Algebraic properties of tensor product of modules over a field
Ahad Rahimi
TL;DR
We study how Cohen–Macaulay-type properties of finitely generated modules transfer along tensor products over a common field: given $A,B$ Noetherian over $\Bbbk$ and $\mathcal{I}= I\otimes_{\Bbbk} B + A\otimes_{\Bbbk} J$, the module $L\otimes_{\Bbbk} N$ over $A\otimes_{\Bbbk} B$ inherits CM, generalized CM, and sequential CM behavior from $L$ over $A$ and $N$ over $B$.$ Central tools include additive formulas for grade and cohomological dimension, a Künneth-type description of local cohomology, and spectral sequence analyses, leading to precise equivalences and filtration-driven results.$ The paper provides explicit expressions for finiteness-dimension and related invariants, and demonstrates when tensor products preserve or reflect CM-related structures, with both global and local (dimension/depth) formulations.$ These results elucidate how homological properties decompose along tensor factors and yield practical criteria for CM and sequential CM in composite ring settings, relevant for research in commutative algebra and its applications.
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.