The structure of quasi-complete intersection ideals

TL;DR

This paper characterizes quasi-complete intersection ideals as those derived from nested complete intersections via flat base change, and introduces minimal two-step Tate complexes with rigidity properties.

Contribution

It provides a new structural understanding of quasi-complete intersection ideals and introduces minimal two-step Tate complexes with rigidity results.

Findings

Quasi-complete intersection ideals are obtained from nested complete intersections by flat base change.

A rigidity statement is established for the minimal two-step Tate complex associated with an ideal.

The minimal two-step Tate complex is exact if and only if the ideal is quasi-complete intersection.

Abstract

We prove that every quasi-complete intersection ideal is obtained from a pair of nested complete intersection ideals by way of a flat base change. As a by-product we establish a rigidity statement for the minimal two-step Tate complex associated to an ideal $I$ in a local ring $R$. Furthermore, we define a minimal two-step complete Tate complex $T$ for each ideal $I$ in a local ring $R$; and prove a rigidity result for it. The complex $T$ is exact if and only if $I$ is a quasi-complete intersection ideal; and in this case, $T$ is the minimal complete resolution of $R/I$ by free $R$-modules.