Background of our Cohomology Computations

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Theory Implementation

Theoretical background

We give a detailed description of the theoretical background of our computations in [2].

Basic approch

We construct finitely many terms of a minimal projective resolution, which allows for a degree-wise approximation of the cohomology ring. A completeness criterion eventually tells us when the approximation is isomorphic to the actual cohomology ring. This basic approach was introduced by Jon Carlson.

Construction of minimal projective resolutions

Minimal projective resolutions are constructed by Gröbner basis techniques that were introduced in [1] by David Green.
— More about this is to be added soon —

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 U ⊂ Rn+1 be a complement for the subspace of decomposable n+1 cochains. A basis for U (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 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 Fn 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

For testing whether the approximation of the cohomology ring is actually complete, we rely on a slightly improved version of the criterion that Dave Benson introduced in [3].
— More about this is to be added soon —

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.

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.
  • 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.


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Simon A. King David J. Green
Fakultät für Mathematik und Informatik Fakultät für Mathematik und Informatik
Friedrich-Schiller-Universität Jena Friedrich-Schiller-Universität Jena
Ernst-Abbe-Platz 2 Ernst-Abbe-Platz 2
D-07743 Jena D-07743 Jena
Germany Germany

E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46184
Fax: +49 (0)3641 9-46162
Office: Zi. 3524, Ernst-Abbe-Platz 2
E-mail: david dot green at uni hyphen jena dot de
Tel: +49 3641 9-46166
Fax: +49 3641 9-46162
Office: Zi 3512, Ernst-Abbe-Platz 2



Last change: 08/25/2009