Set-Theoretically Perfect Ideals and Residual Intersections

TL;DR

This paper investigates residual intersections in algebraic rings satisfying Serre's condition, establishing their properties, bounds on multiplicity, and cohomological characteristics, with implications for their geometric connectedness.

Contribution

It introduces a free approach to residual intersections, providing a uniform upper bound for their multiplicity and analyzing their cohomological and geometric properties.

Findings

Residual intersections admit free approaches with perfect subideals.

A uniform upper bound for the multiplicity of residual intersections is established.

Residual intersections are cohomologically complete intersections in positive characteristic.

Abstract

This paper studies algebraic residual intersections in rings with Serre's condition \( S_{s} \). It demonstrates that residual intersections admit free approaches i.e. perfect subideal with the same radical. This fact leads to determining a uniform upper bound for the multiplicity of residual intersections. In positive characteristic, it follows that residual intersections are cohomologically complete intersection and, hence, their variety is connected in codimension one.