Weakly Gorensteinness of tensor algebras and Morita algebras

TL;DR

This paper investigates the conditions under which tensor algebras and Morita algebras are left weakly Gorenstein, establishing equivalences based on properties of component algebras and modules.

Contribution

It characterizes the weakly Gorenstein property for tensor algebras and Morita algebras, providing explicit criteria and module descriptions.

Findings

Tensor algebra $A\otimes B$ is weakly Gorenstein iff $A$ is, given $\operatorname{gl.dim} B<\infty$.

Morita algebra $\Lambda$ is weakly Gorenstein iff $A$ and $B$ are.

Upper triangular matrix algebra $T_n(A)$ is weakly Gorenstein iff $A$ is.

Abstract

An algebra $A$ is left weakly Gorenstein if any semi-Gorenstein-projective left $A$-modules is Gorenstein-projective. The weakly Gorensteinness of two kinds of algebras are answered. Using the method of the monomorphism category, it is proved that the tensor algebra $A\otimes B$ with ${\rm gl.dim} B< \infty$ is left weakly Gorenstein if and only if so is $A$. For a class of Morita algebras $\Lambda=\begin{pmatrix}\begin{smallmatrix} A & N \\ M & B \\ \end{smallmatrix}\end{pmatrix}$, the (semi-)Gorenstein-projective left $\Lambda$-modules are computed and described; and then it is proved that $\Lambda$ is left weakly Gorenstein if and only if so are $A$ and $B$. As an application, the upper triangular matrix algebra $T_n(A)$ is left weakly Gorenstein if and only if so is $A$.