Cohomology, Computations, and Commutative Algebra Jon F

Cohomology, Computations, and Commutative Algebra Jon F. Carlson

roup cohomology has roots that go back Another example is the cohomology ring of a to the early part of the last century. The semidihedral 2-group. The semidihedral group of topic has played a significant role in order 16 has the form several mathematical areas such as num- G =g,h| g8 = h2 = 1,hgh= g3. Gber theory, algebraic topology, and representation theory. Yet only in the last decade, with Again, if k is a field of characteristic 2, then its co- the aid of modern computers, have many examples homology ring has the form been calculated. The results of the calculations H∗(G, k) = k[z, y, x, w]/(z3,zy,zx,x2 + y 2w). have provided insight into theoretical developments, and they have stimulated interest in a whole It is interesting to notice in this case that the struc- new collection of issues involving the structure of ture of the cohomology ring is independent of the order of the group. All semidihedral 2-groups have cohomology rings. isomorphic mod 2 cohomology rings. The cohomology ring H∗(G, k) of a finite group Other examples are more complicated. There is G is a finitely generated algebra over its coeffi- a group of order 128 whose cohomology ring (with cients. We usually write it using generators and coefficients in a field of characteristic 2) has the relations, as in form ∗ H∗(G, k) k[z,y,x]/(zy). H (G, k) = k[z, y, x, w, v, u, t, s]/I This example is the cohomology ring of a dihedral where I is the ideal generated by the polynomials 2-group in the case that the coefficient ring k is a zy, y2,z3 + yx, zx + yx, yw, z2v + zt, field of characteristic 2. It is the quotient of the polynomial ring k[z.y.x] by the ideal generated by zy. x2,yu,yt,z2u + z2t + xt, xu, Hence it looks like the coordinate ring of the union 2 2 + + 2 2 + + + + 2 of two planes (the z-x plane and the y-x plane) that z w zvt t ,xw zvu zvt ut t , intersect in a line. and + + + 2 + 2 + + 2 Jon F. Carlson is professor of mathematics at the zwu zvu zvt z s u ut t . University of Georgia, Athens. His email address is You have probably guessed that this last exam- [email protected]. ple is a computer calculation. The wonders of mod- This research was supported in part by a grant from ern technology have permitted us to generate data the NSF. on a scale that we could only have dreamed about The author thanks Lisa Townsley for suggestions on thirty years ago. The output of other computer- this paper. generated examples can be found in the the book

426 NOTICES OF THE AMS VOLUME 52, NUMBER 4 [10] or on the web page http://www.math.uga. Properties of Cohomology Rings edu/~lvalero/cohointro.html. Experimental To begin, we outline some of the basic structure evidence has aided the development of the subject, of cohomology rings. This structure is measured and it has also provoked some new problems. In with notions such as dimension, depth, and prime this paper we discuss a few of the problems, show ideal spectra. We review some of these ideas in the why they are important in a more general mathe- discussion. matical setting, and explore the connections to Graded Commutativity and Elementary Abelian other areas of algebra and mathematics. In partic- Groups ular, the cohomology rings seem to have some very A cohomology ring H∗(G, k) is a k-algebra and is special properties which illustrate some basic graded in the sense that it is a direct sum concepts of commutative algebra. ∗ n The motivation for considering the cohomol- H (G, k) = H (G, k) ogy rings of finite groups comes from two very n≥0 different directions. To someone in algebraic of vector spaces over the field k. The product topology, the cohomology ring H∗(G, k) is an respects degrees in that example of the cohomology ring of a space. Specif- Hr (G, K) · Hs (G, k) ⊆ Hr+s (G, k). ically, it is the cohomology ring of the classifying space BG of the group G. The classifying spaces We say the ring is graded commutative because of groups occupy an important place in modern the elements of odd degree anticommute. That is, homotopy theory. For example, Lannes [18] has if x is in degree m and y is in degree n, then yx = shown that the mod p cohomology rings for ele- (−1)mnxy. In particular, if the degree of x is odd mentary abelian p-groups are injective modules and the characteristic of the field k is not 2, then over the Steenrod algebra of reduced power oper- x2 = 0. An element is homogeneous provided it has ations. Another direction is the study of modular a homogeneous degree, i.e., it is contained in r representation theory of finite groups. It is a fact H (G, K) for some r. that the geometry associated to the cohomology An important example to consider is the coho- ring controls and expresses much of the homo- mology ring of an elementary abelian group of n Z Z n logical algebra of kG-modules. order p , G ( /p ) . This is an abelian group in There are several equivalent ways to define group which every nonidentity element has order p. The cohomology. For one thing, it is the topological cohomology rings of these groups are as close as = cohomology of the classifying space mentioned possible to actual polynomial rings. If p 2, then −1 = 1 and H∗(G, k) is genuinely commutative. So above. For an algebraist, the usual classical defin- we have that ition is stated in terms of projective resolutions and ∗ homomorphisms. This is also the definition that we H (G, k) k[z1,...,zn] have used in the computer experiments [9, 15]. is actually a polynomial ring generated by elements That is, the algorithms for the computation follow in degree 1. If the characteristic of k is p>2, then the definition almost exactly. the graded commutativity requires that the poly- In this exposition we delay the actual definition ∗ nomial generators be in degree 2. Hence, H (G, k) of the cohomology ring until the discussion of the has the form computations in the section on Computing Group ∗ Cohomology. Instead, we concentrate on the prop- H (G, k) k[y1,...,yn] ⊗ Λ(z1,...,zn) erties of the cohomology rings and how these prop- where Λ is an exterior algebra. The generators z erties reflect the properties of the group. The dis- i are in degree 1, while the y occur in degree 2. The cussion ends up being something of a primer in i elements zi are in every prime or maximal ideal of certain aspects of modern commutative algebra. H∗(G, k), and therefore, they are contained in the Both the problems that we encounter and the ques- Jacobson radical Rad H∗(G, k) which is the inter- tions that we want to answer are expressed in the section of all maximal ideals. It follows that language of commutative algebra and algebraic ∗ ∗ H (G, k)/ Rad H (G, k) k[y1,...,yn] is a polyno- geometry. In addition, some of the tools we employ mial ring even in the case where p is odd. to analyze the experimental data are developments Functorial Properties of modern commutative and noncommutative al- Group cohomology is a functor. This is a compli- gebra. cated way of saying that homomorphisms of groups Throughout the paper, G denotes a finite group induce homomorphisms of group cohomology. and k a field. By a “cohomology ring” we mean a In particular, suppose that H ⊆ G is a subgroup. mod p cohomology ring in which the field k has Then there is a homomorphism induced from the finite characteristic p. Moreover, if G is a p-group inclusion called the restriction, for some prime p, then it should be assumed that ∗ → ∗ the field k has characteristic p. resG,H :H (G, k) H (H,k).

APRIL 2005 NOTICES OF THE AMS 427 Moreover, if H is a normal subgroup of G, then the cyclic center generated by the element e = (gh)2 and quotient map G → G/H induces a homomorphism two maximal elementary abelian subgroups, called the inflation, E =g,e and F =h, e. The cohomology rings of G ∗ → ∗ these two groups can be given as infG/H :H(G/H, k) H (G, k). H∗(E,k) k[u, v] H∗(F,k). These maps have many nice features. For one, each is transitive. That is, for example, if H ⊆ K ⊆ G, The restriction maps have the following form (with then resK,H ◦ resG, K = resG, H . In addition, both the suitable choice of the variables). The restriction restriction and inflation maps are homomorphisms resG, E sends z,y,x to 0,u,v(u + v), respectively, of graded commutative k-algebras. That is, they are while resG, F sends z,y,x to u, 0,v(u + v). The in- linear in k and preserve both the gradings and tersection of the kernels of the restrictions is thus product structures. the ideal (zy) =I⊆P = k[z,y,x]. Hence there is no In general, the inflation map can have a large element in H∗(G, k) P/I that is in the kernel of kernel and may be very far from being surjective. both of these restrictions. The variety of H∗(G, k) Likewise, a restriction map may have a large is union of the z-x and y-x planes in k3 . The maps kernel. However, restriction maps must be nearly on varieties have the form surjective. The way we say this is that H∗(H,k) ∗ = + must be finitely generated as a module over the resG, E(a, b) (0,a,b(a b)), ∗ ∗ = + image resG, H (H (G, k)). resG, F (a, b) (a, 0,b(a b)), Varieties and Quillen’s Theorem and hence we can calculate that V (k) is the union For the purposes of this section we assume that k G of the images is algebraically closed. A very important theorem = ∗ ∪ ∗ in group cohomology says that the cohomology VG(k) resG, E(VE (k)) resG, F (VF (k)). rings are finitely generated. The theorem was proved independently by Evens [14] and Venkov In fact, this is the situation in general. The follow- [21] more than forty years ago. The theorem tells ing theorem was proved by Quillen more than us in the case p = 2 that H∗(G, k) P/I where P is thirty years ago. a polynomial ring in a finite number of variables Theorem. Let G be any finite group. Then and I is an ideal in P. When p is odd, the theorem ∗ ∗ = ∗ implies that H (G, k)/ Rad H (G, k) (which is a com- VG(k) resG, E(VE (k)) mutative ring) can be written as P/I. This means E∈MEA that cohomology rings are noetherian and the max- where the union is over the set MEA of all maximal ideals can be classified by varieties. If P is a polynomial ring in n generators, then the variety imal elementary abelian p-subgroups of G. I of P/ is simply the set of simultaneous zeros of Dimension the polynomials that generated I. In the case that In general, the dimension of a ring or a variety is a G is an elementary abelian subgroup of order pn fairly complicated notion. However, with Quillen’s as above, the variety of H∗(G, k) is the affine space theorem in view, calculating the dimension of a kn. We denote the variety of H∗(G, k) by V (k). This G cohomology ring is an easy exercise. Perhaps it is is the maximal ideal spectrum of the ring. The n ∗ naive, but still it seems obvious that k should have points are the maximal ideals in H (G, k) and a dimension n. Thus it would seem that dimension, subset of VG(k) is closed if it is the set of all maximal ideals that contain a particular ideal. called “Krull dimension”, of the polynomial ring The functoriality of the preceding section k[x1,...,xn] should be n. All of this is correct, and moreover, Quillen’s Theorem implies that the Krull extends naturally to the varieties. For example, if ∗ H is a subgroup of G, then the restriction homo- dimension of H (G, k) is the largest integer r such morphism induces a map on varieties that G has an elementary abelian subgroup of order pr. This number is called the p-rank of G. ∗ → resG, H : VH (k) VG(k). Usually the Krull dimension of a commutative The point is that if m is a maximal ideal in H∗(H,k), ring R is defined as the length " of the longest ∗ strictly increasing chain then its inverse image {u ∈ H (G, k)| resG, H (u) ∈ m} is a maximal ideal in H∗(G, k). It should be noted that p ⊂ p ⊂···⊂p ⊂ R because H∗(H,k) is finitely generated over the image 1 2 " ∗ ∗ resG, H (H (G, k)), the map resG, H on varieties is fi- of proper prime ideals in R. Under the circum- nite-to-one. Now consider the dihedral group, men- stances that we encounter, the Krull dimension is tioned at the beginning of the paper. The dihedral also the length of the longest sequence of alge- group of order 8 has a presentation in the form braically independent elements in R. Such sequences G =g,h | g2 = h2 = 1 = (gh)4 . The group has a are of particular concern to us in later sections.

428 NOTICES OF THE AMS VOLUME 52, NUMBER 4 ∗ Depth and Detection H (G, k)/(u1) will be annihilated by all elements of We saw in the last section that the Krull dimension positive degree. Consequently, there is no regular of the cohomology ring H∗(G, k) is the p-rank of G. sequence of length 2. This is one of the consequences of Quillen’s The- It is not difficult to prove that this situation orem. In this section we consider another ring- happens in general. The depth of a commutative theoretic invariant that seems to be only partly graded ring R is never larger than the Krull discernible from the group structure. We are dimension of R/p where p is any associated prime speaking here of the depth of the cohomology ideal. ring. To define the depth of a ring, we first need On the other hand, the depth of a ring need not the notion of a regular sequence. be controlled by the existence of associated primes. As an example, consider the subring R of the poly- Depth and Regular Sequences nomial ring k[z,y] that is generated by the mono- Suppose that R is a finitely generated graded mials y,z2,z3,zy. That is, R is the subring of commutative k-algebra such as H∗(G, k). A regular k[z,y] consisting of all polynomials in which the sequence for R is a sequence u1,u2,...,ur of coefficient on z is zero. This time R is an integral homogeneous elements of positive degree having domain, because k[z,y] is an integral domain. the following properties. First, u1 should be a Hence, every element in R is regular. But it is not regular element, meaning that multiplication by u1 possible to choose a regular sequence of length 2. is an injective map or that the annihilator of u1 in For example, if we choose y as the first element of { } ≤ ≤ R is 0 . Then for each 2 i r, we must have that the regular sequence, then the element zy (which ui is a regular element on the quotient ring is not in the ideal (y) ⊆ R) is annihilated by every R/(u1,...,ui−1). That is, multiplication by ui on the element of positive degree in R/(y). quotient ring is an injective homomorphism of R- The depth is one of the ways in which the mod modules. The definition of depth is the following. p cohomology rings of finite groups are special. It ∗ Definition. The depth of a finitely generated graded is never the case that H (G, k) has depth 0. In fact, commutative k-algebra R is the length r of the there is a theorem by Duflot [12] which says that longest regular sequence of elements in R. the depth of a cohomology ring is at least as large as the p-rank of the center. Indeed, it can be shown In fact, a sequence u1,...,ur is regular if and only that if u1,...,ur is a sequence of homogeneous if the ring R is a free module over the polynomial elements whose restriction to the center of a Sylow subring k[u1,...,ur ] ⊆ R. Thus, if a sequence of p-subgroup of G is a regular sequence, then elements in R is regular, then any permutation of u1,...,ur is a regular sequence. the elements in the sequence is also regular. More- In addition, for all the known computations of over, it can be proved that if u1,...,us is a regular cohomology rings of finite groups, there is no sequence, then it can be extended to a regular example of a cohomology ring where the depth is sequence of maximal length. not determined by the existence of an associated For a ring such as a graded polynomial ring, the prime. In every case that we know of, there is a depth is again equal to the Krull dimension. That nonzero element u in H∗(G, k) such that the anni- is, for the ring k[z,y,x] the sequence z,y,x is a hilator p of u is a prime ideal with H∗(G, k)/p regular sequence. Such rings are said to be Cohen- having Krull dimension equal to the depth of Macaulay. Certainly, the depth of any ring is never H∗(G, k). So we can ask if this is always the case. greater than the Krull dimension. However, the depth ∗ Question. Does every cohomology ring H (G, k) can be less than the Krull dimension, and this can have an associated prime p such that the depth of happen for several reasons. An example to consider ∗ H (G, k)/p coincides with the Krull dimension of is the cohomology ring of the semidihedral group ∗ H (G, k)? over a field of characteristic 2 as in the introduction. It has the form H∗(G, k) k[z, y, x, w]/I where I is There are a few partial results. The answer is the ideal generated by z3,zy,zxand x2 + y 2w. Note affirmative if the p-rank of G is one [8]. The same that the generators z, y, x, w are in degrees 1, 1, 3, happens if the depth is equal to the p-rank of the and 4, so that the relation x2 + y 2w is homogeneous center of a Sylow p-subgroup of G [16]. Beyond that, in degree 6. In this case, H∗(G, k) has depth 1 while not much is known. Indeed, there does not seem its Krull dimension is 2. The point is that the annihi- to be much known about the class of rings with this lator of the element z2 is the ideal A = (z,y,x). Prime property. ideals that are the annihilators of elements in a ring Detecting Cohomology are called associated primes for the ring. In this case The question raised above is not an idle one. It is ∗ an element of positive degree in H (G, k) is regular if intimately involved with at least one study that and only if it is not contained in A. So no matter what is central to many of the efforts to compute regular element u1 may be chosen for the first ele- group cohomology. This is the question of detec- ment in a regular sequence, the class of z2 in tion of cohomology. We say that a collection S of

APRIL 2005 NOTICES OF THE AMS 429 subgroups of G detects the cohomology of G, Computing Group Cohomology provided the intersections of the kernels of the In this section, we describe the computer calcula- S restriction maps to the subgroups in is zero: tions for cohomology. Embedded in the algorithms ∗ ∗ is the definition of group cohomology and the Kernel(resG, H :H (G, k) → H (H,k)) ={0}. H∈S product structure. At the present time, there are at least two different implementations of the We should point out that if S is a Sylow p-subgroup algorithms. The older implementation [8, 10] by the of G, then the restriction map res is always in- G, S author and his collaborators relies almost entirely jective. Consequently, any collection of subgroups on linear algebra methods in the early stages of the that contains a Sylow subgroup or any subgroup that computation. The newer implementation by David contains a Sylow subgroup is a detecting family. Green [15] uses noncommutative Gröbner bases. We saw above that for G a dihedral 2-group, the Green’s programs run somewhat faster and (more two elementary abelian subgroups detect the coho- importantly) are less extravagant with memory mology of G. On the other hand, for some p-groups usage. The basic algorithms are the same for both there is no family of detecting subgroups of G that systems. The first system is built into the computer does not include G itself. An example is the quater- algebra system MAGMA [6], while Greens’s system nion group of order 8, which can be presented as is a stand-alone package. G =g,h | g2 = h2,g4 = 1,hgh−1 = g−1 . The computer programs are designed to compute the cohomology rings only of p-groups. This Its cohomology ring has the form is arguably the hardest problem, since the first H∗(G, k) k[z,y,x]/(z2 + zy + y 2,z2y + zy2) step of most cohomology calculations is to compute the cohomology of the Sylow p-subgroup where the generators z,y,x have degrees 1, 1, and of the group. The main reasons for considering 4. All of the subgroups of order 4 in G are cyclic p-groups have more to do with the reality of com- and the restriction of any element in degree 2 or ∗ puter capabilities. On the one hand, while the co- any element in degree 3 in H (G, k) to any one of homology of p-groups is important, the groups these subgroups is zero. are also small, and the projective modules are also The existence of detecting families has certainly small and easily constructible. The Mathieu group played a role in several complicated cohomology M has 95,040 elements, but its Sylow subgroup calculations (see, for example, [1]). The point is 12 has order 64 [3]. that if a detecting family is known, then the ideal of relations among the cohomology generators can The other reason that makes the calculation of be found by simply taking the intersection of the the cohomology of p-groups seem practical is that kernels of the restrictions to the members of the the group rings are local rings. If G is a p-group detecting family. It reduces the computational and k is a field of characteristic p, then in kG we − pn = pn − = problem to finding the generators of cohomology have that (x 1) x 1 0 for any element x n and computing their restriction to the elements in of order p in G. So the augmentation ideal, which − ∈ the detecting family. This is often a big improve- is the ideal spanned by all x 1 for x G is a ment over trying to compute the relations among nilpotent ideal of codimension 1 in kG. It is the the generators directly. radical of kG, and kG/ Rad(kG) k. It follows The connection between depth and detection from this fact that every projective kG-module is comes from the following theorem [7]. free and is isomorphic to a direct sum of copies of kG. Consequently, the projective modules are easy Theorem. Suppose that the cohomology ring to construct on the computer. ∗ H (G, k) has depth d. Then the cohomology is de- For the rest of this section we assume that G is tected on the centralizers of the elementary abelian a p-group and that k is the prime field k = Fp. p-subgroups of rank d. Since only the characteristic of the field really The question that often arises is the converse. matters, for computational purposes we use the smallest field possible. Question. Suppose that the cohomology of G is Projective Resolutions detected on the centralizers of the elementary The first step in the cohomology computation is abelian p-subgroups of rank r. Is it necessary that the construction of a projective resolution for the the depth of H∗(G, k) be at least r? trivial kG-module k kG/ Rad(kG). The projective Adem and Karagueuzian [2] have shown this to resolution is an exact sequence of the form be true in the case that the rank of the center of a ∂ ∂ ∂ ε ··· → P →3 P →2 P →1 P → k → 0 Sylow p-subgroup of G coincides with the p-rank 3 2 1 0 of G. The answer to the question is yes for all of where each Pi is a projective kG-module. In prac- the cohomology rings that we have computed. tice, we take a minimal resolution of k, meaning

430 NOTICES OF THE AMS VOLUME 52, NUMBER 4 ⊆ that P0 kG and that for every i, ∂(Pi ) Rad(Pi−1). linear equations to obtain each ζi in succession. This insures that in the induced complex In each degree n, the matrices of the monomials ∂∗ of degree n in all previously calculated generators 0 → Hom (P ,k) → Hom (P ,k) kG 0 kG 1 are computed so as to avoid computing chain ∗ ∗ ∂ ∂ maps for elements that are not generators for the → HomkG(P2,k) → ... cohomology. all of the maps are zero maps. The definition of the Once we have computed the chain maps for the group cohomology is precisely the cohomology of generators of cohomology, we want the relations. n = n this complex. That is, H (G, k) H (HomkG(P∗,k)), So suppose the generators are elements of the set th the n cohomology group of this complex. Because {z,y,x,...}. Then for each n we compute the in- we have used a minimal resolution, we have that → n ∗ duced maps Pn/ Rad(kG)Pn k for all monomials H (G, k) = HomkG(Pn,k) (Pn/ Rad(kG)Pn) , where ∗ of degree n in the generators. Each such map is a the () indicates the k-dual. (1 × n)-matrix since the null space of the matrix ∂→n The calculation of a typical map Pn Pn−1 in with these matrices (vectors) as rows is the set of the projective resolution proceeds by reasonably all relations among the monomials of degree n. easy steps. First, in order that the sequence be Of course, we obtain far too many relations exact, the image of ∂n should be the kernel of ∂n−1 . from the computation. For practical reasons, some So we compute the kernel Kn of ∂n−1. If we have rep- analysis is required. resented ∂ − as a matrix, then we are only com- n 1 Gröbner Bases puting a null space. Next, we find a basis for At this point we are given a polynomial ring and a Rad(kG)K = ∈ (x − 1)K . Of course, in practice n x G n huge collection of polynomials that generate the we need only to let the sum range over a set of generators for G. Third, we find a complementary ideal of relations. The programs described in [8] basis u ,...,u for Rad(kG)K in K . Finally, we generate many relations in each degree. For ex- 1 tn n n ample, if x2 − zy is a relation in degree 2, then in let Pn be the direct sum of tn copies of kG and construct ∂ as the map which takes the identity degree 3 the system would generate the relation n 2 − 2 2 − 2 3 − element 1 in the ith copy of kG to the element zx z y as well as yx zy and x zyx. We would like to have a minimal set of generators for ui ∈ Kn ⊆ Pn−1. Various time- and space-saving tricks are built the ideal of relations, so we need some technique for analyzing the data. In addition we need to into the program. For example, the spaces Kn and calculate ring-theoretic properties such as depth, Pn are never constructed as actual kG-modules but rather are just k-vector spaces on which the action regular sequences, and associated primes. The of an element x ∈ G is determined by the multi- method that does all of this is Gröbner bases. plication of x on kG. The algorithm for obtaining a minimal set of relations is very simple. Suppose that S = Products and Relations {s |i = 1,...,t} are the relations that have been There are several equivalent ways of defining the i computed. First we order the set by degree, so that cup product structure on group cohomology. One Degree(s ) ≤ Degree(s ) whenever i

APRIL 2005 NOTICES OF THE AMS 431 Gröbner basis for an ideal I is a collection S=SI map. The inflation map from a quotient group of of polynomials that generate I and also has the G is constructed similarly. property that the leading term of any f in I is Other properties of cohomology rings, such as divisible by the leading term of some element of the depth or the associated primes can be computed S. It is a theorem that every ideal has a Gröbner using the Gröbner basis machinery. Some of these basis [11]. calculations can be very time consuming when the The membership test goes as follows. Let f be rings are complicated. any element of P. If the leading term of f is not Are We Almost There Yet? divisible by the leading term of any element of S, The reader must have noticed by this point that we then we are done. Otherwise, there exists an ele- are computing the cohomology degree by degree. ment s ∈SI and a monomial α ∈ P such that the The problem with this approach should be obvious. leading term of f − αs is lower in the ordering If we only compute to degree 7, then any genera- than that of f. So now replace f by f − αs and tor or any relation in degree 8 will be completely repeat until either f = 0 or it is shown that f is not lost, ignored. On the other hand, the number of in I. steps that we can compute is finite. Not only do the Computing a Gröbner basis for an ideal can be projective resolutions grow to absorb all available difficult and time consuming. For one thing, the memory, but computing time also becomes a fac- Gröbner basis depends heavily on the chosen order tor. A typical calculation of the cohomology ring on the monomials. Consider the example of the of one of the groups of order 64 (assuming the ideal I=x2,z2 + yx⊆k[z,y,x]. Suppose that group has rank 3 or more) would use a hundred or the ordering is chosen as graded lexicographic more megabytes of RAM after fifteen to twenty and that z>y>x. Then it can be seen that the steps. In general, the time to compute each step Gröbner basis for I consists of the two generators would exceed the sum of all the time to compute x2 and z2 + yx which have leading terms x2 and z2 . the previous steps. The memory requirements also On the other hand, if the ordering is chosen so that increase dramatically with each step. z

432 NOTICES OF THE AMS VOLUME 52, NUMBER 4 positive degree in H∗(G, k). It is always possible to a ring and as a module over the polynomial sub- choose a homogeneous set of parameters with the ring k[z,y] are in degrees less than m + n. Likewise, property that the first d elements in the sequence it can be shown that the ideal of relations among is a regular sequence of length equal to the depth the generators is generated in degrees at most d of H∗(G, k). 2(m + n). The basic idea behind most of the tests for com- Quasiregular Sequences and Regularity pletion can be illustrated with the following What the sample test shows is that, while we can example. Here we present only a sketch. More not expect a cohomology ring to have a regular details can be found in Theorem 14.5.2 of [10]. sequence which is as long as its Krull dimension, A Sample Test for Completion we might always be able to find a quasiregular Suppose that the cohomology ring of G has Krull sequence. The notion of a quasiregular sequence dimension 2. Then we can choose a homogeneous was introduced by the author and Benson in [5]. The set of parameters z,y such that the element z is a definition follows. regular element. Let m, n be the degrees of z and Definition. Suppose that H is a graded commutative, y, respectively. Now choose a minimal projective finitely generated k-algebra. We say that a sequence resolution (P∗,ε) of the trivial module k as above. Then the cohomology element z is represented by of homogeneous elements x1,...,xn in degrees n1,...,nn is a quasiregular sequence, provided that a cocycle fz : Pm → k. The statement that fz is a cocycle means that the composition for each i we have that the map ∂ f j j+ni z xˆi :(H/(x1,...,xi−1)) → (H/(x1,...,xi−1)) Pm+1 → Pm → k is the zero map. Hence, fz induces a homomorphism of multiplication by xi is injective for all degrees f : Pm/∂(Pm+1) → k whose kernel we denote as Lz . (*) j ≥ n +···+n − . So we have an exact sequence 1 i 1 If the cohomology ring has a quasiregular se- → → f→z → 0 Lz Pm/∂(Pm+1) k 0, quence of length equal to its Krull dimension, then and there is a corresponding long exact sequence there is a straightforward formula for how far one

∗ has to compute in order to obtain a complete set →δ r (fz→) r m ... ExtkG(k, k) ExtkG(Ω (k),k) of generators and relations for the ring. So we must ask the following question. → r →δ r−1 → ExtkG(Lz ,k) ExtkG (k, k) ... . Question. Does every mod p cohomology ring of r r Now ExtkG(k, k) H (G, k) and by degree shifting a finite group have a quasiregular sequence? Extr (Ωm(k),k) Hm+r (G, k). Moreover the map kG In the event that the depth H∗(G, k) is one less (f )∗ is given by multiplication by z. Because z is z than the Krull dimension, then a quasiregular se- a regular element, (f )∗ is injective, and the con- z quence always exists. This is proved by an argument necting homomorphisms δ are all zero maps. Hence which is very similar to that in the example. Benson we have an exact sequences [4] has recently shown that the answer is affirma- ∗ → r (fz→) r+m → r → tive in the case where the depth is no more than two 0 H (G, k) H (G, k) ExtkG(Lz ,k) 0. less than the Krull dimension. A partial result in the Consequently, the cokernel of multiplication by z ∗ codepth 2 case had also been proved by Okuyama in degrees at least m is the cohomology ExtkG(Lz ,k). and Sasaki [19]. However, Benson’s work goes much Now here is the key. The module Lz must be a further and involves the notion of regularity. periodic module, that is, a module whose coho- In [4], Benson considers cohomology rings in the mology repeats regularly after some fixed number context of some construction from local cohomol- of stages. The reason basically is that there is only ogy. In particular, he considers the Castelnuovo- one parameter left after factoring out z (Proposi- Mumford regularity of the rings. The definition of tion 9.7.3 of [10]). Moreover, because y is the only this form of regularity is rather complicated, and parameter left, multiplication by y must induce an we refer to Benson’s paper or to [13] for a detailed isomorphism on the cohomology of Lz . So we have explanation. All we will say here is that it is related to that the cohomology of a certain Koszul complex associ- f ∗ :Hr+m(G, k)/z Hr (G, k) y ated to a homogeneous set of parameters. Such + + + → Hr m n(G, k)/z Hr n(G, k) complexes are doubly graded and the regularity is the maximum value of the internal degrees such that is an isomorphism for all r ≥ 0, where f ∗ is mul- y the cohomology is nonzero. Benson has conjectured tiplication by y. The result is that every element of an affirmative answer to the following question. Hr+m+n(G, k) can be written in the form zα + yβ for some α ∈ Hr+n(G, k) and some β ∈ Hr+m(G, k). It Question. Is the regularity of H∗(G, k) equal to follows that all of the generators of H∗(G, k) as both zero?

APRIL 2005 NOTICES OF THE AMS 433 Benson shows that if the answer is yes, then the [5] D. J. BENSON and J. F. CARLSON, Projective resolutions and cohomology ring has a quasiregular sequence. Poincaré duality complexes, Trans. Amer. Math. Soc. Moreover, he shows that it has what he calls a 132 (1994), 447–488. strongly quasiregular sequence—a quasiregular [6] W. BOSMA and J. CANNON, Handbook of Magma Functions, Magma Computer Algebra, Sydney, 2002. sequence as in the definition above except that ∗ ≥ i−1 − [7] J. F. CARLSON, Depth and transfer maps in the the condition ( ) is replaced by j t=1(nt 1). cohomology of groups, Math. Z. 218 (1995), 461–468. Moreover, he shows there is an explicit formula in [8] ——— , Problems in the computation of group terms of the regularity of the computed cohomol- cohomology, Progr. Math. 173 (1999), 107–120. ogy ring that determines whether the computation [9] ——— , Calculating group cohomology: Tests for is complete. That is, if Rn is the computation of completion, J. Symbolic Comput. 31 (2001), 229–242. ∗ H (G, k) calculated to degree n, and if Rn satisfies [10] J. CARLSON, L. TOWNSLEY, L. VALERO-ELIZONDO, and Benson’s formula, then it must be the case that M. ZHANG, Cohomology Rings of Finte Groups, Kluwer, ∗ Dordrecht, 2003. Rn H (G, k). [11] D. A. COX, J. LITTLE, and D. O’SHEA, Ideals, Varieties So What Is True? and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, The investigations of the questions about depth, de- Springer-Verlag, New York, 1997. tection, and regularity share one curious property. [12] J. DUFLOT, Depth and equivariant cohomology, Com- The theoretical developments seem to be right at ment. Math. Helv. 56 (1981), 627–637. the edge of the experimental evidence. That is to say, [13] D. EISENBUD and S. GOTO, Linear free resolutions and we can prove most of what we can see. Moreover, minimal multiplicities, J. Algebra 88 (1984), 89–133. seeing the evidence and proving the theorems seem [14] L. EVENS, The cohomology ring of a finite group, to go hand in hand. Trans. Amer. Math. Soc. 101 (1961), 224–239. A case in point is the study of the quasiregular [15] D. J. GREEN, Gröbner Bases and the Computation of Group Cohomology, Lecture Notes in Math., vol. 1828, sequences and regularity. We can prove the results Springer-Verlag, Berlin, 2003. we want as long as the codepth of the cohomology [16] ——— , On Carlson’s conjecture in group cohomology, ring is at most 2. This is also approximately what Math. Z. 244 (2003), 711-723. we have computed. There are almost no computed [17] , private communication, 2004. ∗ ——— examples of groups G such that H (G, k) has [18] J. LANNES, Sur les espaces fonctionnels dont la source codepth more than 2. Green recently reported that est le classificant d’un p-groupe abélien élémentaire, he computed the mod 2 cohomology of codepth 3 Publ. Math. Inst. Hautes Études Sci. 75 (1992), 135–244. of a group of order 128 [17]. In this case he got an [19] T. OKUYAMA and H. SASAKI, private communication. affirmative answer to Benson’s question (the reg- [20] D. QUILLEN, The spectrum of an equivariant coho- ∗ mology ring, I, II, Ann. of Math. 94 (1971), 549–602. ularity of H (G, k) equal to zero), but one example [21] B. B. VENKOV, Cohomology algebras of some arbitrary is hardly strong evidence. classifying spaces, Dokl. Akad. Nauk SSSR 127 (1959), The questions on depth and associated primes 943–944. (Russian) were developed partly by looking at experimental evidence. The computations have also aided in eliminating some of the earlier speculations. For example, it has now been shown that Question 3.3 of [7] has a negative answer. Counterexamples (if they exist) to any of conjectures will likely lie in large and complicated groups, and the computations become more expensive in terms of time and memory. At the same time both the equipment and the software are improving and perhaps will some day provide us with the insights to answer some of the questions.

References [1] A. ADEM, J. F. CARLSON, D. KARAGUEUZIAN, and R. J. MILGRAM, The cohomology of the Sylow 2-subgroup of Higman-Sims, J. Pure Appl. Algebra 164 (2001), 275–305. [2] A. ADEM and D. KARAGUEUZIAN, Essential cohomology of finite groups, Comment. Math. Helv. 72 (1997), 101–109. [3] A. ADEM, J. MAGINNIS, and R. J. MILGRAM, The geometry and cohomology of M12, J. Algebra 139 (1991), 90–133. [4] D. J. BENSON, Dickson invariants and regularity in group cohomology, preprint.

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