Auslander-type conditions and weakly Gorenstein algebras
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913465042468864 |
|---|---|
| author | Huang, Zhaoyong |
| author_facet | Huang, Zhaoyong |
| contents | Let $R$ be an Artin algebra. Under certain Auslander-type conditions, we give some equivalent characterizations of (weakly) Gorenstein algebras in terms of the properties of Gorenstein projective modules and modules satisfying Auslander-type conditions. As applications, we provide some support for several homological conjectures. In particular, we prove that if $R$ is left quasi Auslander, then $R$ is Gorenstein if and only if it is (left and) right weakly Gorenstein; and that if $R$ satisfies the Auslander condition, then $R$ is Gorenstein if and only if it is left or right weakly Gorenstein. This is a reduction of an Auslander--Reiten's conjecture, which states that $R$ is Gorenstein if $R$ satisfies the Auslander condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_05468 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Auslander-type conditions and weakly Gorenstein algebras Huang, Zhaoyong Rings and Algebras Representation Theory 16E65, 16E10, 18G25 Let $R$ be an Artin algebra. Under certain Auslander-type conditions, we give some equivalent characterizations of (weakly) Gorenstein algebras in terms of the properties of Gorenstein projective modules and modules satisfying Auslander-type conditions. As applications, we provide some support for several homological conjectures. In particular, we prove that if $R$ is left quasi Auslander, then $R$ is Gorenstein if and only if it is (left and) right weakly Gorenstein; and that if $R$ satisfies the Auslander condition, then $R$ is Gorenstein if and only if it is left or right weakly Gorenstein. This is a reduction of an Auslander--Reiten's conjecture, which states that $R$ is Gorenstein if $R$ satisfies the Auslander condition. |
| title | Auslander-type conditions and weakly Gorenstein algebras |
| topic | Rings and Algebras Representation Theory 16E65, 16E10, 18G25 |
| url | https://arxiv.org/abs/2408.05468 |