Computational Cohomology for basic algebras

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Experience shows that advances in the field of computational homological algebra lead to new theoretical insights, and that theoretical advances in turn result in better computational methods. By combining the work of several authors – including the applicant’s previous DFG project on the cohomology of p-groups – we will develop methods and software to compute cohomology rings and Ext-algebras for a variety of groups and basic algebras, using noncommutative Gröbner bases to construct minimal resolutions. This will allow us to compute a series of interesting test cases: both in group cohomology and for conjectures about finite dimensional algebras such as the Strong No Loops conjecture. The software will make use of the systems Gap and Singular, and will be made available as a Sage package. In addition to the design and implementation of suitable algorithms and the evaluation of the computational results obtained, we will also seek to generalise known degree bounds for cohomology rings to the case of Ext-algebras.