Minimal quasi-complete intersection ideals

TL;DR

This paper investigates the structure of quasi-complete intersection (q.c.i.) ideals in local rings, providing conditions for when all q.c.i. ideals are principal and presenting examples of non-principal minimal q.c.i. ideals with unique properties.

Contribution

It establishes conditions under which q.c.i. ideals are necessarily principal and constructs examples of minimal q.c.i. ideals that are non-principal and not embedded.

Findings

Conditions guaranteeing all q.c.i. ideals are principal

Existence of minimal non-principal q.c.i. ideals

Examples of embedded q.c.i. ideals without principal q.c.i. ideals

Abstract

A quasi-complete intersection (q.c.i.) ideal of a local ring is an ideal with "free exterior Koszul homology"; the definition can also be understood in terms of vanishing of Andr\'e-Quillen homology functors. Principal q.c.i. ideals are well understood, but few constructions are known to produce q.c.i. ideals of grade zero that are not principal. This paper examines the structure of q.c.i. ideals. We exhibit conditions on a ring $R$ which guarantee that every q.c.i. ideal of $R$ is principal. On the other hand, we give an example of a minimal q.c.i. deal $I$ which does not contain any principal q.c.i. ideals and is not embedded, in the sense that no faithfully flat extension of $I$ can be written as a quotient of complete intersection ideals. We also describe a generic situation in which the maximal ideal of $R$ is an embedded q.c.i. ideal that does not contain any principal q.c.i. ideals.