Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2020
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2007.09174 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909268259635200 |
|---|---|
| author | Lyle, Justin Montaño, Jonathan Sather-Wagstaff, Keri |
| author_facet | Lyle, Justin Montaño, Jonathan Sather-Wagstaff, Keri |
| contents | A commutative Noetherian ring $R$ is said to be Tor-persistent if, for any finitely generated $R$-module $M$, the vanishing of $\operatorname{Tor}_i^R(M,M)$ for $i\gg 0$ implies $M$ has finite projective dimension. An open question of Avramov, et. al. asks whether any such $R$ is Tor-persistent. In this work, we exploit properties of exterior powers of modules and complexes to provide several partial answers to this question; in particular, we show that every local ring $(R,\mathfrak{m})$ with $\mathfrak{m}^3=0$ is Tor-persistent. As a consequence of our methods, we provide a new proof of the Tachikawa Conjecture for positively graded rings over a field of characteristic different from 2. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_09174 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Exterior powers and Tor-persistence Lyle, Justin Montaño, Jonathan Sather-Wagstaff, Keri Commutative Algebra Rings and Algebras A commutative Noetherian ring $R$ is said to be Tor-persistent if, for any finitely generated $R$-module $M$, the vanishing of $\operatorname{Tor}_i^R(M,M)$ for $i\gg 0$ implies $M$ has finite projective dimension. An open question of Avramov, et. al. asks whether any such $R$ is Tor-persistent. In this work, we exploit properties of exterior powers of modules and complexes to provide several partial answers to this question; in particular, we show that every local ring $(R,\mathfrak{m})$ with $\mathfrak{m}^3=0$ is Tor-persistent. As a consequence of our methods, we provide a new proof of the Tachikawa Conjecture for positively graded rings over a field of characteristic different from 2. |
| title | Exterior powers and Tor-persistence |
| topic | Commutative Algebra Rings and Algebras |
| url | https://arxiv.org/abs/2007.09174 |