Weakly Gorensteinness of tensor algebras and Morita algebras

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 $A$.