On Artin algebras arising from Morita contexts

TL;DR

This paper explores the structure and properties of Morita rings derived from Artin algebras, focusing on subcategory finiteness, global dimension bounds, and Gorenstein conditions, with specific results for rings with zero bimodule homomorphisms.

Contribution

It provides new insights into the module categories, global dimensions, and Gorenstein properties of Morita rings, especially when bimodule homomorphisms are zero.

Findings

Characterization of finite subcategories in module categories of Morita rings

Bounds for global dimension based on component algebras

Identification of Gorenstein-projective modules in specific Morita rings

Abstract

We study Morita rings $\Lambda_{(\phi,\psi)}=\bigl({smallmatrix} A &_AN_B_BM_A & B {smallmatrix}\bigr)$ in the context of Artin algebras from various perspectives. First we study covariant finite, contravariant finite, and functorially finite subcategories of the module category of a Morita ring when the bimodule homomorphisms $\phi$ and $\psi$ are zero. Further we give bounds for the global dimension of a Morita ring $\Lambda_{(0,0)}$, regarded as an Artin algebra, in terms of the global dimensions of $A$ and $B$ in the case when both $\phi$ and $\psi$ are zero. We illustrate our bounds with some examples. Finally we investigate when a Morita ring is a Gorenstein Artin algebra and then we determine all the Gorenstein-projective modules over the Morita ring with $A=N=M=B=\Lambda$, where $\Lambda$ is an Artin algebra.