Homology, Homotopy and Applications
Volume 24 (2022)
Generalized persistence and graded structures
Pages: 27 – 53
We investigate the correspondence between generalized persistence modules and graded modules in the case the indexing set has a monoid action. We introduce the notion of an action category over a monoid graded ring. We show that the category of additive functors from this category to the category of Abelian groups is isomorphic to the category of modules graded over the set with a monoid action, and to the category of unital modules over a certain smash product. Furthermore, when the indexing set is a poset, we provide a new characterization for a generalized persistence module being finitely presented.
persistence module, graded module, action category, smash product, finitely presented
2010 Mathematics Subject Classification
13E15, 16D90, 16W50
The second author was supported in part by the Finnish Cultural Foundation.
Received 22 October 2020
Received revised 8 February 2021
Accepted 11 February 2021
Published 30 March 2022