Auslander-Reiten Conjecture, Finite $C$-Injective Dimension of $\operatorname{Hom}$, and vanishing of $\operatorname{Ext}$
| Autor: | Mendoza-Rubio, Victor D., Jorge-Pérez, Victor H. |
|---|---|
| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Let $R$ be a Noetherian local ring, and let $C$ be a semidualizing $R$-module. In this paper, we present some results concerning to vanishing of $\operatorname{Ext}$ and finite injective dimension of $\operatorname{Hom}$. Additionally, we extend these results in terms of finite $C$-injective dimension of $\operatorname{Hom}$. We also investigate the consequences of some of these extensions in the case where $R$ is Cohen-Macaulay, and $C$ is a canonical module for $R.$ Furthermore, we provide positive answers to the Auslander-Reiten conjecture for finitely generated $R$-modules $M$ such that $\mathcal{I}_C\operatorname{-id}_R(\operatorname{Hom}_R(M,R))<\infty$ or $M \in \mathcal{A}_C(R)$ with $\mathcal{I}_C \operatorname{-id}_R(\operatorname{Hom}_R(M,M))<\infty$. Moreover, we derive a number of criteria for a semidualizing $R$-module $C$ to be a canonical module for $R$ in terms of the vanishing of $\operatorname{Ext}$ and the finite $C$-injective dimension of $\operatorname{Hom}$. Comment: 16 pages. The first version of this paper was submitted to a scientific journal on June 25, 2023 and subsequently rejected. Currently, the paper is submitted to another journal. Ghosh and Dey achieved some similar results in arXiv:2310.10607. Both papers were developed independently |
| Databáze: | arXiv |
| Externí odkaz: |
|