Background of our Cohomology Computations

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Theory Implementation

Theoretical background

We describe here the theoretical background for our computation of modular cohomology rings of finite groups.

Basic approch

Let G be a finite group and let U be a subgroup of G that contains a Sylow p-subgroup S of G. Usually, we take U = NG(Z(S)), or U=S. We first compute the cohomology ring H(S;Fp). Using the stable element method, one can identify H(U;Fp) as a subring of H(S;Fp) and finally H(G;Fp) as a subring of H(U;Fp). That way, we can approximate the cohomology ring structure in increasing degree. For computing H(S;Fp), an approximation of the ring structure is based on the construction of a minimal projective resolution.

In 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 groups

For computing H(S;Fp), we construct finitely many terms of a minimal projective resolution, which allows for a degree-wise approximation of the cohomology ring.

Construction of minimal projective resolutions

Minimal projective resolutions are constructed by Gröbner basis techniques that were introduced in [1] by David Green.

Approximation of the ring structure

Now let a resolution R be given. A d-cochain C gives rise to a chain map C: R→R mapping Ri to Ri-d for i ≥ d. The components of C can be computed with standard homological algebra techniques. The cup product of two cohomology classes given by cochains is given by the composition of the associated chain maps.

Assume that we have a minimal list of generators of the cohomology ring out to degree n, and let Fn be the free graded commutative algebra generated by them. Let In < Fn be the ideal generated by the algebraic relations out to degree n that hold between the cohomology generators. Then, the n-th approximation of the cohomology ring is the quotient An = Fn ⁄ In.

For obtaining the next approximation, we compute the degree n+1 standard monomials of In in Fn using a homogeneous Gröbner basis of In 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 In+1.

Choice of new generators

Let V < Rn+1 be a complement for the subspace of decomposable n+1 cochains. A basis for V (if non-zero) 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.
New generators are eventually chosen by the following procedure:

  1. Start with a basis for the subspace of V with nilpotent restriction to all maximal elementary abelian subgroups. It is known that these generators are nilpotent.
  2. Extend this to a basis for the subspace of V with nilpotent restriction to the greatest central elementary abelian subgroup. The generators introduced in this step have no significant property, hence, we call these generators "boring".
  3. Finally, extend this to a basis for V. This last step yields so-called Duflot generators.

That choice of generators has three benefits:

  • In the completeness criterion, we will use the Duflot generators anyway.
  • We equip Fn with a monomial order that takes into account the number of nilpotent respectively Duflot generators occuring in a monomial. It was observed in [1] by David Green that this monomial order magically simplifies the computations and often yields relatively small Gröbner bases.
  • If there are nilpotent or boring generators in degree d then we can expect new relations at least out to degree 2d. Hence, we will not try to prove completeness before degree 2d. This heuristics is not perfect, but saves a massive amount of resources.

Completeness criteria for modular cohomology rings of prime power groups

For 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 An a filter-regular 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 p-elementary abelian subgroups. In [4], we suggest a method that usually yields smaller parameters:
  1. The Duflot generators yield a regular sequence. If one extends it by lifts of (powers of) Dickson elements in the cohomology rings of a complement of the greatest central p-elementary abelian subgroup in the maximal p-elementary abelian subgroups, a filter regular system of parameters for the cohomology ring results.
  2. Moreover, if a parameters can be expressed as a non-trivial product, possibly after modification by a nilpotent element, then it can be replaced by any non-skalar factor.
  3. The last parameter can be replaced by any parameter of smaller degree, since filter-regularity is automatic for the last parameter.
In that way, we often obtain filter regular parameters x1,...,xr that often are of fairly small degrees.

Modified Benson Criterion

In [4], we show that, under certain conditions, in addition to the xi there exist filter regular parameters y1,...,yr for the cohomology ring of G with coefficients in a finite extension of Fp, so that the degree sum is smaller than the degree sum of x1,...,xr.

An upper bound for the Mumford regularity of An can be computed using the explicitly constructed parameters x1,...,xr. Assume that n is larger than the sum of the Mumford regularity of An plus the degree sum of y1,...,yr minus the p-rank of the group (or at least as large if it is known that the cohomology ring is of depth at least two). Then, An is isomorphic to the cohomology ring [4].

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 x1,...,xr is small enough, we apply the following criterion due to Peter Symonds.

Symonds′ Criterion

Let x1,...,xr be a (not necessarily filter regular) homogeneous system of parameters for An that gives rise to parameters for the cohomology ring. If n is larger than the degree sum of the parameters minus the p-rank of the group and if it is at least as large as the maximal degree of a module generator of An, considered as a module over the parameters, then An is isomorphic to the cohomology ring [6].

The 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 groups

Let G be a finite group and let U be a subgroup of G that contains a Sylow p-subgroup S of G (possibly, U=S). Assume that H(U;Fp) is already known. Each double coset for U in G gives rise to a stability condition, which is a pair of maps from H(U;Fp) to the cohomology of a subgroup of U. Then, H(G;Fp) can be identified with the subring of H(U;Fp) for which the two maps coincide, for each double coset. See, for example, Prop. 3.8.2 in [2].

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 Hn(U;Fp), etc.

A completion criterion for the cohomology generators

It is known that H(U;Fp) is a finitely generated module over H(G;Fp). Using that the transfer map is a module homomorphism, one can show that a minimal generating set for H(G;Fp) has degree bounded by the top degree of a module generator of H(U;Fp). But this top degree can eventually be computed from the approximation. If one knows that there will be no further generators in degree n, the computation becomes much easier: It is then not needed to compute the stable subspace of Hn(U;Fp) by means of the stability conditions.

The construction of parameters

It 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:

  • Lift (powers of) the Dickson elements in the maximal elementary abelian subgroups of S (not just in a complement of C). This is always possible, and yields a filter-regular homogeneous system of parameters for the cohomology ring.
  • Test whether the parameters constructed for the Sylow subgroup happen to be stable. This is not always the case. But if they are, they yield filter regular parameters for the cohomology ring of G.
  • Try to simplify it by factorisation and by replacing the last parameter.

Then, we try to prove the existence of smaller parameters as in the case of prime power groups.

The Hilbert-Poincaré criterion

The Poincaré series of H(G;Fp) is a rational function whose degree is at most minus the depth of H(G;Fp). The depth is bounded from below by the depth of H(U;Fp).

Moreover, if y1,...,yr is a homogeneous system of parameters of degrees d1,...,dr for the cohomology, one can write this rational function with a denominator that is given as a product of 1-tdi.

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 Fp has parameters in degrees d1,...,dr and if n is at least the sum of the di minus the depth of H(U;Fp), then An is isomorphic to the cohomology ring if and only if its Poincaré series multiplied with the product of 1-tdi is a polynomial of degree at most the sum of the di minus the depth of H(U;Fp).

This criterion has two advantages: We can use our existence result for small parameters, and it is easy to use.

Acknowledgment

We are grateful to Peter Symonds for interesting discussions, in particular for suggesting to us to use the Poincaré series for a completeness criterion.

References

  1. D. J. Benson: Dickson invariants, regularity and computation in group cohomology. Illinois J. Math. 48 (2004), pp. 171–197.
  2. D. J. Benson: Representations and cohomology. I. Cambridge Studies in Advanced Math., vol. 30. Cambridge University Press, Cambridge, second edition, 1998.
  3. D. J. Green: Gröbner bases and the computation of group cohomology. Lecture Notes in Mathematics, 1828. Springer-Verlag, Berlin, 2003.
  4. D. J. Green, S. A. King: The computation of the cohomology rings of all groups of order 128. Preprint, Jan 2010. To appear in J. Algebra.
  5. S. A. King: Completeness criteria for modular cohomology rings of non prime power groups. Preprint, April 2010.
  6. P. Symonds: On the Castelnuovo-Mumford Regularity of the Cohomology Ring of a Group. Preprint, June 2009.

Theory Implementation

Implementation

Our 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. Sage was initiated by William Stein in 2005. Sage′s programming language is Cython, which stands for Compiled Python.

It is of particular importance for us that the Computer Algebra Systems Singular and Gap are included in Sage via interfaces, and that Cython allows for an easy inclusion of C-Code. These features are used as follows.

  • Gap is used for computing the maximal elementary abelian subgroups and other relevant data. The groups were obtained from the Small Groups library and the Atlas of finite groups.
  • David Green wrote C-routines for computing minimal projective resolutions for modular group rings of prime power groups.
  • C-MeatAxe is used for most of the linear algebra computations over finite fields.
  • Simon King wrote several Cython modules for Sage, providing methods for homological algebra computations (cup product, restriction to maximal elementary abelian subgroups, etc).
  • Singular is used for all computations in (graded) commutative algebras: Computing a Gröbner basis of the relation ideal, detecting relations, partially for constructing simultaneous lifts of the subgroup Dickson invariants, for detecting filter-regular systems of parameters and for computing their filter degree type. In our implementation of the stable element method, H(U;Fp) is represented as a quotient ring in Singular.

Acknowledgment

We are greatful to Mathieu Dutour Sikiric and to the Gap support team for pointing out how to efficiently use Gap.


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Simon King
Department of Mathematics and Computer Science
Friedrich-Schiller-Universität Jena
07737 Jena
GERMANY
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46163
Fax: +49 (0)3641 9-46162
Office: Zi. 3529, Ernst-Abbe-Platz 2



Last change: 10/02/2008