Computing residue currents of monomial ideals using comparison formulas

TL;DR

This paper provides a detailed description of residue currents associated with Artinian monomial ideals using cellular resolutions, extending previous work and employing comparison formulas to connect algebraic and analytic properties.

Contribution

It introduces a complete characterization of residue currents for monomial ideals via cellular resolutions, expanding the understanding of their structure and applications.

Findings

Explicit description of residue currents for Artinian monomial ideals

Extension of previous results to more general cellular resolutions

Connection to fundamental cycle factorization in the monomial case

Abstract

Given a free resolution of an ideal $\mathfrak{a}$ of holomorphic functions, one can construct a vector-valued residue current, $R$, which coincides with the classical Coleff-Herrera product if $\mathfrak{a}$ is a complete intersection ideal and whose annihilator ideal is precisely ~$\mathfrak{a}$. We give a complete description of $R$ in the case when $\mathfrak{a}$ is an Artinian monomial ideal and the resolution is the hull resolution (or a more general cellular resolution), extending previous results by the second author. The main ingredient in the proof is a comparison formula for residue currents due to the first author. By means of this description we obtain in the monomial case a current version of a factorization of the fundamental cycle of $\mathfrak{a}$ due to Lejeune-Jalabert.