David J. Green

Cohomology
→Theory
→Implementation

Jena:

Faculty

External links:

Singular

Gap

Background of our Cohomology Computations

Theoretical background

Basic approch

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 U 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 U 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 U. This last step yields Duflot regular generators.

That choice of generators has three benefits:

• In the completeness criterion, we will use Duflot regular generators anyway.
• We equip F_n with a monomial order that takes into account the number of nilpotent respectively Duflot regular 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 2⋅d. Hence, we will not try to use the completeness criterion before degree 2⋅d. This heuristic saves a massive amount of resources.

Benson′s completeness criterion

References

1. D. J. Green: Gröbner bases and the computation of group cohomology. Lecture Notes in Mathematics, 1828. Springer-Verlag, Berlin, 2003.
2. D. J. Green, S. A. King: The computation of the cohomology rings of all groups of order 128. J. of Algebra 325 (2011), pp. 352–363. DOI 10.1016/j.jalgebra.2010.08.016.
3. D. J. Benson: Dickson invariants, regularity and computation in group cohomology. Illinois J. Math. 48 (2004), pp. 171–197.

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.
• 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.