Cohen–Macaulayness and sequentially Cohen–Macaulayness of monomial ideals

Abstract

In this paper, we give a characterization for Cohen–Macaulay rings $R/I$ where $I\subset R=K\[y\_1, \dots, y\_n]$ is a monomial ideal which satisfies bigsize $I$ = size $I$. Next, we let $S=K\[x\_1,\ldots,x\_m, y\_1,\ldots,y\_n]$ be a polynomial ring and $I\subset S$ a monomial ideal. We study the sequentially Cohen–Macaulayness of $S/I$ with respect to $Q=(y\_1,\ldots,y\_n)$. Moreover, if $I\subset R$ is a monomial ideal such that the associated prime ideals of $I$ are in pairwise disjoint sets of variables, a classification of $R/I$ to be sequentially Cohen–Macaulay is given. Finally, we compute grade$(Q, M)$ where $M$ is a sequentially Cohen–Macaulay $S$-module with respect to $Q$.