Exact structures for persistence modules
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910771609337856 |
|---|---|
| author | Blanchette, Benjamin Brüstle, Thomas Hanson, Eric J. |
| author_facet | Blanchette, Benjamin Brüstle, Thomas Hanson, Eric J. |
| contents | We discuss applications of exact structures and relative homological algebra to the study of invariants of multiparameter persistence modules. This paper is mostly expository, but does contain a pair of novel results. Over finite posets, classical arguments about the relative projective modules of an exact structure make use of Auslander-Reiten theory. One of our results establishes a new adjunction which allows us to ``lift'' these arguments to certain infinite posets over which Auslander-Reiten sequences do not always exist. We give several examples of this lifting, in particular highlighting the non-existence and existence of resolutions by upsets when working with finitely presentable representations of the plane and of the closure of the positive quadrant, respectively. We then restrict our attention to finite posets. In this setting, we discuss the relationship between the global dimension of an exact structure and the representation dimension of the incidence algebra of the poset. We conclude with our second novel contribution. This is an explicit description of the irreducible morphisms between relative projective modules for several exact structures which have appeared previously in the literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_01790 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Exact structures for persistence modules Blanchette, Benjamin Brüstle, Thomas Hanson, Eric J. Algebraic Topology Computational Geometry Representation Theory 55N31, 16G20, 18G25 (primary), 16E10, 16E20, 16S50, 19A49 (secondary) We discuss applications of exact structures and relative homological algebra to the study of invariants of multiparameter persistence modules. This paper is mostly expository, but does contain a pair of novel results. Over finite posets, classical arguments about the relative projective modules of an exact structure make use of Auslander-Reiten theory. One of our results establishes a new adjunction which allows us to ``lift'' these arguments to certain infinite posets over which Auslander-Reiten sequences do not always exist. We give several examples of this lifting, in particular highlighting the non-existence and existence of resolutions by upsets when working with finitely presentable representations of the plane and of the closure of the positive quadrant, respectively. We then restrict our attention to finite posets. In this setting, we discuss the relationship between the global dimension of an exact structure and the representation dimension of the incidence algebra of the poset. We conclude with our second novel contribution. This is an explicit description of the irreducible morphisms between relative projective modules for several exact structures which have appeared previously in the literature. |
| title | Exact structures for persistence modules |
| topic | Algebraic Topology Computational Geometry Representation Theory 55N31, 16G20, 18G25 (primary), 16E10, 16E20, 16S50, 19A49 (secondary) |
| url | https://arxiv.org/abs/2308.01790 |