On a representation of the fundamental class of an ideal due to Lejeune-Jalabert

TL;DR

This paper provides an explicit description of a differential form representing the fundamental class of a Cohen-Macaulay ideal, specifically for generic monomial ideals with a Scarf resolution, refining previous results.

Contribution

It offers a new explicit formula for the differential form in Lejeune-Jalabert's residue representation for generic monomial ideals.

Findings

Explicit description of the differential form for Scarf resolutions

New proof of Lejeune-Jalabert's result in the monomial ideal case

Refinement of the fundamental class representation

Abstract

Lejeune-Jalabert showed that the fundamental class of a Cohen-Macaulay ideal $\mathfrak a\subset \mathcal O_0$ admits a representation as a residue, constructed from a free resolution of $\mathfrak a$, multiplied by a certain differential form coming from the resolution. We give an explicit description of this differential form in the case where the free resolution is the Scarf resolution of a generic monomial ideal. As a consequence we get a new proof and a refinement of Lejeune-Jalabert's result in this case.