Relative weak global Gorenstein dimension, AB-contexts and model structures
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_ | 1866917713623908352 |
|---|---|
| author | Bennis, Driss Maaouy, Rachid EL Rozas, Juan Ramon Garcia Oyonarte, Luis |
| author_facet | Bennis, Driss Maaouy, Rachid EL Rozas, Juan Ramon Garcia Oyonarte, Luis |
| contents | In this paper we introduce and study the weak Gorenstein global dimension of a ring $R$ with respect to a left $R$-module $C$. We provide several characterizations of when this homological invariant is bounded. Two main applications are given: first, we prove that the weak Gorenstein global dimension of $R$ relative to a semidualizing $(R,S)$-bimodule $C$ can be computed either by the ${\rm G_C}$-flat dimension of the left $R$-modules or right $S$-modules, just like the (absolute) weak global dimension. As a consequence, a new argument for solving Bennis' conjecture is obtained. As a second application, we give a concrete description of the weak equivalences in the ${\rm G_C}$-flat model structure recently found by the authors. In order to prove this result, an interesting connection between abelian model structures and AB-weak contexts is proved. This connection leads to a result that can be applied to obtain abelian model structures with a simpler description of trivial objects. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_05228 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Relative weak global Gorenstein dimension, AB-contexts and model structures Bennis, Driss Maaouy, Rachid EL Rozas, Juan Ramon Garcia Oyonarte, Luis Commutative Algebra Category Theory Rings and Algebras 18N40, 16E10, 16E65 In this paper we introduce and study the weak Gorenstein global dimension of a ring $R$ with respect to a left $R$-module $C$. We provide several characterizations of when this homological invariant is bounded. Two main applications are given: first, we prove that the weak Gorenstein global dimension of $R$ relative to a semidualizing $(R,S)$-bimodule $C$ can be computed either by the ${\rm G_C}$-flat dimension of the left $R$-modules or right $S$-modules, just like the (absolute) weak global dimension. As a consequence, a new argument for solving Bennis' conjecture is obtained. As a second application, we give a concrete description of the weak equivalences in the ${\rm G_C}$-flat model structure recently found by the authors. In order to prove this result, an interesting connection between abelian model structures and AB-weak contexts is proved. This connection leads to a result that can be applied to obtain abelian model structures with a simpler description of trivial objects. |
| title | Relative weak global Gorenstein dimension, AB-contexts and model structures |
| topic | Commutative Algebra Category Theory Rings and Algebras 18N40, 16E10, 16E65 |
| url | https://arxiv.org/abs/2304.05228 |