PROBLEMS IN THE CALCULATION OF GROUP COHOMOLOGY

JON F. CARLSON

1. Introduction

For several years now, I have occupied myself with a project to compute the modp cohomology of finite p-groups using computer algebra. To date, programs have been written to build minimal projective resolutions for modules over finite p-groups and create chain maps between resolutions representing cohomology classes. The cup products can be computed by treating them as compositions of chain maps. With this machinery it is possible to find a set of generators and relations for a cohomology ring H^{∗}(G, k) in the case that G is a finite p-group and k is a finite field of characteristic p. There are, of course, practical problems that arise if the group or the field is too big. However even for groups and fields of reasonable size, we encounter the problems of producing an output in a form that is usable for interpretation in theoretical investigations and of verifying that the output gives a complete description of the cohomology ring of the group. The description of the algorithms for the actual computation of the cohomology is given in the literature [9, 10, 11]. In this paper, I will focus on some of the problems which still need work.

The problem with such calulations is that they require techniques and methods (algorithms) from a wide variety of algebraic areas. In the beginning the computations are almost all linear algebra though, it is helpful to have an adept implementation of groups and of modules for group algebras available. In the later stages, particularly the verification of the complete calculation, the computation requires reasonably sophisticated applications from commutative algebra. One approach that could be taken is to write all of the code from scratch. But this does not seem practical given the wide range of techniques which are used in the advanced stage of some of the applications that are needed. My personal choice has been to write all of my programs in MAGMA [7], which offers both the module theory and the commutative algbra. I owe a great debt of gratitude to John Cannon and his group in Sydney (particularly Allen Steel) for all of the help that they have given me. At the same time I have had to write many of my own routines, even for the commutative algebra portion of the calculations, and it is clear to me that there are many improvements that can be made both in my program and in the general state of what is available for a project like this one. Hence I would be grateful for any help or suggestions from the readers.

It should also be mentioned that David Green has written stand-alone versions of some of the programs [15].

So far, the programs have played a crutial role in the calculation of the mod-2 cohomology of several groups, namely the Janko groups J_{2} and J_{3} [12], and the Higman-Sims group HS [3]. The latter calculation is still in progress but some resolution is expected soon. In addition, Dave Benson and I are working on a project to compute the cohomology of the double cover 2M_{12} of the Mathieu group M_{12}. The scheme for this project would be to identify the elements of the computed cohomology of the Sylow 2-subgroup of M_{12} with the generators in the presentation of [2] and then use the inflation map routines which have been recently included in the cohomology package. Then it should be possible to complete the calculation using the hypercohomology spectral sequence.

One of the major aims of the project has been the computation of the mod-2 cohomology rings for the groups of order 64. The rings for the 51 groups of order 32 were computed by Rusin [17] several years ago and they have served as test cases and general inspiration for several investigations (e.g. see [1, 6]). It is hoped that the new calculations will play a similar role. As of the writing of this report and the EuroConference in Essen, more than 90% of the cohomology rings of the 267 groups of order 64 had been successfully computed. Only 20 or so problem cases remained to be done. The obstructions to finishing the last few cases are varied and mostly involve technical questions in computational commutative algebra. The actual calculation of the cohomology ring – as far as it goes – is reasonably straightforward, and breaks down in only a few cases because the Groebner basis calculations of the relations ideal is too complicated. More often the problems arise in the verification that the computation is complete, that all generators and relations for the cohomology ring have been found in the course of the calculation. The test for completion is rather tortured and is based on obtaining positive answers to some questions which are only conjectural, but seem to hold for all of the examples that have been computed. The tests invoke mostly computer applications that deal only with the output in the form of generators and relations for a quotient of a polynomial ring.

There are three main aims to this note. First I want to give some explanation and discussion of the conjectures (questions) on which the test for the completion of the calculation is based. Most of these can be verified using Groebner basis techniques on the output of the calculation. However in Section 3 we note that there are some problems with a few of the verifications. The second aim is to give some discussion of the computational problems. In some cases it may be a matter of finding a correct and efficient algorithm for making the calculation. In other cases it is possible to circumvent the difficulties by employing a different test for the completion of the calculation. So the third aim of the note is to outline a couple of tests for the completion which are not already recorded in the literature. This we do in Section 4.

It should be emphasized that the output of the calculation is being made public as it is completed. It is posted on my web page

http://www.math.uga.edu/^{∼}jfc/groups/cohomology.html

Some explanation of the project and the problems is given there. In addition, anyone who is interested is welcome to get the programs, written in MAGMA code, which run the calculation.

For what follows, the reader is urged to consult the general reference books [4, 14] for basic information on modules and cohomology and the book [16] for the details of the commutative algebra.

2. Questions and conjectures

For notation assume that G is a finite p-group and that k is a field of characteristic p. We also use k to denote the one-dimensional trivial kG-module. Let H^{∗}(G, k) be the cohomology ring of G. It is known that H^{∗}(G, k) is a finitely generated algebra over k and that its Krull dimension is equal to the p-rank of G. More specifically a theorem of Quillen (see [4]) says that the minimal prime ideals of H^{∗}(G, k) are precisely the radicals of the kernels of the restrictions to the elementary abelian psubgroups of G. Moreover we know by Duflot’s theorem [13] that the depth of the cohomology ring is at least equal to the p-rank of the center of G. The depth of a ring is defined to be the length of the longest regular sequence of elements of the ring. Indeed we know (see also [8]) that if r is the p-rank of the center of G and if ζ_{1}, …, ζ_{r} is a sequence of elements of H^{∗}(G, k) whose restriction to the center of G is a system of parameters then ζ_{1}, …, ζ_{r} is a regular sequence for the cohomology ring. In the case that the cohomology ring has a regular sequence of length equal to its Krull dimension then, by definition, the cohomology is Cohen-Macaulay.

A great deal of what we wish to discuss in this section is concerned with the existence of sequences of parameters with special properties. A homogeneous system of parameters for H^{∗}(G, k) is a sequence of homogeneous elements ζ_{1}, …, ζ_{n} with the property that the subring of H^{∗}(G, k) generated by the elements is a polynomial subring and the algebra H^{∗}(G, k) is a finitely generated module over the polynomial subring k[ζ_{1}, …, ζ_{n}]. In particular it is required that the length of any homogeneous system of parameters be equal to the Krull dimension of the cohomology ring. If V_{G}(k) is the maximal ideal spectrum of H^{∗}(G, k) and if V_{G}(ζ_{i}) is the set of all maximal ideals containing the ideal generated by ζ_{i} then it can be seen that ζ_{1}, …, ζ_{n} is a homogeneous system of parameters if and only if n is the Krull dimension and ∩_{1}^{n}V_{G}(ζ_{i}) = {0}, where 0 denotes the maximal ideal of all elements of positive degree.

Question 2.1. Does the cohomology ring H^{∗}(G, k) have a quasi-regular sequence? [5,10]

A sequence ζ_{1}, …, ζ_{n} of homogeneous elements is a quasi-regular sequence provided

(i) ζ_{1} is a regular element, i.e. multiplication by ζ_{1} is an injective map from H^{∗}(G, K) to itself and

(ii) for every i > 1 multiplication by ζ_{i}

ζ_{i} : (H^{∗}(G, k)/(ζ_{1}, … ζ_{i−1}))_{m} ⟶ (H^{∗}(G, k)/(ζ_{1}, … ζ_{i−1}))_{m+deg(ζ_{i})}

is injective whenever the degree m is at least equal to deg(ζ_{1}) + … + deg(ζ_{i−1}).

From the standpoint of performing computer calculations the advantage of having a quasi-regular sequence is clear from Proposition 10.3 of [5]. Specifically, if H^{∗}(G, k) has a quasi-regular sequence ζ_{1}, …, ζ_{n} then all of the ring is generated by elements in degrees ∑_{1}^{n} deg(ζ_{i}). This holds because the Koszul complex described in the Proposition has no homology and, in particular the last term which is H^{m}(G, k)/ ∑ ζ_{i} · H^{m−deg(ζ_{i})}(G, k) is zero provided m is large enough. So the only problem with finding the entire cohomology ring is making certain that a basis for the ideal of relations is computed. A different proof of this fact can be reasonably easily deduced from the discussion in Section 4. Note that in the paper [10], the Condition G (3.4) is essentially the existence of a quasi-regular sequence.

The essential cohomology of G is defined to be intersection of the kernels of the restriction maps res_{G,H} : H^{∗}(G, k) ⟶ H^{∗}(H, k) for all maximal subgroups H of G. So it is the ideal of the cohomology ring consisting of all elements which are not detected on any subgroup of G. Let E = E(G, K) denote the essential cohomology of G. It should be clear that it is a graded ideal.

Proposition 2.2. (See Theorem 2.3 of [9]) If the essential cohomology is nonzero, then the dimension of its annihilator equal to the p-rank of the center of G.

One motivation for wanting to know the proposition is clear. If the essential cohomology is zero, then the cohomology ring H^{∗}(G, k) is determined entirely by its restrictions to the maximal subgroups of G. Thus the essential cohomology is the part that is not determined by the restrictions. By an induction, we can assume the cohomology of the maximal subgroups is known.

The dimension of the annihilator J of E is the same as the dimension of the variety V_{G}(J) or the Krull dimension of the ring H^{∗}(G, k)/J. The Proposition concerns the existence of an associated prime 𝔭 in H^{∗}(G, k) whose dimension is equal to the depth of the cohomology ring. That is, if E ≠ 0, then E has an element whose annihilator is prime and has dimension equal to the p-rank of the center of G, which is the depth. In some sense this is part of a more general question.

Question 2.3. Does the cohomology ring always have an associated prime 𝔭 whose dimension is equal to the depth of H^{∗}(G, k)? [9]

For arbitrary commutative rings it is easy to find examples for which the last question does not have an affirmative answer. For example, suppose that R = k[x, y, u, v]/(u^{2}, v^{2}, uv, xu − yv). Then the ideal generated by u and v as a k[x, y]- module looks like the augmentation ideal of k[x, y]. Thus the ring has depth one, although the Krull dimension is two and every nonzero element of k[x, y] is a regular element.

It is not hard to see that minimum dimension of any associated prime is less than the depth. So the question (2.3) is really about the converse. In the case that the depth d of H^{∗}(G, k) is equal to the p-rank r of G then the cohomology ring is Cohen-Macaulay and there is no question at all. As to the ideal E, Adem and Karagueuzian [1] have shown that in this case the essential cohomology is nonzero if and only if the group has a unique central maximal elementary abelian p-subgroup. But this does not address the question of how we determine the depth.

Another way of expressing this is to say that the real issue is detection. That is, we say that a collection of subgroups of G detects the cohomology of G if no element of H^{∗}(G, k) is in the kernel of the restricton to all of the subgroups in the collection. The theorem of [1] affirms in a roundabout way that the cohomology of centralizers of the maximal elementary abelian subgroups are precisely the detectors of the cohomology of G in the Cohen-Macaulay case. In general, we know [9] that if the depth of the cohomology ring is d then the cohomology is detected on the centralizers of the elementary abelian subgroups of rank d. The question again is the converse.

Question 2.4. Suppose the depth of G is d and d < r where r is the p-rank of G. Is it necessary that the intersection of the kernels of the restrictions to the centralizers of the elementary abelian p-subgroups of rank d + 1 is not zero? Must the annihilator of that intersection have dimension d?

An affirmative answer to (2.4) would also imply an affirmative answer to (2.3). The answer is yes if the group has a unique maximal elementary abelian subgroup which is not central. This is because there is a nonzero element of

H^{1}(G, k) ≅ k ⊗ H^{1}(G, ℤ/p) ≅ k ⊗ Hom(G, ℤ/p)

corresponding a maximal subgroup (the kernel of the homomorphism) which contains the centralizer of of the maximal elementary abelian. That element restricts to zero on the centralizer. The same argument will work in the case of a semi-dihedral group where there is a maximal subgroup which contains the centralizers of all of the maximal elementary abelian subgroups. However the reader should be careful of what this means. For example, the group of order 32 (p = 2) with Hall-Senior number 33 has two maximal elementary abelian subgroups of orders 8 and 16. So the Krull dimension of the cohomology is 4, but the depth is 3. The cohomology is actually detected on the elementary abelians, but not on the centralizer of the only elementary abelian subgroup of rank 4.

Affirmative answers to these questions would be a great aid in the computational problem. However at least until the issues are settled, it is necessary to ask another question.

Question 2.5. Assume that E ≠ 0 and the dimension of the annihilator of E is d, the p-rank of the center of G. Let ζ_{1}, …, ζ_{d} be a regular sequence of maximal length, then is E a free module over the polynomial subring k[ζ_{1}, …, ζ_{d}]?

An affirmative answer here allows us to establish limits on the maximum degrees of a minimal set of relations among the generators of the cohomology ring. In particlular, we can show the following.

Theorem 2.6 [10] Suppose that the generators and relations for the cohomology ring H^{∗}(G, k) have been calculated out to degree N and that the following conditions hold.

i) Questions 2.1 and 2.5 have affirmative answers for the quotient of the computed polynomial ring by the ideal of computed generators.

ii) N is larger than the sum of the degrees of the elements in a homogeneous system of parameters for the computed cohomology.

iii) N is larger than the degrees of the generators of the kernels of the restriction of the computed cohomology to the maximal subgroups of G.

Then the computation is complete and all generators of the cohomology ring and the ideal of relations have been computed.

In the rest of the paper we want to consider some of the problems involved in verifying the hypotheses of the theorem.

3. Some problems in the computations

As we said in the introduction, a major purpose of this note is the discussion of some of the difficulties which arise in the computation of the group cohomology. In several cases the problems have been become most obvious in the effort to compute all of the cohomology rings of the groups of order 64. For this project a few of the difficult situations have occurred in several different cases and there has been the opportunity to compare methods and results. In some cases, the problems have been solved in the sense that the computation has been successful even though it might be slow.

The method of computation is outlined in [11] and [10] and we will not discuss it much here. Suffice it to say that there are two main types of operations used. Initially the calculation is almost all linear algebra. This part includes the construction of the projective resolution, the chain maps for the minimal generators of the cohomology ring, and the set of all relations among the generators in degrees within the range of the computation. It also includes the constructions of the restriction maps to subgroups. The linear algebra portion can be very memory intensive because of the size of the modules and spaces and can be also time consuming when the resolutions get large. On the other hand it is very reliable and reasonably predictable. As long as the size of the computation is not too large for the machine, then it always finishes and it usually finishes in a reasonable amount of time.

The real difficulties in the computations come up in the second phase, which is mostly commutative algebra applications. I must admit that this is the area that I know least about and some of the problem that have arisen have clearly been due to my ignorance of the proper use of the methods. I am still trying to separate the problems which can be solved by intelligent use of available methods from those that are simply hard computations or for which no really good method has been discovered.

It is a fact that some Groebner basis computations are very difficult. Still a few of the successes seem quite amazing. Consider the example of the group number 153 of order 64. It is isomorphic to the Sylow 2-subgroup of the Suzuki group Sz(8). The cohomology ring has Krull dimension 3 and is the quotient of a polynomial ring on 20 generators by an ideal that is minimally generated by 136 relations. All computations including the kernels of the restrictions all maximal subgroups and the unique maximal elementary abelian subgroup and the nilradical and essential cohomology were performed in approximately 9 days on an SUN ULTRA 2200. However other cases with fewer generators and relations have not gone as well. The search for a solution for the last few groups of order 64 may finally degenerate into a search for any Groebner basis package that can do the job. It is also possible that the right ordering on the variables might make a big difference.

In the rest of the section let me just mention a few computational difficulties which, I hope, have reasonable solutions. The ultimate aim is to develop a collection of programs which could run through the cohomology rings of most or all of a set of groups without a glitch. At the present time the program runs until something goes wrong and it crashes. The information which is posted on my web page is the result of care in adjusting and restarting the computation. However this is certainly not the first time that we will run the cohomology of the groups of order 64. As we figure out more questions and conjectures about the cohomology rings we will find new ideas to test and add more pieces to the computation. It would be very desirable to have it run smoothly and not require a lot of monitoring.

3.1. Systems of parameters and regular sequences. The problem is to find an efficient method of computing a system of parameters in low degrees for a ring which is the quotient of a polynomial ring P by and ideal I. Strangely the problem is made more difficult by our desire to keep the field small, and to have the system begin with a regular sequence of maximal length. The last condition is important in the verification that the computed cohomology ring has a quasi-regular sequence (see 2.1). The method that is currently in use in the project is a search that begins by looking at the generators of the cohomology and then looks at more complicated expressions in the generators. At each step of the search we simply look for elements which have the property that factoring out the element decreases the Krull dimension by one. In this way we get a system of parameters. However we may not have a regular sequence of maximal length at the beginning of the system.

The problem is illustrated by the following examples. Suppose that P = k[x, y, u] for a field k and that I = (u^{2}, xyu). Then it is easy to see that P/I has Krull dimension 2 and that the sequence x, y is a system of parameters for P/I. However neither x nor y is a regular element since x annihilates yu and y annihilates xu. Still the depth of the ring is one because x + y is a regular element. In the case that k = GF (2), then there is an example that is even worse. Suppose that P = k[x, y, u] and I = (u^{2}, xy(x + y)u). Again x, y is a system of parameters, but no element of degree one is a regular element. To find a regular element we must go to degree 2 (x^{2} + xy + y^{2} is regular).

So we see that to find regular elements it is necessary to avoid the associated primes in the ring. It would be nice to have an implementation of an algorithm to find associated primes. However it might also be expensive to recompute the associated primes at each stage. The algorithm currently in use in the project involves extending the field and checking random elements when a problem with the regular sequence is encountered. It works well in small cases, but is slow and probably inefficient in others.

3.2. Free generators for the essential cohomology. One of the checks for the completion of the cohomology ring calculation is a verification that the essential cohomology E is a finitely generated free module over the polynomial ring k[ζ_{1}, …, ζ_{d}] where d is the depth and ζ_{1}, …, ζ_{d} is a regular sequence of maximal length in the computed cohomology. Magma does not currently have an algorithm that would test the freeness of a module over a polynomial ring that is presented in this fashion. The program which is being used in the cohomology project is rather naive and doomed to failure in certain circumstances. Basically it is the following.

Suppose that R is the computed cohomology ring. Then we know that R is itself a finitely generated module over the polynomial ring k[ζ_{1}, …, ζ_{d}] (see Theorem 10.3.4 of [14]). Any k-basis of R/(ζ_{1}, …, ζ_{d}) is a free k[ζ_{1}, …, ζ_{d}]-basis of R. Likewise any k-basis of E/(ζ_{1}, …, ζ_{d}) is a set of generators of E as a k[ζ_{1}, …, ζ_{d}]-module. Hence if the k-basis of E/(ζ_{1}, …, ζ_{d}) is linearly independent in R/(ζ_{1}, …, ζ_{d}) then E is a free module as desired.

The test outlined above is reasonably easy to implement, but it tells you nothing when it fails. Moreover in some cases it is certain to fail. For suppose that p = 2 and G ≅ ℤ/2 × H where H is a 2-group. Then H^{∗}(G, k) ≅ H^{∗}(ℤ/2, k) ⊗ H^{∗}(H, k) and H^{∗}(ℤ/2, k) ≅ k[η] is a polynomial ring in one variable. Now suppose that there is an element β in the essential cohomology of H. Then 1 ⊗ β is not in the essential cohomology of G because H is one of the maximal subgroups. However η ⊗ β is in the essential cohomology of G. So the problem is that η is a regular element and the essential cohomology is all contained in η ⊗ H^{∗}(H, k).

The programs that run at the moment simply ignore the failures of the test. I have been checking the cases of failure by hand. In the example of the last paragraph, it can be seen that if the answer to (2.5) is yes for H, then it is the same for G. Indeed, if enough steps have been computed to get the cohomology of H then the calculation is complete. However it would be much better to have an efficient implementation of an algorithm for testing when a module is free.

In some other cases, the completion of the calculation can be checked by other test such as those in the next section.

4. Two exceptional cases

We just noticed in the last section that there are problems sometimes in verifying the hypothesis of Question 2.5. Fortunately the aforementioned problems seem only to have occurred in situations where there exist alternate tests for the completion of the calculations. However the alternate tests are not well exposed in the literature. The aim of this section is to state the requirements of the alternate tests and sketch the proofs. The proofs rely on some module theoretic methods, so we need to introduce some notation. As a general reference for the homological algebra we refer the reader to the book by Benson [4].

As before assume that G is a p-group but now let k be an algebraically closed field of characteristic p. If M is a kG-module let ϕ : P ⟶ M be a projective cover of M. That is, P is a minimal projective kG-module such that there exists such a homomorphism ϕ which is surjective. The kernel of ϕ is denoted Ω(M). For n > 1 we define inductively Ω^{n}(M) = Ω(Ω^{n−1}(M)). Similarly we let Ω^{−1}(M) be the cokernel of an injective hull θ : M ⟶ Q and for n > 1 let Ω^{−n}(M) = Ω^{−1}(Ω^{−n+1}(M)). It should be remembered that projective kG-modules are also injective.

In the case that M = k the trivial kG-module, then we have an exact sequence

0 ⟶ Ω^{n}(k) ⟶ P_{n−1} ⟶ … ⟶ P_{0} ⟶ k ⟶ 0

where P_{∗} is a part of a minimal projective resolution of k. Then it can be seen that, because of the minimality, any element ζ of H^{∗}(G, k) ≅ Ext^{∗}(k, k) is uniquely represented by and a homomorphism (cocycle) ζ : Ω^{n}(k) ⟶ k. So if ζ is not zero then we have an exact sequence

0 ⟶ L_{ζ} ⟶ Ω^{n}(k) ⟶^{ζ_{1}} k ⟶ 0 (1)

Throughout the section we assume that G has the property that the p-rank of the center Z(G) of G is at least r − 1, where r is the p-rank of G. So there exists a homogeneous system of parameters ζ_{1}, …, ζ_{r} for H^{∗}(G, k) such that the sequence ζ_{1}, …, ζ_{r−1} is regular. Let n_{i} be the degree of ζ_{i} for all I.

Our hypothesis that ζ_{1} is regular means that in the long exact sequence

… ⟶ Ext_{kG}^{n}(k, k) ⟶ Ext_{kG}^{n}(Ω^{n_{1}}(k), k) ⟶ Ext_{kG}^{n}(L_{ζ_{1}}, k) ⟶ Ext_{kG}^{n+1}(k, k) ⟶ … the module Ext_{kG}^{n}(Ω^{n_{1}}(k), k) ≅ Ext_{kG}^{n+n_{1}}(k, k) ≅ H^{n+n_{1}}(G, k) and the map ζ̂_{1} is injective for all n ≥ 0. It follows that

Ext_{kG}^{n}(L_{ζ_{1}}, k) ≅ H^{n+n_{1}}(G, k)/ζ_{1} · H^{n}(G, k)

for all n ≥ 0. In fact we can do the same thing for Tate cohomology and get that Êxt_{kG}^{−1}(L_{ζ_{1}}, k) ≅ H^{n_{1}−1}(G, k) since ζ_{1} · H^{−1}(G, k) ≤ 0 [6]. For convenience of notation let U_{1} = Ω^{−n_{1}}(L_{ζ_{1}}). By degree shifting in the cohomology have that

Ext_{kG}^{m}(U_{1}, k) ≅ Ext_{kG}^{m}(Ω^{−n_{1}}(L_{ζ_{1}}), k) ≅ Ext^{m−n_{1}}(L_{ζ_{1}}, k) ≅ H^{m}(G, k)/ζ_{1} · H^{m−n_{1}}(G, k)

for m ≥ n_{1} − 1.

Now consider ζ_{2}. Assume that r > 2 so that ζ_{2} is the second element of a regular sequence. If we tensor the exact sequence of modules (4.1) corresponding to ζ_{2} with L_{ζ_{1}} we get a sequence

0 ⟶ L_{ζ_{1}} ⊗ L_{ζ_{2}} ⟶ L_{ζ_{1}} ⊗ Ω^{n_{2}}(k) ⟶^{1⊗ζ_{2}} L_{ζ_{1}} ⊗ k ⟶ 0

Now notice that L_{ζ_{1}} ⊗ Ω^{n_{2}}(k) ≅ Ω^{n_{2}}(L_{ζ_{1}}) ⊕ P, for some projective module P. So with careful degree shifting, and by letting U_{2} = Ω^{−n_{1}}(L_{ζ_{1}}) ⊗ Ω^{−n_{2}}(L_{ζ_{2}}), we have that

Ext_{kG}^{m}(U_{2}, k) ≅ H^{m}(G, k)/(ζ_{1} · H^{m−n_{1}}(G, k) + ζ_{2} · H^{m−n_{2}}(G, k))

for all m ≥ n_{1} + n_{2} − 2.

Obviously we can continue this process as long as the elements in the sequence are regular. Our first result in this direction is the following.

Theorem 4.1. Suppose that the p-rank of Z(G) is equal to the p-rank, r, of G. Then the cohomology ring H^{∗}(G, k) is generated by elements in degrees at most s = ∑_{1}^{r}(n_{i} − 1). Moreover if t is the largest degree of any generator then the ideal of relations for the cohomology ring is generated in degrees at most s + t.

Proof. The proof relies on the varieties of the modules in question. Recall that V_{G}(k) is the maximal ideal spectrum of the cohomology ring H^{∗}(G, k) and if M is a kGmodule then V_{G}(M) is the set of maximal ideals which contain the annihilator J(M) of Ext_{kG}^{∗}(M, M) in H^{∗}(G, k). The most relevant facts are that V_{G}(M) = {0} if and only if M is projective and the Tensor Product Theorem which says that the variety of the tensor product of two modules is the intersection of their varieties. Another important result is that V_{G}(L_{ζ}) = V_{G}(Ω^{m}(L_{ζ})) = V_{G}(ζ), the set of all maximal ideals which contain ζ ∈ H^{∗}(G, k), for all m. See [4] or [14] for details.

For the case in question, the important thing to notice is that the cohomology ring H^{∗}(G, k) is Cohen-Macaulay [13] and the sequence ζ_{1}, …, ζ_{r} is a regular sequence. Also

U_{r} = ⨂_{i=1}^{r} Ω_{−n_{i}}(L_{ζ_{i}}).

Consequently V_{G}(U_{r}) = ∩_{1}^{r}V_{G}(ζ_{i}) = {0}, since ζ_{1}, …, ζ_{r} is a system of parameters for the cohomology ring. Hence U_{r} is projective, and Ext_{kG}^{m}(U_{r}, k) = {0} for all m > 0. So from the above argument we conclude that

H^{m}(G, k) = ζ_{1} · H^{m−n_{1}}(G, k) + … + ζ_{r} · H^{m−n_{r}}(G, k)

whenever m > s. This proves the first part of the theorem.

For the rest we need to recall that the cohomology ring is a free module over the polynomial subring k[ζ_{1}, …, ζ_{r}] (see Theorem 10.6.2 of [14]). So if α_{1}, …, α_{t} is a set of generators for this module, then a minimal set of relations can be given in the form

α_{i} · α_{j} = ∑_{1}^{t} η_{ijk}α_{k}

where the coefficients η_{ijk} are in k[ζ_{1}, …, ζ_{r}].

The second case that we want to consider is the case in which the codepth of H^{∗}(G, k) is known to be at most one. That is, suppose that the p-rank of Z(G) is r − 1. Then we know that

Ext_{kG}^{m}(U_{r−1}, k) ≅ H^{m}(G, k)/ ∑_{1}^{r−1} ζ_{i} · H^{m−n_{i}}(G, k)

for all m ≥ ∑_{1}^{r−1}(n_{i} −1). This time the thing to notice is that V_{G}(U_{r−1}) = ∩_{1}^{r−1}V_{G}(ζ_{i}) is a variety of dimension one because ζ_{1}, …, ζ_{r} is a system of parameters. As a consequence U_{r−1} is a periodic kG-module and because V_{G}(ζ_{r}) ∩ V_{G}(U_{r−1}) = {0} multiplication by ζ_{r} induces an isomorphism

ζ_{r} : Ext_{kG}^{m}(U_{r−1}, k) ⟶ Ext_{kG}^{m+n_{r}}(U_{r−1}, k)

for all m > 0. All of this is made reasonably clear if we tensor the sequence (4.1) for ζ = ζ_{r} with the module U_{r−1}. That is, U_{r−1} ⊗ L_{ζ_{r}} is projective, hence injective, because its variety is {0}, and the sequence splits. It follows that

ζ_{r} : H^{m}(G, k)/ ∑_{1}^{r−1} ζ_{i} · H^{m−n_{i}}(G, k) ⟶ H^{m+n_{r}}(G, k)/ ∑_{1}^{r−1} ζ_{i} · H^{m+n_{r}−n_{i}}(G, k) (2)

is an isomorphism for m ≥ s′ = ∑_{1}^{r−1}(n_{i} − 1). Hence we have proved the following

Proposition 4.3. [5] Suppose that ζ_{1}, …, ζ_{r−1} is a regular sequence for H^{∗}(G, k) where r = p-rank(G). Then ζ_{1}, …, ζ_{r} is a quasi-regular sequence.

Our aim is to prove the following.

Theorem 4.4. Suppose that the p-rank of the center of G is r − 1 wbere r is the prank of G. Let ζ_{1}, … ζ_{r} be a system of parameters for the computed cohomology such that ζ_{1}, … ζ_{r−1} is a regular sequence and the restrictions of ζ_{1}, … ζ_{r−1} to the center of G is a regular sequence for H^{∗}(Z(G), k). Assume also that we have an isomorphism

ζ_{r} : R^{m}/ ∑_{1}^{r−1} ζ_{i} · R^{m−n_{1}} ⟶ R^{m+n_{r}}/ ∑_{1}^{r−1} ζ_{i} · R^{m+n_{r}−n_{i}}

Let s := ∑_{1}^{r}(n_{i} − 1) where n_{i} = deg(ζ_{i}). Then if the cohomology of G has been calculated to degree s + w where w is the maximum of the degrees of the generators of the cohomology, then the computation is complete.

Proof. It should be clear that if the hypothesis of (4.3) is satisfied then all of the generators of H^{∗}(G, k) can be taken to lie in degrees at most s. What about the relations? For convenience of notation let R denoted the computed cohomology ring. So we have a ring homomorphism θ : R ⟶ H^{∗}(G, k).

The thing to notice is that both H^{∗}(G, k) and R are free modules over the polynomial subring Q = k[ζ_{1}, …, ζ_{r−1}]. So if {β_{i}} is a (necessarily countable) set of free generators of R as a module over Q then we have a complete though infinite set of relations of the form

β_{i} · β_{j} = ∑_{k} μ_{ijk}β_{k} (3)

for μ_{ijk} ∈ Q. Now because of the isomorphism we may assume that β_{i} = ζ_{r}β_{j} whenever i > s. Hence by the hypothesis and the isomorphism (4.3) the elements {θ(β_{i})} form a basis for H^{∗}(G, k) as a module over Q. Hence θ is an isomorphism of free Q-modules. So it is also injective and a ring isomorphism.

Remark 4.6. The hypothesis for the last theorem may seem a bit complicated, but all of the items in it are easy to check in the computer calculation. Moreover they are built into the programs that are currently being run.

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Department of Mathematics, University of Georgia, Athens, GA 30602 E-mail address: jfc@sloth.math.uga.edu