A Descent Theorem for Formal Smoothness

Abstract

Abstract. Let u be a local homomorphism of noetherian local rings forming part of a commutative square vf = gu. We give some conditions on the square which imply that u is formally smooth. This theorem encapsulates a variety of (apparently unrelated) results in commutative algebra greatly improving some of them: Greco's theorem on descent of quasi-excellence property by finite surjective morphisms, Kunz's characterization of regular local rings in positive characteristic by means of the Frobenius homomorphism (and in fact the relative version obtained by André and Radu), etc. In the second part of the paper, we study a similar question for the complete intersection property instead of formal smoothness, giving also some applications.In this paper we obtain a descent result for formal smoothness (and regularity). This single result (Theorem 1) has as particular cases a variety of (a priori unconnected) results in commutative algebra. Some of these particular cases greatly improve known important results (the descent of quasi-excellence by finite surjective morphisms [20] (Corollary 25). Since the main result is rather technical, we confine ourselves in this introduction to discuss these applications in more detail.A. Extension of Greco's theorem. If f : R → S is a finite local homomorphism of noetherian local rings such that Spec(S) → Spec(R) is surjective, Greco [20] showed that if S is quasi-excellent then so is R (in particular, quasi-excellence descends along finite flat local homomorphisms). He also obtained an example [20, Proposition 4.1] which shows that this theorem cannot be extended to the case where f is of finite type instead of finite, even when R and S are local domains of dimension 1 (the problem is concentrated in the local condition, i.e., the geometric regularity of the formal fibers, since the J-2 property always descends and ascends by surjective morphisms of finite type [20, Corollary 2.4]). So the problem for finite type homomorphisms was left aside and he proceeded instead to consider proper surjective morphisms of schemes instead, a problem which was finally solved in [14], [24].We reconsider here the problem for homomorphisms of finite type (but with an adequate formulation of the hypothesis on the surjectivity between spectra), and we will prove that it holds. Note first that ifR andŜ are the completions of R and S at their maximal ideals, the surjectivity of Spec(S) → Spec(R) is equivalent to

Some Homological Criteria for Regular, Complete Intersection and Gorenstein Rings

“…iii) The good properties of André-Quillen homology (for example, the ones stated above and its behaviour under tensor products [1, Proposition 5.21]) allow us to obtain new examples of h 2 -vanishing homomorphisms from others.Remarks 7. (i)It can be proved that if f : A → A is contracting, then for any n 0, there exists s (depending on n) such that f s has the h n -vanishing property, that is, it induces the zero map H n (A, k, k) → H n (A, k, k)[19, Proposition 10]. However, our ad hoc proof given here for the case in which n = 2, besides being simpler and shorter, gives a better idea of the relationship between contracting and h 2 -vanishing.…”