Homological approximations in persistence theory
| Autor: | Blanchette, Benjamin, Brüstle, Thomas, Hanson, Eric J. |
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| Rok vydání: | 2021 |
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| Zdroj: | Canadian Journal of Mathematics 76 (2024), no. 1, 66-103 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.4153/S0008414X22000657 |
| Popis: | 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. They are also thankful to an anonymous referee for their thorough reading of this paper and suggestions for improvement. Comment: v3: accepted manuscript, v2 (sizable update): added numerous references, reorganized paper, added new section on motivation and related work (Section 3), expanded upon the relationship between homological invariants and dimensions of hom-spaces (Theorem 1.1), extended main Theorem 1.2 (formerly Theorem 1.1), corrected errors in comparisons to other invariants (Section 7). 24 pages |
| Databáze: | arXiv |
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