Complete intersections of dimension zero: variations on a theme of Wiebe

TL;DR

This paper explores the structure of zero-dimensional complete intersection ideals in noetherian local rings, establishing a correspondence with certain matrices and analyzing their properties to understand the ring's structure.

Contribution

It introduces a correspondence between C.I.0-ideals and x-nice matrices, providing criteria for recognizing such ideals and analyzing their chains and minimal generators.

Findings

C.I.0-ideals correspond to x-nice matrices and their factorizations.

When the ring is a zero-dimensional complete intersection, C.I.0-ideals are of the form (0:bA).

The ideals yA and (0:yA) are C.I.0 if and only if (0:yA) is principal.

Abstract

Wiebe's criterion, which recognizes complete intersections of dimension zero among the class of noetherian local rings, is revisited and exploited in order to provide information on what we call C.I.0-ideals (those such that the corresponding quotient is a complete intersection of dimension zero) and also on chains of C.I.0-ideals. A correspondence is established between C.I.0-ideals and a certain kind of matrices which we call $x$-nice, and a chain of C.I.0-ideals corresponds to a factorization of some $x$-nice matrix. When the local ring $A$ itself is a complete intersection of dimension zero, a C.I.0-ideal is necessarily of the form $(0:bA)$ for some $b\in A$. Some criteria are provided to recognize whether an ideal $(0:bA)$ is C.I.0 or not. When $y$ is a minimal generator of the maximal ideal of $A$, it is also proved that the ideals $yA$ and $(0:yA)$ are C.I.0 simultaneously and that this is the case exactly when the ideal $(0:yA)$ is principal. These C.I.0-ideals of the form $(0:yA)$, $y$ being a minimal generator of the maximal ideal, are investigated. They are of interest because the smallest nonnull C.I.0-ideal in a strict chain of C.I.0-ideals of the maximal length is necessarily of that form, and their existence has some implications for a realization of the ring, i.e. for the way the ring can can be viewed as a quotient of a regular local ring.