2007/09/01 by Manoj Kummini, Kummini, Manoj
Mathematics · #13D02 (Primary) #13F55 #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.0709.0031
openalex publication_date 2007/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this note, we study Serre's property (Si), 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-(Si)-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 (Si) 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 (S2) is equivalent to being locally connected in codimension 1.