Algebraic properties of tensor product of modules over a field

Abstract

Let $A$ and $B$ be commutative Noetherian algebras over an arbitrary field $\mathbb{k}$ such that $A{\otimes }_{\mathbb{k}}B$ 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 $A{\otimes }_{\mathbb{k}}B$-module $L{\otimes }_{\mathbb{k}}N$, 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 $L{\otimes }_{\mathbb{k}}N$ with respect to the ideal $I{\otimes }_{\mathbb{k}}B+A{\otimes }_{\mathbb{k}}J$, in terms of the corresponding properties for $L$ and $N$ with respect to $I$ and $J$, respectively.

19 pages, 1 figure, to appear in Journal of Commutative Algebra