Relative Gorenstein Dimensions over Triangular Matrix Rings

Driss Bennis,Juan Ramón García Rozas,Rachid El Maaouy,Luis Oyonarte

doi:10.3390/math9212676

Driss Bennis, Juan Ramón García Rozas + Show 2 more

Open Access

https://doi.org/10.3390/math9212676

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Journal: Mathematics	Publication Date: Oct 22, 2021
Citations: 3	License type: CC BY 4.0

Affiliation: Mohammed V University, University of Almería

Abstract

Let A and B be rings, U a (B,A)-bimodule, and T=A0UB the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We first study how to construct w-tilting (tilting, semidualizing) over T using the corresponding ones over A and B. We show that when U is relative (weakly) compatible, we are able to describe the structure of GC-projective modules over T. As an application, we study when a morphism in T-Mod is a special GCP(T)-precover and when the class GCP(T) is a special precovering class. In addition, we study the relative global dimension of T. In some cases, we show that it can be computed from the relative global dimensions of A and B. We end the paper with a counterexample to a result that characterizes when a T-module has a finite projective dimension.

Highlights

Introduction dimensionsLet A and B be rings and U be a ( B, A)-bimodule
The main objective of the present paper is to study relative Gorenstein homological notions (w-tilting, relative Gorenstein projective modules, relative Gorenstein projective dimensions, and the relative global projective dimension) over triangular matrix rings
We introduce C-compatible ( B, A)-bimodules for a T-module C

Summary

Preliminaries

Throughout this paper, all rings are associative (not necessarily commutative) with identity, and all modules are, unless otherwise specified, unitary left modules. The bimodule B U A is said to be C-compatible if the following two conditions hold: The complex U ⊗ A X1 is exact for every exact sequence in A-Mod: X1 : · · · → P11 → P10 → C10 → C11 → · · ·. There exists a Hom A (−, Add A (C1 ))-exact exact sequence in A-Mod: where the P1i ’s are all projective, G1 ∼. By Lemma 2, the functor p preserves finitely generated modules, so we only need to prove the statement for w-tilting. S ⊗ R X is HomS (−, AddS (C2 ))-exact; 2 We end this section with an example of a w-tilting module that is neither projective nor injective.

Relative Gorenstein Projective Modules

Relative Global Gorenstein Dimension