An argument principle for generalised point residues

Abstract

We define a point residue for any Artinian O-modules via Hermitian free resolutions, generalising the one dimensional residue and the classical multivariate Grothendieck point residue. We consider various definitions of multiplicity for such modules and prove a residue formula connecting the algebraic multiplicity to our residues. Our result can be seen as a generalisation both of the argument principle and a corresponding result for Grothendieck residues. It is also a special case of a recent result for Andersson-Wulcan currents proven by Lärkäng and Wulcan.