they are easily checked in a computational situation.

Suppose that ζ_{1}, …, ζ_{r} is a sequence of homogeneous elements in R. Associated to the sequence is a Koszul complex

K(ζ_{1}, …, ζ_{r}; R) : 0 ⟶ K_{0} ⟶ K_{1} ⟶ ⋯ ⟶ K_{r} ⟶ 0

in which each K_{i} = R^{t} is a direct sum of t = ( \begin{matrix}n \\ i \end{matrix} ) copies of the ring R. For

our purposes it is sufficient to define the Koszul complex as a tensor product K(ζ_{1}, …, ζ_{r}; R) = K^{(1)} ⊗ K^{(2)} ⊗ ⋯ ⊗ K^{(r)} where K^{(i)} is given by

K^{(i)} = K(ζ_{i}; R) : 0 ⟶ K_{0}^{(i)} ⟶^{ζ̂_{i}} K_{1}^{(i)} ⟶ 0

for K_{0} ≅ K_{1} ≅ R and boundary map is multiplication by ζ_{i}. For notation let u_{i} ∈ K_{0}^{(i)}, v_{i} ∈ K_{1}^{(i)} be the identity elements so that ζ̂_{i}(au_{i}) = aζ_{i}v_{i}. Of course, the tensor products are taken over the base field k, and the differential on the whole Koszul complex is the usual thing for a tensor product of complexes.

The reader will note here that we are writing the Koszul complex as a cochain complex rather than the more standard chain complex. The reason for this should be come clear in the last section. Because of this notation the “homology” of Koszul complex should be cohomology and so we write H^{n} rather than H_{n}. We hope this causes no confusion.

Proposition 3.3. Suppose that ζ_{1}, …, ζ_{r} is a sequence of homogeneous elements in R. Assume that ζ_{1}, …, ζ_{d} is a regular sequence for R. Then for all i, we have that

H^{i+d}(K(ζ_{1}, …, ζ_{r}; R)) ≅ H^{i}(K(ζ_{d+1}, …, ζ_{r}; R/(ζ_{1}, …, ζ_{d})))

Proof: We have a chain map of complexes

K(ζ_{1}, R) : 0 ⟶ R ⟶^{ζ_{1}} R ⟶ 0
↓μ μ_{0}↓ μ_{1}↓
C(R/ζ_{1}) : 0 ⟶ 0 ⟶ R/ζ_{1} ⟶ 0

where C(R/ζ_{1}) is the complex with only one nonzero term R/ζ_{1} in degree 1. Here μ_{1} is the natural quotient map. Because ζ_{1} is regular, μ induces an isomorphism on cohomology and the kernel of μ is an exact complex (R ⟶^{ID} R) of projective R modules. Then, letting K′ = K^{(2)} ⊗ ⋯ ⊗ K^{(r)}, we have an exact sequences of complexes

0 ⟶ kerμ ⊗ K′ ⟶ K(ζ_{1}; R) ⊗ K′ ⟶ C(R/ζ_{1}) ⊗ K′ ⟶ 0

which implies a long exact sequence on cohomology. It is straightforward to show that H^{∗}(kerμ ⊗ K′) = 0. Hence it follows that

H^{i}(K(ζ_{1}, …, ζ_{r}; R)) = H^{i}(K(ζ_{1}; R) ⊗ K′)
≅ H^{i}(C(R/ζ_{1}) ⊗ K′)
≅ H^{i−1}(K(ζ_{2}, …, ζ_{r}); R/ζ_{1}).

The rest follows by induction.

If we assume that the elements ζ_{1}, …, ζ_{r} are homogeneous then the Koszul complex K(ζ_{1}, …, ζ_{r}; R) and its cohomology are doubly graded. For this paper we grade the complex in such a way that the differentials are degree zero maps (i. e. bidegree (1,0) maps). That is, the identity element in degree 0, u_{1} ⊗ ⋯ ⊗ u_{r} ∈ K^{0,0} is in degree (0, 0). So for any i, ζ_{i}u_{1} ⊗ ⋯ ⊗ u_{i−1} ⊗ v_{i} ⊗ ⋯ ⊗ u_{r} is in degree (1, 0).

Thus u_{1} ⊗ ⋯ ⊗ u_{i−1} ⊗ v_{i} ⊗ ⋯ ⊗ u_{r} ∈ K^{1,−m_{i}} and v_{1} ⊗ ⋯ ⊗ v_{r} ∈ K^{r,− ∑ m_{i}} where m_{i} = deg(ζ_{i}). In this regard, the isomorphism of the last proposition requires a double degree shift

H^{i+d,j−w}(K(ζ_{1}, …, ζ_{r}; R)) ≅ H^{i,j}(K(ζ_{d+1}, …, ζ_{r}; R/(ζ_{1}, …, ζ_{d})))

where w = ∑_{i=1}^{d} m_{i}.

The condition which we wish to investigate is the following. The condition holds for R = H^{∗}(G, k) in all of the cases which we have been able to check.

Condition G 3.4. Let ζ_{1}, …, ζ_{r} be a homogeneous system of parameters for R. Then

H^{∗,j}(K(ζ_{1}, …, ζ_{r}; R)) = 0

whenever j ≥ 0.

Proposition 3.5. If R = H^{∗}(G, k) is Cohen-Macaulay or if H^{∗}(G, k) has depth τ − 1 then Condition G holds for R.

Proof: See [3]. The proof there uses the hypercohomology spectral sequence which is discussed in Section 5.

Remark: For most groups where the group cohomology is in the computable range it is unlikely that one would need to compute a Koszul complex with r > 3. This is particularly true in view of Propositon 3.3. In the low rank cases there is an easy implementation of the complex. If r = 1, then H^{1}(K(ζ_{1}; R)) = R/ζ_{1} = Q/(J + Rζ_{1}) while

H^{0}(K(ζ_{1}; R)) = Ann_{R}(ζ_{1}) = [J : Rζ_{1}]/J
= {x ∈ Q|x · ζ_{1} ∈ J}/J

For r = 2 things are slightly more complicated but we can filter the cohomology according to lines w_{1} ⊗ w_{2} where w ∈ {u, v}. That is suppose the line for w has the form I_{1}/I_{2} for I_{1} and I_{2} ideals in Q as above. Then the line corresponding to w_{1} ⊗ u_{2} ≅ I′_{1}/I_{2} where I′_{1} = I_{n} ∩ Ann_{R/I_{2}}(ζ_{2}). On the other hand, the line corresponding to w_{1} ⊗ v_{2} is I_{1}/I′_{2} where I′_{2} = I_{2} + ζ_{j}R. It is easy to see that this works for H^{0}(K) and H^{2}(K). The degree 1 cohomology is actually filtered: {0} ⊆ line of u_{1} ⊗ v_{2} ⊆ H^{1}(K). That is, the quotient of H^{1}(K) by the line of u_{1} ⊗ v_{2}, is the line of v_{1} ⊗ u_{2} as computed above.

This process can be iterated at least once more to get useful information. That is, when r = 3, the cohomology of the Koszul complex injects into the modules filtered as above. However, we may not have an isomorphism. Because our interest is in determining when the cohomology is zero, this may still be helpful.

4. Relations

In this section we consider the conjecture on the relations ideal in the cohomology ring. We assume that the ring R satisfies the Standard Hypothesis (i.e., R = R_{N} as in (2.2)) to degree N with respect to H^{∗}(G, k). Notice that for any subgroup H ⊆ G it makes sense to speak of the restriction map θ_{H} : R ⟶ H^{∗}(H, k). This restriction is the composition of the natural map R ⟶ H^{∗}(G, k) followed by the restriction, res_{G,H}, on cohomology.

Condition R 4.1. : Let E be the intersection of the kernels of the restriction maps θ_{H} : R ⟶ H^{∗}(H, k) for all maximal subgroups H of G. Then E = {0} unless the depth of R is equal to the p-rank of Z(G). Moreoever there exists a regular sequence y_{1}, …, y_{d} in H^{∗}(G, k) with y_{1}, …, y_{d} ∈ R and a regular sequence in R and d = p-

rank (Z(G)). If E ≠ 0 then E = ∑_{i=1}^{s} T \overline{α_{i}} is a free module over T = k[y_{1}, …, y_{d}] with

a finite number of generators \overline{α_{i}} = α_{i} + J, α_{i} ∈ Q and deg(α_{i}) ≤ N for i = 1, …, s.

In the case that R = H^{∗}(G, k), E is known as the essential cohomology. In all of the cases that we have checked H^{∗}(G, k) satisfies Condition R. Finally, in the case that the depth of R is equal to the p-rank of the center of G we should notice that if we choose a regular sequence for R then we can guarantee that it is regular sequence for H^{∗}(G, k) by making certain that its restriction is a regular sequence for H^{∗}(Z(G), k) by Proposition 3.2.

Theorem 4.2. Assume that the ring R satisfies Conditon R. Let {μ_{i}|i = 1, …, t} be a minimal set of generators for the kernel J′ of the composition

Q ⟶ R ⟶^{θ_{H}} ⊕ ∑_{H} H^{∗}(H, k)

where the sum is over all maximal subgroups H of G. For each i = 1, …, t, there exist elements f_{i1}, …, f_{is} ∈ T such that

σ_{i} = μ_{i} − ∑_{j=1}^{s} f_{ij}α_{j} ∈ J.

For each i = 1, …, m and j = 1, …, s there exist g_{ijk} ∈ T such that

τ_{ij} = x_{i}α_{j} − ∑_{k=1}^{s} g_{ijk}α_{k} ∈ J.

Then the set {σ_{i}, τ_{jk}|i = 1, …, t, j = 1, …, n, k = 1, … s} generates the ideal J.

Proof: The existence of the elements f_{ij} and g_{ijk} is a consequence of Condition R. In fact, because E is a free T -module these coefficients are unique.

Now suppose that y ∈ J. Then y ∈ J′ and there exist elements b_{i} ∈ Q such that

y = ∑_{i=1}^{t} b_{i}μ_{i}. Hence

y − ∑_{i=1}^{t} b_{i}σ_{i} = y − ∑_{i=1}^{t} b_{i}(μ_{i} − ∑_{j=1}^{s} f_{ij}α_{j})
= ∑_{j=1}^{s} (∑_{i=1}^{t} b_{i}f_{ij}) α_{j} ∈ J

Now notice that for each j, ∑_{i=1}^{s} b_{i}f_{ij} ∈ Q and Q = T + (x_{1}, …, x_{m}) as vector spaces.

So, modulo the ideal U = (τ_{ij}), x_{i}α_{j} ≡ ∑_{k=1}^{s} g_{ijk}α_{k}. In particular we must have that

∑_{j=1}^{s} (∑_{i=1}^{t} b_{i}f_{ij}) α_{j} = ∑_{k=1}^{s} t_{k}α_{k} + u

for t_{k} ∈ T, k = 1, …, s and u ∈ U. However U ⊆ J and \overline{α}_{1}, …, \overline{α}_{s} are generators

of a free T -module. Therefore t_{k} = 0 for all k and y − ∑_{i=1}^{t} b_{i}σ_{i} = u ∈ U as desired.

This proves the theorem.

Theorem 4.3. Suppose that R satisfies the Standard Hypothesis to degree N and also satisfies Condition R. Suppose that N is at least equal to the maximum of the degrees of a set of homogeneous generators for the ideal J′ = ∩J_{H} ≤ Q, where each J_{H} is the kernel of the composition Q ⟶ R ⟶^{θ_{H}} H^{∗}(H, k) for H a maximal subgroup of G. Then J = I ∩ (x_{1}, …, x_{m}) and if β ∈ I and deg(β) < min{deg(x_{j})|j = m + 1, …, n} then β ∈ J.

Proof: The condition which we impose on N relative to the degrees of the generators of J′, is meant to insure that the ideal J′ has really been computed. That is, we want to know that the relators {σ_{i}} of the previous theorem are elements of J. On the other hand the elements \overline{α_{i}} are generators for E ≅ J′/J as a T -module and should not be confused with the generators of J′ as an ideal of Q.

As noted in (2.2), N < deg(x_{j}) for all j > m. Suppose that β ∈ I is homogeneous and that deg(β) < deg(x_{j}) for all j > m. Then β ∈ Q. Also for all maximal subgroups H we have that β ∈ J_{H} since Q ∩ I ⊆ J_{H}. Hence β ∈ J′. If J′ = J then we are done. Otherwise this implies that there exist elements t_{k} ∈ T such that β − ∑ t_{k}α_{k} ∈ J. Now β ∈ I and J ⊆ I and so ∑ t_{k}α_{k} ∈ I. We need to show that t_{k} must be zero for all k.

Now recall that T = k[y_{1}, …, y_{d}] where y_{1}, …, y_{d} is a regular sequence in H^{∗}(G, k). Hence H^{∗}(G, k) is a free module over T (see (10.3.4) of [11]). By the hypothesis the images of the elements α_{1}, …, α_{t} are k-linearly independent in H^{∗}(G, k)/(y_{1}, …, y_{d})H^{∗}(G, k). The last statement is a consequence of the fact that these elements are k-linearly independent in R/(y_{1}, …, y_{d})R which, in degrees up to N, coincides with H^{∗}(G, k)/(y_{1}, …, y_{d}).

Therefore α_{1}, …, α_{t} are free generators of H^{∗}(G, k) as a T -module. Hence ∑_{k=1}^{s} t_{k}α_{k} ∈

I implies that t_{k} = 0 for k = 1, …, s.

5. A test for completion

Let R be a finite dimensional k-algebra satisfying the Standard Hypothesis (2.2). In this section we want to lay down the rules for a test for when R = H^{∗}(G, k). A major ingredient in the proof is the hypercohomology spectral sequence. The spectral sequence is discussed in some detail in [3] and [6], though the reader will notice some differences in the presentation here. First we need some notation.

For a kG-module M let

⋯ ⟶ P_{2} ⟶^{∂_{2}} P_{1} ⟶^{∂_{1}} P_{0} ⟶^{ϵ} M ⟶ 0

be a minimal projective resolution. For n > 0 we define Ω^{n}(M) to be ker(∂_{n−1}) = ∂_{n}(P_{n}). The minimality of the resolution assures that Ω^{n}(M) has no projective submodule or quotient modules. Similarly we can define Ω^{−n}(M) using injective resolutions. If N is another kG-module then we let P Hom_{kG}(M, N) be the subset of Hom_{kG}(M, N) consisting of all homomorphisms which factor through projective modules. Let

\underline{Hom}_{kG}(M, N) = Êxt_{kG}^{0}(M, N) = Hom_{kG}(M, N)/P Hom_{kG}(M, N)

The following results are standard (see [2], [7] or [11]).

Lemma 5.1. i) For n > 0, m > 0,

\underline{Hom}_{kG}(M, Ω^{−n}(N)) ≅ \underline{Hom}_{kG}(Ω^{n}(M), N) ≅ Ext_{kG}^{n}(M, N),

ii) Ω^{m}(M) ⊗ Ω^{n}(N) ≅ Ω^{m+n}(M ⊗ N) ⊕ proj,

iii) Ext_{kG}^{m}(M, Ω^{−n}(N)) ≅ Ext_{kG}^{m+n}(M, N)) ≅ Ext_{kG}^{n}(Ω^{m}(M), N).

Here ⊕proj means the direct sum with some projective module.

Now suppose that ζ ∈ H^{n}(G, k). Then ζ is represented (uniquely) by a cocycle ζ : Ω^{n}(k) ⟶ k, which we also call ζ. Let i : Ω^{n}(k) ⟶ P_{n−1} be the inclusion. Because of the minimality of the projective resolution, P_{n−1} is actually the injective hull of Ω^{n}(k). Then we get an exact sequence

0 ⟶ Ω^{n}(k) ⟶^{α} k ⊕ P_{n−1} ⟶ L_{ζ} ⟶ 0

where α(x) = (ζ(x), i(x)) for x ∈ Ω^{n}(k) and L_{ζ} is the natural quotient.

Now suppose that ζ_{1}, …, ζ_{r} is a sequence of nonzero homogeneous elements of positive degree in H^{∗}(G, k). Let m_{i} = deg(ζ_{i}) for all i. For each i we can form the exact sequence

0 ⟶ Ω^{m_{i}}(k) ⟶^{α_{i}} k ⊕ proj ⟶ L_{i} ⟶ 0

as above with L_{i} = L_{ζ_{i}}. From this we get a complex

C^{(i)} : 0 ⟶ C_{1}^{(i)} ⟶^{∂_{1}^{(i)}} C_{0}^{(i)} ⟶ 0

where C_{1}^{(i)} ≅ Ω^{m_{i}}(k), C_{0}^{(i)} = k ⊕ proj and ∂_{1}^{(i)} = α_{i} as in the exact sequence. Clearly the homology of C^{(i)} is concentrated in degree 0 and H_{0}(C^{(i)}) ≅ L_{i}.

Now define C = C^{(1)} ⊗ ⋯ ⊗ C^{(r)} to be the usual tensor product of complexes with ⊗ = ⊗_{k}. Notice that by the Künneth formula

H_{j}(C) = \{ \begin{matrix}L_{1} ⊗ ⋯ ⊗ L_{r} & if j = 0 \\ 0 & if j ≠ 0 \end{matrix}

Proposition 5.2. If ζ_{1}, …, ζ_{τ} is a homogeneous set of parameters for H^{∗}(G, k) then H_{0}(C) ≅ L_{1} ⊗ ⋯ ⊗ L_{τ} is a projective module.

Proof: This is a standard support variety argument. The point is that V_{G}(L_{i}) = V_{G}(ζ_{i}) (for example see (5.9.1) of [2]) and by the Tensor Product Theorem ((5.7.1) of [2])

V_{G}(L_{1} ⊗ ⋯ ⊗ L_{r}) = V_{G}(ζ_{1}) ∩ ⋯ ∩ V_{G}(ζ_{r}) = {0}.

The complex C has the form

0 ⟶ C_{r} ⟶ ⋯ ⟶ C_{1} ⟶ C_{0} ⟶ 0

where C_{0} ≅ k ⊕ proj, C_{1} ≅ ∑_{i=1}^{r} Ω^{m_{i}}(k) ⊕ proj, C_{2} ≅ ∑_{i<j} Ω^{m_{i}+m_{j}}(k) ⊕ proj, etc. by

Proposition 5.1. Also C_{n} ≅ Ω^{w}(k) ⊕ proj, where w = ∑ m_{i}.

Definition 5.3. Let (P_{n}, ϵ) be a minimal projective resolution of k. The hypercohomology spectral sequence is the spectral sequence of the double complex

E_{0}^{p,q} = Hom_{kG}(C_{p} ⊗ P_{q+1}, Ω(k))

for p ≥ 0 and q ≥ −1.

Theorem 5.4. The spectral sequence converges to the total cohomology of the total complex which in degree s is Ext_{kG}^{s}(L_{1} ⊗ ⋯ ⊗ L_{r}, Ω(k)). In particular if ζ_{1}, …, ζ_{r} is a homogeneous system of parameters then E_{∞}^{p,q} = 0 if either p > 0 or q > −1 while

E_{∞}^{0,−1} = Hom_{kG}(L_{1} ⊗ ⋯ ⊗ L_{r}, Ω(k)).

Proof: Note that the spectral sequences converges because it has only a finite number of columns and hence E_{r+1} = E_{∞}. The easiest way to see the theorem is to run the “other” spectral sequence of taking the differentials in the reverse order. That is, first take the differential (∂ ⊗ 1)^{∗}, i.e., the one induced by the differential on C. Because every P_{q} is projective, the E_{1} page of this complex is

E_{1}^{p,q} = Hom_{kG}(H_{p}(C) ⊗ P_{q+1}, Ω(k)).

But H_{p}(C) = 0 if p ≠ 0 and H_{0}(C) = L_{1} ⊗ ⋯ ⊗ L_{r}. Consequently E_{1} has only one nonzero column in p = 0 and there is only one further differential which is (1 ⊗ ∂)^{∗}. But L_{1} ⊗ ⋯ ⊗ L_{r} ⊗ P_{∗} is a projective resolution of L_{1} ⊗ ⋯ ⊗ L_{r}. Hence

E_{∞}^{p,q} = E_{2}^{p,q} = \{ \begin{matrix}Ext_{kG}^{q+1}(L_{1} ⊗ ⋯ ⊗ L_{r}, Ω(k)) & ifp = 0 \\ 0 & ifp ≠ 0. \end{matrix}

The last statement follows from Proposition 5.2.

Theorem 5.5. For q ≥ 0 the E_{1}-page of the spectral sequence coincides with the Koszul complex K(ζ_{1}, …, ζ_{n}; H^{∗}(G, k)). That is

E_{1}^{p,q} = K^{p,q}(ζ_{1}, …, ζ_{n}; H^{∗}(G, k))

for q ≥ 0 and the differential induced by the boundary map on C, E_{1}^{p,q} ⟶ E_{1}^{p+1,q} is the same as the differential in the Koszul complex.

Proof: Notice first that for each p, (C_{p} ⊗ P_{∗}, 1 ⊗ ϵ) is a projective resolution of C_{p}. Hence

E_{1}^{p,q} ≅ Ext_{kG}^{q+1}(C_{p}, Ω(k)) ≅ Ext_{kG}^{q}(C_{p}, k) for q ≥ 0.

Then for p = 0, C_{0} = k ⊕ proj and

E_{1}^{0,q} ≅ Ext^{q}(k, k) ≅ H^{q}(G, k)

for q ≥ 0. For C_{1} = ∑_{i=1}^{r} Ω^{m_{i}}(k) ⊕ proj and

E_{1}^{1,q} ≅ Ext_{kG}^{q}(∑_{i=1}^{r} Ω^{m_{i}}(k), k) ≅ ∑_{i=1}^{r} H^{q+m_{i}}(G, k)

etc. We leave it to the reader to check all of the details.

Remark: The proof of the first statement of Proposition 3.5 can be derived from the last theorem. That is, if ζ_{1}, …, ζ_{τ} is a regular sequence and a homogeneous set of parameters for H^{∗}(G, k) then by Proposition 3.3, E_{2}^{p,q} = 0 for q ≥ 0 and p ≠ τ. But further differentials on the spectral sequence must be zero. Hence E_{2}^{τ,q} = E_{∞}^{τ,q} for q ≥ 0. On the other hand, by Theorem 5.4, E_{∞}^{τ,q} = 0 for q ≥ 0.

The following is our main theorem.

Theorem 5.6. Suppose that R satisfies the Standard Hypothesis to degree N, and that R also satsifies both Condition G and Condition R. Assume further that N is at least equal to the maximum of the degrees of a set of homogeneous generators for the ideal J′ = ∩J_{H} ≤ Q, where each J_{H} is the kernel of the composition Q ⟶ R ⟶^{θ_{H}} H^{∗}(H, k) for H a maximal subgroup of G. Suppose that there exist elements w_{1}, …, w_{τ} ∈ Q such that w_{1} + I, …, w_{τ} + I is a homogeneous system of parameters

for H^{∗}(G, k) and such that deg(w_{i}) ≥ 2 for all i and N ≥ ∑_{i=1}^{τ} deg(w_{i}). Then

R = H^{∗}(G, k).

Proof: We need to show that Q = P, that is, that there are no extra generators in H^{∗}(G, k) and that I = J, i.e., that there are no undiscovered relations. Note that all generators and relations for R lie in degree N or less. By Theorem 4.3 any new relation for H^{∗}(G, k) must lie in a degree which is at least equal to the smallest degree of a new generator. Thus if there are no new generators then there are no new relations and we are done.

So assume that there is a new generator x_{m+1} ∈ P with deg(x_{m+1}) = w > N. That is, w is the least degree in which there is a generator for P which is not in Q. Let ζ_{1}, …, ζ_{τ} ∈ R be a homogeneous set of parameters with the property

that deg(ζ_{i}) ≥ 2 for all i and ∑_{i=1}^{τ} deg(ζ_{i}) ≤ N. Now consider the hypercohomology

spectral sequence defined by ζ_{1}, …, ζ_{r}. Note that the class of x_{m+1} in E_{1}^{τ,v}, v =

w − ∑_{i=1}^{τ} deg(ζ_{i}) > 0 survives to the E_{2}-page of the spectral sequence.

Now remember that E_{∞}^{τ,v} = 0 and hence it is necessary that the class of x_{m+1} be hit by something on some eariler page of the spectral sequence. That is, for some t, 2 ≤ t ≤ τ there is an element μ ∈ E_{t}^{τ−t,v+t−1} such that

d_{t}(μ) = γ

where γ ∈ E_{t}^{τ,v} is the class of x_{m+1}. But now

E_{1}^{τ−t,v+t−1} ≅ ∑_{|S|=τ−t} H^{v+t−1+deg(S)}(G, k)

where the sum is over all subsets S ⊆ {1, …, τ} such that the number of elements in S is τ − t and deg(S) = ∑_{i∈S} deg(ζ_{i}). From this we get that

E_{1}^{τ−t,v+t−1} = ∑_{|S|=τ−t} R^{v+t−1+deg(S)}

because

v + t − 1 + deg(S) = w − ∑_{i=1}^{τ} deg(ζ_{i}) + t − 1 + ∑_{i∈S} deg(ζ_{i})
= w + t − 1 − ∑_{i∉S} deg(ζ_{i}) < w

since deg(ζ_{i}) ≥ 2 and there are t of the elements in {1, …, τ} which are not in S. The same also hold for E_{1}^{τ−t−1,v+t−1} and E_{1}^{τ−t+1,v+t−1}. Hence this portion of the E_{1}-page of the spectral sequence for H^{∗}(G, k) coincides with corresponding portion of the E_{1}-page of the hypercohomology spectral sequence for R, which is the Koszul complex K(ζ_{1}, …, ζ_{r}; R). Hence E_{2}^{τ−t,v+t−1} = E_{t}^{τ−t,v+t−1} = 0. Therefore we have a contradiction.

References

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

  E-mail address: jfc@sloth.math.uga.edu; http://www.math.uga.edu/∼jfc