On the stable property of projective dimension

TL;DR

This paper introduces monomial ideals with stable projective dimension, generalizing Cohen-Macaulayness, and explores their properties, relations, and specific classes like polymatroidal ideals.

Contribution

It defines and studies monomial ideals with stable projective dimension, extending Cohen-Macaulay concepts and characterizing classes like polymatroidal ideals.

Findings

Stable projective dimension is preserved under certain localizations.

Relations between stable projective dimension and Cohen-Macaulayness are established.

Characterizations of polymatroidal ideals with this property are provided.

Abstract

We introduce the concept of monomial ideals with stable projective dimension, as a generalization of the Cohen-Macaulay property. Indeed, we study the class of monomial ideals $I$, whose projective dimension is stable under monomial localizations at monomial prime ideals $\fp$, with $\height \fp\geq \pd S/I$. We study the relations between this property and other sorts of Cohen-Macaulayness. Finally, we characterize some classes of polymatroidal ideals with stable projective dimension.