Simon King′s home page:
Mathematics:
Cohomology
Jena:
External links: 
Background of our Cohomology ComputationsBack to the Cohomology Rings
Theoretical backgroundWe describe here the theoretical background for our computation of modular cohomology rings of finite groups.Basic approchLet G be a finite group and let U be a subgroup of G that contains a SylowIn both cases, a completeness criterion eventually tells us when the approximation is isomorphic to the actual cohomology ring. This basic approach was introduced by Jon Carlson. The case of prime power groupsFor computingConstruction of minimal projective resolutionsMinimal projective resolutions are constructed by Gröbner basis techniques that were introduced in [1] by David Green.Approximation of the ring structureNow let a resolution R_{✱} be given. A dcochain C gives rise to a chain map C_{✱}: R_{✱}→R_{✱} mapping R_{i} to R_{id} for
Assume that we have a minimal list of generators of the cohomology ring out to degree n, and let
F_{n} be the free graded commutative algebra generated by them. Let
I_{n} < F_{n} be the ideal generated by the algebraic relations out to degree n
that hold between the cohomology generators. Then, the nth approximation of the
cohomology ring is the quotient A_{n} = For obtaining the next approximation, we compute the degree n+1 standard monomials of I_{n} in F_{n} using a homogeneous Gröbner basis of I_{n} up to degree n+1. The standard monomials can be represented by cochains, and elementary linear algebra then yields either decomposable n+1 cochains or new relations in degree n+1. In that way, we obtain I_{n+1}. Choice of new generatorsLet V < R_{n+1} be a complement for the subspace of decomposable n+1 cochains. A basis for V (if nonzero) yields new cohomology generators in degree n+1. We take advantage of a particular choice of such basis.
Our strategy for choosing new generators relies
on the restriction maps to the greatest central elementary subgroup and to the maximal elementary abelian subgroups.
Hence, these subgroups have to be computed in the very beginning. Then, the restriction maps
have to be lifted to degree n+1.
That choice of generators has three benefits:
Completeness criteria for modular cohomology rings of prime power groupsFor proving completeness in the case of a prime power group, we alternatively use two methods. The first is an improved version of the criterion that Dave Benson introduced in [1]. The second criterion was introduced by Peter Symonds in [6]. For both criteria, it is essential to construct in A_{n} a filterregular homogeneous system of parameters that is known to yield a homogeneous system of parameters for the cohomology ring. Benson suggests to construct such parameters by lifting (powers of) Dickson invariants from the cohomology rings of maximal pelementary abelian subgroups. In [4], we suggest a method that usually yields smaller parameters:
Modified Benson CriterionIn [4], we show that, under certain conditions, in addition to the x_{i} there exist filter regular parameters y_{1},...,y_{r} for the cohomology ring of G with coefficients in a finite extension of F_{p}, so that the degree sum is smaller than the degree sum of x_{1},...,x_{r}.
An upper bound for the Mumford regularity of A_{n}
can be computed using the explicitly constructed parameters
x_{1},...,x_{r}. Assume that n is
larger than the sum of the Mumford regularity of
A_{n} plus the degree sum of
y_{1},...,y_{r} minus the prank of
the group (or at least as large if it is known that the cohomology
ring is of depth at least two). Then, A_{n} is
isomorphic to the cohomology The advantage of this criterion is that it can already apply in quite low degree, due to using our existence result for parameters of small degrees. The disadvantage is that the computation of an upper bound for the Mumford regularity (via a computation of the filter degree type of the parameters) can be quite difficult. Therefore, if the degree of the explicitly constructed parameters x_{1},...,x_{r} is small enough, we apply the following criterion due to Peter Symonds. Symonds′ CriterionLet x_{1},...,x_{r} be a (not necessarily filter regular) homogeneous system of parameters for A_{n} that gives rise to parameters for the cohomology ring. If n is larger than the degree sum of the parameters minus the prank of the group and if it is at least as large as the maximal degree of a module generator of A_{n}, considered as a module over the parameters, then A_{n} is isomorphic to the cohomologyThe advantage of this criterion is that it is quite easy to use. The disadvantage is that it depends on the explicit construction of parameters of small degrees. The case of non prime power groupsLet G be a finite group and let U be a subgroup of G that contains a Sylow
Now, an approximation in increasing degree is constructed
analogously to the case of prime power groups, computing a
complement of the decomposable elements in the stable subspace
of A completion criterion for the cohomology generatorsIt is known thatThe construction of parametersIt is, in general, not true that the Duflot generators form a regular sequence; for example, if the Sylow subgroup is elementary abelian then every generator is a Duflot generator. Nevertheless, it is easy to find a sequence of elements in the cohomology ring approximation that restricts to a maximal regular sequence in the greatest central elementary abelian subgroup C of a Sylow subgroup S.However, the Dickson elements in the cohomology rings of a complement of C in the maximal elementary abelian subgroups of S can, in general, not be lifted to elements of the cohomology of G. Therefore, we proceed as follows:
Then, we try to prove the existence of smaller parameters as in the case of prime power groups. The HilbertPoincaré criterionThe Poincaré series ofMoreover, if y_{1},...,y_{r} is a homogeneous system of parameters of degrees d_{1},...,d_{r} for the cohomology, one can write this rational function with a denominator that is given as a product of 1t^{di}.
From this, one can show [5]: If the cohomology ring has no
generators above degree n (which can be tested as we have
explained above) and the cohomology ring with coefficients
in a finite extension field of F_{p}
has parameters in degrees
d_{1},...,d_{r} and if n is at
least the sum of the d_{i} minus the depth of
This criterion has two advantages: We can use our existence result for small parameters, and it is easy to use. AcknowledgmentWe are grateful to Peter Symonds for interesting discussions, in particular for suggesting to us to use the Poincaré series for a completeness criterion.References
ImplementationOur implementation is based on the broad range of free open source Computer Algebra software that is accessible with Sage, and on some further pieces of mathematical software.
It is of particular importance for us
that the Computer Algebra Systems
AcknowledgmentWe are greatful to Mathieu Dutour Sikiric and to the Gap support team for pointing out how to efficiently use Gap.
Back to the Cohomology Rings

Last change: 10/02/2008