Homology, Homotopy and Applications

Volume 24 (2022)

Number 1

Generalized persistence and graded structures

Pages: 27 – 53

DOI: https://dx.doi.org/10.4310/HHA.2022.v24.n1.a2

Authors

Eero Hyry (Faculty of Information Technology and Communication Sciences, Tampere University, Tampere, Finland)

Markus Klemetti (Faculty of Information Technology and Communication Sciences, Tampere University, Tampere, Finland)

Abstract

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.

Keywords

persistence module, graded module, action category, smash product, finitely presented

2010 Mathematics Subject Classification

13E15, 16D90, 16W50

Received 22 October 2020

Received revised 8 February 2021

Accepted 11 February 2021

Published 30 March 2022