Algebraic properties of tensor product of modules over a field

Abstract

Let $A$ and $B$ be commutative Noetherian algebras over an arbitrary field $\Bbbk$ such that $A \otimes_\Bbbk B$ is Noetherian. We consider ideals $I$ and $J$ of $A$ and $B$, respectively, as well as nonzero finitely generated modules $L$ and $N$ over $A$ and $B$, respectively. In this paper, we investigate certain algebraic properties of the $A \otimes_\Bbbk B$-module $L\otimes_{\Bbbk} N$, which are often inherited from the properties of the $A$-module $L$ and the $B$-module $N$. Specifically, we provide characterizations for the Cohen-Macaulayness, generalized Cohen-Macaulayness, and sequentially Cohen-Macaulayness of $L\otimes_{\Bbbk} N$ with respect to the ideal $I \otimes_\Bbbk B + A \otimes_\Bbbk J$, in terms of the corresponding properties for $L$ and $N$ with respect to $I$ and $J$, respectively.