Residual Intersections are Koszul-Fitting ideals

TL;DR

This paper explores the structure of residual intersections in commutative rings, revealing their connection to Koszul-Fitting ideals and showing how DG-algebra structures influence their generators.

Contribution

It establishes the equivalence of disguised and algebraic residual intersections over Cohen-Macaulay rings with sliding depth, using Koszul homology and spectral sequences.

Findings

Disguised residual intersections are the same as algebraic residual intersections under certain conditions.

DG-algebra structures of Koszul homologies determine generators of residual intersections.

Buchsbaum-Eisenbud complexes can be derived from Koszul-ch spectral sequences.

Abstract

One describes generators of disguised residual intersections in any commutative Noetherian rings. It is shown that, over Cohen-Macaulay rings, the disguised residual intersections and algebraic residual intersections are the same, for ideals with sliding depth. This coincidence provides structural results for algebraic residual intersections in quite general setting. It is shown how the DG-algebra structure of Koszul homologies affects the determination of generators of residual intersections. In the midway it is shown that the Buchsbaum-Eisenbud family of complexes can be derived from the Koszul-\v{C}ech spectral sequence. This interpretation of Buchsbaum-Eisenbud families has crucial rule to establish the above results.