4 - Complete intersections

Summary

The main purpose of this chapter is to prove the descent of the complete intersection property by flat local homomorphisms (4.3.8), which has as a consequence the localization theorem for complete intersections (4.3.9): if (A,m,K) is a complete intersection ring, p a prime ideal of A, then A_p is complete intersection. This is another important result which appears without proof in Matsumura's book [Mt, end of Section 21]. The case when A is a quotient of a regular ring follows easily from the same localization property for regular rings (Serre's theorem). The difficult part, solved by Avramov [Av1], is to reduce the problem to this case.

We follow some papers by Avramov. We first need to present Gulliksen's proof [GL] of the existence of minimal DG algebra resolutions (4.1.7). This result is used to prove Main Lemma 4.2.1 following [Av1]. We do not know any easier proof of this lemma (or, equivalently, of (4.2.2)). Finally, we characterize complete intersection rings in terms of homology modules in order to prove the main theorems (4.3.8), (4.3.9). In (higher) André–Quillen homology theory, complete intersections are characterized by the vanishing of an H₃ module [An1, 6.27]. Since we want to avoid these higher homology modules, we characterize them by counting dimensions of the lower homology modules (4.3.5) as in [Av2, Section 3].

Avramov's Lemma 4.2.1 is very powerful (as an example we give in (4.4.2) an alternative proof of Kunz's characterization of regularity in characteristic p [Ku] using this lemma).