Homological Algebra for Persistence Modules

Peter Bubenik, Nikola Milicevic

Published 2019-05-14Version 1

We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module and sheaf tensor product and Hom bifunctors as well as their derived functors, Tor and Ext, and give explicit computations for interval modules. We give a classification of injective, projective, and flat interval modules. We state Kunneth theorems and universal coefficient theorems for homology and cohomology of chain complexes of persistence modules in both the sheaf and graded modules settings. We give a Gabriel-Popescu theorem for persistence modules.