The Improved New Intersection Theorem revisited

TL;DR

This paper proves a generalized version of the Improved New Intersection Theorem, providing a sharper bound on the length of complexes with specific torsion properties in local rings, advancing the understanding of intersection theory.

Contribution

It introduces a generalized theorem that improves the bound on the length of complexes with I-torsion homology, extending previous results by Avramov, Iyengar, and Neeman.

Findings

The length of the complex is at least dim R - dim R/I.

The theorem applies to complexes with I-torsion homology in positive degrees.

It refines the bounds established in earlier intersection theorems.

Abstract

We prove a generalized version of Evans and Griffith's Improved New Intersection Theorem: Let I be an ideal in a local ring R. If a finite free R-complex, concentrated in nonnegative degrees, has I-torsion homology in positive degrees, and the homology in degree 0 has an I-torsion minimal generator, then the length of the complex is at least dim R - dim R/I. This improves the bound ht I obtained by Avramov, Iyengar, and Neeman in 2018.