Cohen–Macaulayness for symbolic power ideals of edge ideals

Abstract

Let S = K [ x 1 , … , x n ] be a polynomial ring over a field K. Let I ( G ) ⊆ S denote the edge ideal of a graph G. We show that the ℓth symbolic power I ( G ) ( ℓ ) is a Cohen–Macaulay ideal (i.e., S / I ( G ) ( ℓ ) is Cohen–Macaulay) for some integer ℓ ⩾ 3 if and only if G is a disjoint union of finitely many complete graphs. When this is the case, all the symbolic powers I ( G ) ( ℓ ) are Cohen–Macaulay ideals. Similarly, we characterize graphs G for which S / I ( G ) ( ℓ ) has (FLC). As an application, we show that an edge ideal I ( G ) is complete intersection provided that S / I ( G ) ℓ is Cohen–Macaulay for some integer ℓ ⩾ 3 . This strengthens the main theorem in Crupi et al. (2010) [3].