A comparison formula for residue currents

Abstract

Given two ideals I and J of holomorphic functions such that I ⊆ J , we describe a comparison formula relating the Andersson-Wulcan currents of I and J . More generally, this comparison formula holds for residue currents associated to two generically exact Hermitian complexes together with a morphism between the complexes.One application of the comparison formula is a generalization of the transformation law for Coleff-Herrera products to Andersson-Wulcan currents of Cohen-Macaulay ideals. We also use it to give an analytic proof by means of residue currents of theorems of Hickel, Vasconcelos and Wiebe related to the Jacobian ideal of a holomorphic mapping.

Elementary construction of residue currents associated to Cohen–Macaulay ideals

“…In [15], the first author described a way of relating the currents R E of Andersson and Wulcan, related to different free resolutions, of possibly different ideals. We consider the particular case when J is a Cohen-Macaulay ideal of codimension p, i.e., O/J has a free resolution (E, ϕ) of length p. We also assume that rank E 0 = 1, which is always possible to choose.…”

“…. , f p ) of codimension p contained in J, for example by taking p generic linear combinations of a set of generators of J, cf., for example [15,Example 2]. If one lets (K, ψ) be the Koszul complex of f , then it is quite elementary homological algebra that one can construct a morphism of complexes a : (K, ψ) → (E, ϕ) which extends the natural surjection π : O/I → O/J, i.e., which is such that the following diagram is commutative: cf., Proposition 2.2 below.…”

“…If one lets (K, ψ) be the Koszul complex of f , then it is quite elementary homological algebra that one can construct a morphism of complexes a : (K, ψ) → (E, ϕ) which extends the natural surjection π : O/I → O/J, i.e., which is such that the following diagram is commutative: cf., Proposition 2.2 below. By [15,Example 3], the current R E can then be described as…”

“…. , g p ) is also a complete intersection ideal of codimension p, and (E, ϕ) is the Koszul complex of g, then R E is also a Coleff-Herrera product, and (1.6) then becomes the transformation law for Coleff-Herrera products, see [15,Remark 2].…”

“…is a morphism of complexes extending the natural surjection π : O/I → O/J. In the appendix of[15], we give an example of how such a morphism can be computed with the help of the computer algebra system Macaulay2.By Theorem 1.1, we then get that g ∈ J if and only if (x 3 − yz)g ∈ I and (y 2 − xz)g ∈ I.…”