The socle module of a monomial ideal

Abstract

For any ideal I in a Noetherian local ring or any graded ideal I in a standard graded K-algebra over a field K, we introduce the socle module Soc(I), whose graded components give us the socle of the powers of I. It is observed that Soc(I) is a finitely generated module over the fiber cone of I. In the case that S is the polynomial ring and all powers of I⊆S have linear resolution, we define the module Soc∗(I), which is a module over the Rees ring of I. For the edge ideal of a graph and for classes of polymatroidal ideals, we study the module structure of their socle modules.