## Section: Software

### AbelianSystems

Participants : Alban Quadrat [correspondent] , Mohamed Barakat [Univ. Kaiserslautern] .

The AbelianSystems package is an implementation of the algorithm developed in [63] , [72] for the computation of the purity filtration [77] , [78] in the powerful `homalg` package of GAP4 dedicated to constructive homological algebra methods, developed by Barakat (University of Kaiserslautern) and his collaborators (http://homalg.math.rwth-aachen.de/ ). This package supersedes the existing `homalg` procedure which computes purity filtrations based on the computation of spectral sequences and the PurityFiltration package which is based on the rather slow Maple GrĂ¶bner basis computation (see SectionÂ
5.6 ). Using the design of the `homalg` package, where the different layers such as the computational engine (which can the computer algebra systems `Singular` , `Macaulay2` , OreModules , `MAGMA` , `SAGE` ), the module-theoretic results, and the homological ones are separated, the AbelianSystems package can be used for the computation of purity filtration of different objects in abelian categories (e.g., purity filtration of sheaf cohomology of projective varieties). This package will be soon available on the `homalg` website and at http://www-sop.inria.fr/members/Alban.Quadrat/AbelianSystems.html .