(n-1)-st Koszul homology and the structure of monomial ideals

TL;DR

This paper explores the (n-1)-st Koszul homology of monomial ideals, revealing its crucial role in understanding their algebraic structure through Stanley and irreducible decompositions.

Contribution

It demonstrates the significance of the (n-1)-st Koszul homology in describing monomial ideals' structure, linking homological and algebraic perspectives.

Findings

(n-1)-st Koszul homology is key to algebraic descriptions

Links between homology and Stanley decompositions established

Connections to irreducible decompositions clarified

Abstract

Koszul homology of monomial ideals provides a description of the structure of such ideals, not only from a homological point of view (free resolutions, Betti numbers, Hilbert series) but also from an algebraic viewpoint. In this paper we show that, in particular, the homology at degree (n-1), with n the number of indeterminates of the ring, plays an important role for this algebraic description in terms of Stanley and irreducible decompositions.