Approximation of versal deformations

Abstract

In Artin’s work on algebraic spaces and algebraic stacks [A2], [A3], a crucial ingredient is the use of his approximation theorem to prove the algebraizability of formal deformations under quite general conditions. The algebraizability result is given in [A2, Thm 1.6], and we recall the statement now (using standard terminology to be recalled later). Theorem 1.1. (Artin) Let S be a scheme locally of finite type over a field or excellent Dedekind domain, and F a contravariant functor, locally of finite presentation, from the category of S-schemes to the category of sets. Let κ be an OS-field of finite type, and ξ0 ∈ F (κ) an element. Assume that there exists a complete local noetherian OS-algebra (A,m) with residue field κ and an element ξ ∈ F (A) lifting ξ0 ∈ F (κ) such that ξ is an effective versal deformation of ξ0. Then there exists a finite type S-scheme X, a closed point x ∈ X with residue field κ, an element ξ ∈ F (X) lifting ξ0 ∈ F (κ) = F (k(x)), and an OS-isomorphism σ : OX,x ' A such that F (σ)(ξ) and ξ coincide in F (A/m) for all n ≥ 0. The isomorphism σ is unique if ξ is an effective universal deformation of ξ0. Remark 1.2. For a scheme S, an OS-field of finite type is a field κ equipped with a finite type map Spec(κ)→ S. When S is locally noetherian, this is equivalent to saying that κ is a finite extension of the residue field k(s) at a locally closed point s ∈ S (see Lemma 2.1). Whereas the techniques in [A3] are extremely conceptual and easy to digest, these methods ultimately depend upon Theorem 1.1, whose proof in [A2] (together with its clarification in [A3, Appendix]) is quite intricate and hard to “grasp”. Moreover, there are two ways in which the proof of this result uses the hypothesis (harmless in practice) that S is locally of finite type over a field or excellent Dedekind domain. First, this condition is needed in Artin’s original form of his approximation theorem [A1]. Second, and perhaps more seriously (in view of Popescu’s subsequent proof of the Artin approximation theorem for arbitrary excellent rings), the detailed analysis in the proof of Theorem 1.1 uses very special properties of fields and Dedekind domains (such as the structure theorem for modules over a discrete valuation ring). The restriction to base schemes locally of finite type over a field or excellent Dedekind domain in Artin’s form of his approximation theorem is also the source of the restriction to finite extensions κ of residue fields at locally closed points of S (see Remark 1.2), rather than at general points of S, when algebraizing formal deformations as in Theorem 1.1. Since arbitrary localization preserves the property of excellence but tends to destroy the property of a map being (locally) of finite type, if one can work in the more general context of excellent base schemes then one can also hope to get algebraization results over arbitrary points of S. In this note, we present a proof of Theorem 1.1 with S permitted to be an arbitrary excellent scheme and κ permitted to be a finite extension of k(s) for an arbitrary point s ∈ S (but of course the point x ∈ X as in Theorem 1.1 can only be taken to be closed if and only if κ is of finite type over OS). In fact, we shall prove a “groupoid” generalization analogous to [A3, Cor 3.2]; see Theorem 1.5. As a consequence, the entirety of [A3] is valid as written for an arbitrary excellent base scheme S. The central technical ingredient we need is the following remarkable result of Popescu: