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Homological approximations in persistence theory

Part of: Computational aspects of associative rings Homological methods Homological algebra

Published online by Cambridge University Press:  12 December 2022

Benjamin Blanchette ,
Thomas Brüstle  and
Eric J. Hanson
Benjamin Blanchette
Affiliation:
Départment de Mathématiques, Université de Sherbrooke, Sherbrooke, QC J1K 2R1, Canada e-mail: Benjamin.Blanchette@USherbrooke.ca
Thomas Brüstle*
Affiliation:
Départment de Mathématiques, Université de Sherbrooke, Sherbrooke, QC J1K 2R1, Canada Department of Mathematics, Bishop’s University, Sherbrooke, QC J1M 1Z7, Canada e-mail: tbruestl@bishops.ca
Eric J. Hanson
Affiliation:
LACIM, Université du Québec à Montréal, Montréal, QC H2L 2C4, Canada e-mail: hanson.eric@uqam.ca
*
e-mail: Thomas.Brustle@USherbrooke.ca
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Abstract

We define a class of invariants, which we call homological invariants, for persistence modules over a finite poset. Informally, a homological invariant is one that respects some homological data and takes values in the free abelian group generated by a finite set of indecomposable modules. We focus in particular on groups generated by “spread modules,” which are sometimes called “interval modules” in the persistence theory literature. We show that both the dimension vector and rank invariant are equivalent to homological invariants taking values in groups generated by spread modules. We also show that the free abelian group generated by the “single-source” spread modules gives rise to a new invariant which is finer than the rank invariant.

Keywords

persistence modulesinvariantsGrothendieck groupsrelative homological algebraexact structures

MSC classification

Secondary: 16E20: Grothendieck groups, $K$-theory, etc. 16Z05: Computational aspects of associative rings 18G35: Chain complexes
Type
Article
Information
Canadian Journal of Mathematics , Volume 76 , Issue 1 , February 2024 , pp. 66 - 103
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

This work was supported by the NSERC of Canada. A portion of this work was completed while E.J.H. was affiliated with the Norwegian University of Science and Technology (NTNU).

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