Sequentially Cohen--Macaulayness of bigraded modules

TL;DR

This paper investigates the property of sequentially Cohen--Macaulay modules over bigraded polynomial rings, providing characterizations and classifications, especially for hypersurface rings, with a focus on modules related to two graded components.

Contribution

It characterizes when tensor products of graded modules are sequentially Cohen--Macaulay with respect to a specific ideal and classifies all hypersurface rings with this property.

Findings

Tensor product modules are sequentially Cohen--Macaulay under certain conditions.

Complete classification of hypersurface rings that are sequentially Cohen--Macaulay with respect to Q.

Provides criteria for sequential Cohen--Macaulayness in bigraded modules.

Abstract

Let $K$ be a field, $S=K[x_1,\ldots,x_m, y_1,\ldots,y_n]$ be a standard bigraded polynomial ring and $M$ a finitely generated bigraded $S$-module. In this paper we study sequentially Cohen--Macaulayness of $M$ with respect to $Q=(y_1,\ldots,y_n)$. We characterize the sequentially Cohen--Macaulayness of $L\tensor_KN$ with respect to $Q$ as an $S$-module when $L$ and $N$ are non-zero finitely generated graded modules over $K[x_1, \dots, x_m]$ and $K[y_1, \dots, y_n]$, respectively. All hypersurface rings that are sequentially Cohen--Macaulay with respect to $Q$ are classified.