The Stacks project

112.5.1 Deformation theory and algebraic stacks

The first three papers by Artin do not contain anything on stacks but they contain powerful results with the first two papers being essential for [ArtinVersal].

• Artin: Algebraic approximation of structures over complete local rings [Artin-Algebraic-Approximation]

  It is proved that under mild hypotheses any effective formal deformation can be approximated: if $F: (\mathit{Sch}/S) \to (\textit{Sets})$ is a contravariant functor locally of finite presentation with $S$ finite type over a field or excellent DVR, $s \in S$, and $\hat{\xi } \in F(\hat{\mathcal{O}}_{S, s})$ is an effective formal deformation, then for any $n > 0$, there exists an residually trivial étale neighborhood $(S', s') \to (S, s)$ and $\xi ' \in F(S')$ such that $\xi '$ and $\hat{\xi }$ agree up to order $n$ (ie. have the same restriction in $F(\mathcal{O}_{S, s} / \mathfrak m^ n)$).

• Artin: Algebraization of formal moduli I [ArtinI]

  It is proved that under mild hypotheses any effective formal versal deformation is algebraizable. Let $F: (\mathit{Sch}/S) \to (\textit{Sets})$ be a contravariant functor locally of finite presentation with $S$ finite type over a field or excellent DVR, $s \in S$ be a locally closed point, $\hat A$ be a complete Noetherian local $\mathcal{O}_ S$-algebra with residue field $k'$ a finite extension of $k(s)$, and $\hat{\xi } \in F(\hat A)$ be an effective formal versal deformation of an element $\xi _0 \in F(k')$. Then there is a scheme $X$ finite type over $S$ and a closed point $x \in X$ with residue field $k(x) = k'$ and an element $\xi \in F(X)$ such that there is an isomorphism $\hat{\mathcal{O}}_{X, x} \cong \hat{A}$ identifying the restrictions of $\xi $ and $\hat{\xi }$ in each $F(\hat A / \mathfrak m^ n)$. The algebraization is unique if $\hat{\xi }$ is a universal deformation. Applications are given to the representability of the Hilbert and Picard schemes.

• Artin: Algebraization of formal moduli. II [ArtinII]

  Vaguely, it is shown that if one can contract a closed subset $Y' \subset X'$ formally locally around $Y'$, then exists a global morphism $X' \to X$ contracting $Y$ with $X$ an algebraic space.

• Artin: Versal deformations and algebraic stacks [ArtinVersal]

  This momentous paper builds on his work in [Artin-Algebraic-Approximation] and [ArtinI]. This paper introduces Artin's criterion which allows one to prove algebraicity of a stack by verifying deformation-theoretic properties. More precisely (but not very precisely), Artin constructs a presentation of a limit preserving stack $\mathcal{X}$ locally around a point $x \in \mathcal{X}(k)$ as follows: assuming the stack $\mathcal{X}$ satisfies Schlessinger's criterion([Sch]), there exists a formal versal deformation $\hat{\xi } \in \mathop{\mathrm{lim}}\nolimits \mathcal{X}(\hat A / \mathfrak m^ n)$ of $x$. Assuming that formal deformations are effective (i.e., $\mathcal{X}(\hat{A}) \to \mathop{\mathrm{lim}}\nolimits \mathcal{X}(\hat A / \mathfrak m^ n)$ is bijective), then one obtains an effective formal versal deformation $\xi \in \mathcal{X}(\hat A)$. Using results in [ArtinI], one produces a finite type scheme $U$ and an element $\xi _ U: U \to \mathcal{X}$ which is formally versal at a point $u \in U$ over $x$. Then if we assume $\mathcal{X}$ admits a deformation and obstruction theory satisfying certain conditions (ie. compatibility with étale localization and completion as well as constructibility condition), then it is shown in section 4 that formal versality is an open condition so that after shrinking $U$, $U \to \mathcal{X}$ is smooth. Artin also presents a proof that any stack admitting an fppf presentation by a scheme admits a smooth presentation by a scheme so that in particular one can form quotient stacks by flat, separated, finitely presented group schemes.

• Conrad, de Jong: Approximation of Versal Deformations [conrad-dejong]

  This paper offers an approach to Artin's algebraization result by applying Popescu's powerful result: if $A$ is a Noetherian ring and $B$ a Noetherian $A$-algebra, then the map $A \to B$ is a regular morphism if and only if $B$ is a direct limit of smooth $A$-algebras. It is not hard to see that Popescu's result implies Artin's approximation over an arbitrary excellent scheme (the excellence hypothesis implies that for a local ring $A$, the map $A^{\text{h}} \to \hat A$ from the henselization to the completion is regular). The paper uses Popescu's result to give a “groupoid” generalization of the main theorem in [ArtinI] which is valid over arbitrary excellent base schemes and for arbitrary points $s \in S$. In particular, the results in [ArtinVersal] hold under an arbitrary excellent base. They discuss the étale-local uniqueness of the algebraization and whether the automorphism group of the object acts naturally on the henselization of the algebraization.

• Jason Starr: Artin's axioms, composition, and moduli spaces [starr_artin]

  The paper establishes that Artin's axioms for algebraization are compatible with the composition of 1-morphisms.

• Martin Olsson: Deformation theory of representable morphism of algebraic stacks [olsson_deformation]

  This generalizes standard deformation theory results for morphisms of schemes to representable morphisms of algebraic stacks in terms of the cotangent complex. These results cannot be viewed as consequences of Illusie's general theory as the cotangent complex of a representable morphism $X \to \mathcal{X}$ is not defined in terms of cotangent complex of a morphism of ringed topoi (because the lisse-étale site is not functorial).