Elementary construction of residue currents associated to Cohen-Macaulay ideals

TL;DR

This paper presents an elementary construction of residue currents for Cohen-Macaulay ideals, providing explicit proofs of their properties and applications in ideal membership problems.

Contribution

It introduces a new elementary method to construct residue currents associated with Cohen-Macaulay ideals, with two distinct proofs of their annihilator properties.

Findings

Residue currents precisely annihilate Cohen-Macaulay ideals.

Elementary construction simplifies previous approaches.

Explicit division formulas aid in ideal membership testing.

Abstract

For a Cohen-Macaulay ideal of holomorphic functions, we construct by elementary means residue currents whose annihilator is precisely the given ideal. We give two proofs that the currents have the prescribed annihilator, one using the theory of linkage, and another using an explicit division formula involving these residue currents to express the ideal membership.