Categorical properties and homological conjectures for bounded extensions of algebras
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916367139078144 |
|---|---|
| author | Qin, Yongyun Xu, Xiaoxiao Zhang, Jinbi Zhou, Guodong |
| author_facet | Qin, Yongyun Xu, Xiaoxiao Zhang, Jinbi Zhou, Guodong |
| contents | An extension $B\subset A$ of finite dimensional algebras is bounded if the $B$-$B$-bimodule $A/B$ is $B$-tensor nilpotent, its projective dimension is finite and $\mathrm{Tor}_i^B(A/B, (A/B)^{\otimes_B j})=0$ for all $i, j\geq 1$. We show that for a bounded extension $B\subset A$, the algebras $A$ and $B$ are singularly equivalent of Morita type with level. Additionally, under mild conditions, their stable categories of Gorenstein projective modules and Gorenstein defect categories are equivalent, respectively. Some homological conjectures are also investigated for bounded extensions, including Auslander-Reiten conjecture, finististic dimension conjecture, Fg condition, Han's conjecture, and Keller's conjecture. Applications to trivial extensions and triangular matrix algebras are given.
In course of proof, we give some handy criteria for a functor between module categories to induce triangle functors between stable categories of Gorenstein projective modules and Gorenstein defect categories, which generalise some known criteria, and hence might be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_21480 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Categorical properties and homological conjectures for bounded extensions of algebras Qin, Yongyun Xu, Xiaoxiao Zhang, Jinbi Zhou, Guodong Representation Theory K-Theory and Homology Rings and Algebras 16E35, 16E30, 16E40, 16E65, 16G50, 16D20 An extension $B\subset A$ of finite dimensional algebras is bounded if the $B$-$B$-bimodule $A/B$ is $B$-tensor nilpotent, its projective dimension is finite and $\mathrm{Tor}_i^B(A/B, (A/B)^{\otimes_B j})=0$ for all $i, j\geq 1$. We show that for a bounded extension $B\subset A$, the algebras $A$ and $B$ are singularly equivalent of Morita type with level. Additionally, under mild conditions, their stable categories of Gorenstein projective modules and Gorenstein defect categories are equivalent, respectively. Some homological conjectures are also investigated for bounded extensions, including Auslander-Reiten conjecture, finististic dimension conjecture, Fg condition, Han's conjecture, and Keller's conjecture. Applications to trivial extensions and triangular matrix algebras are given. In course of proof, we give some handy criteria for a functor between module categories to induce triangle functors between stable categories of Gorenstein projective modules and Gorenstein defect categories, which generalise some known criteria, and hence might be of independent interest. |
| title | Categorical properties and homological conjectures for bounded extensions of algebras |
| topic | Representation Theory K-Theory and Homology Rings and Algebras 16E35, 16E30, 16E40, 16E65, 16G50, 16D20 |
| url | https://arxiv.org/abs/2407.21480 |