Complex Matching Distance and Stability for Minimal Projective Resolutions, with Applications to Persistence
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915932925853696 |
|---|---|
| author | Asashiba, Hideto Patel, Amit K. |
| author_facet | Asashiba, Hideto Patel, Amit K. |
| contents | We develop a stability theory for minimal projective resolutions of $\mathbf{P}$-modules, where $\mathbf{P}$ is a finite metric poset. We use the Gülen-McCleary distance on $\mathbf{P}$-modules together with a new complex matching distance on bounded complexes of finitely generated projective $\mathbf{P}$-modules. The latter yields an extended metric on homotopy classes of such complexes and restricts to minimal projective resolutions. Our main theorem shows that this induced distance on minimal projective resolutions is bounded above by the Gülen-McCleary distance.
As an application, we pass to the interval poset and kernel construction, interpreting persistence diagrams as minimal projective resolutions of kernel modules. This gives a corresponding stability inequality, which in the one-parameter case recovers classical bottleneck stability and in the multiparameter case extends to signed diagrams arising from minimal projective resolutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15726 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Complex Matching Distance and Stability for Minimal Projective Resolutions, with Applications to Persistence Asashiba, Hideto Patel, Amit K. Representation Theory Algebraic Topology Category Theory 16G20, 18G10, 55N31, 62R40 We develop a stability theory for minimal projective resolutions of $\mathbf{P}$-modules, where $\mathbf{P}$ is a finite metric poset. We use the Gülen-McCleary distance on $\mathbf{P}$-modules together with a new complex matching distance on bounded complexes of finitely generated projective $\mathbf{P}$-modules. The latter yields an extended metric on homotopy classes of such complexes and restricts to minimal projective resolutions. Our main theorem shows that this induced distance on minimal projective resolutions is bounded above by the Gülen-McCleary distance. As an application, we pass to the interval poset and kernel construction, interpreting persistence diagrams as minimal projective resolutions of kernel modules. This gives a corresponding stability inequality, which in the one-parameter case recovers classical bottleneck stability and in the multiparameter case extends to signed diagrams arising from minimal projective resolutions. |
| title | Complex Matching Distance and Stability for Minimal Projective Resolutions, with Applications to Persistence |
| topic | Representation Theory Algebraic Topology Category Theory 16G20, 18G10, 55N31, 62R40 |
| url | https://arxiv.org/abs/2602.15726 |