Are complete intersections complete intersections?

TL;DR

This paper investigates whether complete intersection rings are always quotients of regular rings by regular sequences, showing this holds in one dimension but not in higher dimensions through a counterexample.

Contribution

It proves that one-dimensional complete intersection domains are quotients of regular local rings by regular sequences, but provides a counterexample in three dimensions.

Findings

One-dimensional complete intersection domains are such quotients.

Counterexample exists in three dimensions.

The property does not hold universally for all complete intersections.

Abstract

A commutative local ring is generally defined to be a complete intersection if its completion is isomorphic to the quotient of a regular local ring by an ideal generated by a regular sequence. It has not previously been determined whether or not such a ring is necessarily itself the quotient of a regular ring by an ideal generated by a regular sequence. In this article, it is shown that if a complete intersection is a one dimensional integral domain, then it is such a quotient. However, an example is produced of a three dimensional complete intersection domain which is not a homomorphic image of a regular local ring, and so the property does not hold in general.