The structure of quasi-complete intersection ideals

Abstract

We prove that every quasi-complete intersection (q.c.i.) 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 q.c.i. ideal; and in this case, T is the minimal complete resolution of R/I by free R-modules.