Mathematics:

Cohomology
→Theory
→Implementation

Jena:

Faculty

David Green

External links:

Singular

Gap

Background of our Cohomology Computations

Theoretical background

Basic approch

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

Construction of minimal projective resolutions

Approximation of the ring structure

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 n-th approximation of the cohomology ring is the quotient A_n = F_n ⁄ I_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 generators

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 F_n 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

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.

Modified Benson Criterion

An upper bound for the Mumford regularity of A_n can be computed using the explicitly constructed parameters x₁,...,x_r. Assume that n is larger than the sum of the Mumford regularity of A_n plus the degree sum of y₁,...,y_r 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, A_n 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,...,x_r is small enough, we apply the following criterion due to Peter Symonds.

Symonds′ Criterion

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

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;F_p), etc.

A completion criterion for the cohomology generators

The construction of parameters

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

Moreover, if y₁,...,y_r is a homogeneous system of parameters of degrees d₁,...,d_r for the cohomology, one can write this rational function with a denominator that is given as a product of 1-t^d_i.

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 d₁,...,d_r and if n is at least the sum of the d_i minus the depth of H^✱(U;F_p), then A_n is isomorphic to the cohomology ring if and only if its Poincaré series multiplied with the product of 1-t^d_i is a polynomial of degree at most the sum of the d_i minus the depth of H^✱(U;F_p).

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

Acknowledgment

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.

Implementation

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.
• SharedMeatAxe 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;F_p) is represented as a quotient ring in Singular.

Acknowledgment