Set-Theoretically Perfect Ideals and Residual Intersections

TL;DR

The paper develops a free-approach framework to study residual intersections in rings satisfying Serre's condition $S_s$, focusing on $r$-minimally generated ideals. It constructs finite free complexes that yield a subideal $\tau$ with $\sqrt{\tau}=\sqrt{J}$, enabling a set-theoretic/ homological understanding of residual intersections even when the ideals lack strong homological properties. A key result is a uniform upper bound for the multiplicity $e(R/J)$ in terms of the residual complete intersection, with equality under precise radical conditions; in positive characteristic, residual intersections are cohomologically complete intersections and connected in codimension one. In the sliding-depth context, the authors show that algebraic residual intersections admit free approaches and arithmetic residual intersections are Cohen–Macaulay in regular local rings, advancing questions about the Cohen–Macaulayness of residual intersections. A structural outcome identifies $\tau$ as $\mathfrak a+\mathrm{Kitt}_0(\mathfrak a,I)$, connecting the free-approach to the Koszul–Fitting theory via Kitt ideals and a boundary approximation construction, and providing a concrete description of the defining ideals governing the residual intersection in this framework.

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.

Theorems & Definitions (49)

• Proposition 1.1
• Theorem 1.2
• Definition 2.1
• Proposition 2.2
• proof
• Definition 2.3
• Definition 2.4
• Proposition 2.5
• proof
• Example 2.6