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 $Soc\left(I\right)$, whose graded components give us the socle of the powers of $I$. It is observed that $Soc\left(I\right)$ is a finitely generated module over the fiber cone of $I$. In the case that $S$ is the polynomial ring and all powers of $I\subseteq S$ have linear resolution, we define the module ${Soc}^{\ast }\left(I\right)$, 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.