Alexander Duality and Serre's Property $(S_i)$ for Square-free Monomial Ideals

Abstract

In this note, we study Serre's property $(S_i)$, and its relation to Alexander duality for monomial ideals in a polynomial ring over a field. We describe ideals that define the non-Cohen-Macaulay- and the non-$(S_i)$-loci of finitely generated modules over regular rings, and show that minimal prime ideals in these loci are homogeneous, in the graded case. We show that a square-free monomial ideal has property $(S_i)$ if and only if its Alexander dual has a linear resolution up to homological degree $i-1$. We prove that for square-free monomial ideals, having property $(S_2)$ is equivalent to being locally connected in codimension 1.

Withdrawn by the author as it was learnt that this result was earlier proved by K. Yanagawa, J. Algebra, vol. 225, no. 2, 2000