The Stacks project

97.1 Introduction

In this chapter we discuss Artin's axioms for the representability of functors by algebraic spaces. As references we suggest the papers [ArtinI], [ArtinII], [ArtinVersal].

Some of the notation, conventions, and terminology in this chapter is awkward and may seem backwards to the more experienced reader. This is intentional. Please see Quot, Section 98.2 for an explanation.

Let $S$ be a locally Noetherian base scheme. Let

\[ p : \mathcal{X} \longrightarrow (\mathit{Sch}/S)_{fppf} \]

be a category fibred in groupoids. Let $x_0$ be an object of $\mathcal{X}$ over a field $k$ of finite type over $S$. Throughout this chapter an important role is played by the predeformation category (see Formal Deformation Theory, Definition 89.6.2)

\[ \mathcal{F}_{\mathcal{X}, k, x_0} \longrightarrow \{ \text{Artinian local }S\text{-algebras with residue field }k\} \]

associated to $x_0$ over $k$. We introduce the Rim-Schlessinger condition (RS) for $\mathcal{X}$ and show it guarantees that $\mathcal{F}_{\mathcal{X}, k, x_0}$ is a deformation category, i.e., $\mathcal{F}_{\mathcal{X}, k, x_0}$ satisies (RS) itself. We discuss how $\mathcal{F}_{\mathcal{X}, k, x_0}$ changes if one replaces $k$ by a finite extension and we discuss tangent spaces.

Next, we discuss formal objects $\xi = (\xi _ n)$ of $\mathcal{X}$ which are inverse systems of objects lying over the quotients $R/\mathfrak m^ n$ where $R$ is a Noetherian complete local $S$-algebra whose residue field is of finite type over $S$. This is the same thing as having a formal object in $\mathcal{F}_{\mathcal{X}, k, x_0}$ for some $x_0$ and $k$. A formal object is called effective when there is an object of $\mathcal{X}$ over $R$ which gives rise to the inverse system. A formal object of $\mathcal{X}$ is called versal if it gives rise to a versal formal object of $\mathcal{F}_{\mathcal{X}, k, x_0}$. Finally, given a finite type $S$-scheme $U$, an object $x$ of $\mathcal{X}$ over $U$, and a closed point $u_0 \in U$ we say $x$ is versal at $u_0$ if the induced formal object over the complete local ring $\mathcal{O}_{U, u_0}^\wedge $ is versal.

Having worked through this material we can state Artin's celebrated theorem: our $\mathcal{X}$ is an algebraic stack if the following are true

  1. $\mathcal{O}_{S, s}$ is a G-ring for all $s \in S$,

  2. $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces,

  3. $\mathcal{X}$ is a stack for the étale topology,

  4. $\mathcal{X}$ is limit preserving,

  5. $\mathcal{X}$ satisfies (RS),

  6. tangent spaces and spaces of infinitesimal automorphisms of the deformation categories $\mathcal{F}_{\mathcal{X}, k, x_0}$ are finite dimensional,

  7. formal objects are effective,

  8. $\mathcal{X}$ satisfies openness of versality.

This is Lemma 97.17.1; see also Proposition 97.17.2 for a slight improvement. There is an analogous proposition characterizing which functors $F : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ are algebraic spaces, see Section 97.16.

Here is a rough outline of the proof of Artin's theorem. First we show that there are plenty of versal formal objects using (RS) and the finite dimensionality of tangent and aut spaces, see for example Formal Deformation Theory, Lemma 89.27.6. These formal objects are effective by assumption. Effective formal objects can be “approximated” by objects $x$ over finite type $S$-schemes $U$, see Lemma 97.10.1. This approximation uses the local rings of $S$ are G-rings and that $\mathcal{X}$ is limit preserving; it is perhaps the most difficult part of the proof relying as it does on general Néron desingularization to approximate formal solutions of algebraic equations over a Noetherian local G-ring by solutions in the henselization. Next openness of versality implies we may (after shrinking $U$) assume $x$ is versal at every closed point of $U$. Having done all of this we show that $U \to \mathcal{X}$ is a smooth morphism. Taking sufficiently many $U \to \mathcal{X}$ we show that we obtain a “smooth atlas” for $\mathcal{X}$ which shows that $\mathcal{X}$ is an algebraic stack.

In checking Artin's axioms for a given category $\mathcal{X}$ fibred in groupoids, the most difficult step is often to verify openness of versality. For the discussion that follows, assume that $\mathcal{X}/S$ already satisfies the other conditions listed above. In this chapter we offer two methods that will allow the reader to prove $\mathcal{X}$ satisfies openness of versality:

  1. The first is to assume a stronger Rim-Schlessinger condition, called (RS*) and to assume a stronger version of formal effectiveness, essentially requiring objects over inverse systems of thickenings to be effective. It turns out that under these assumptions, openness of versality comes for free, see Lemma 97.20.3. Please observe that here we are using in an essential manner that $\mathcal{X}$ is defined on that category of all schemes over $S$, not just the category of Noetherian schemes!

  2. The second, following Artin, is to require $\mathcal{X}$ to come equipped with an obstruction theory. If said obstruction theory “commutes with products” in a suitable sense, then $\mathcal{X}$ satisfies openness of versality, see Lemma 97.22.2.

Obstruction theories can be axiomatized in many different ways and indeed many variants (often adapted to specific moduli stacks) can be found in the literature. We explain a variant using the derived category (which often arises naturally from deformation theory computations done in the literature) in Lemma 97.24.4.

In Section 97.26 we discuss what needs to be modified to make things work for functors defined on the category $(\textit{Noetherian}/S)_{\acute{e}tale}$ of locally Noetherian schemes over $S$.

In the final section of this chapter as an application of Artin's axioms we prove Artin's theorem on the existence of contractions, see Section 97.27. The theorem says roughly that given an algebraic space $X'$ separated of finite type over $S$, a closed subset $T' \subset |X'|$, and a formal modification

\[ \mathfrak {f} : X'_{/T'} \longrightarrow \mathfrak {X} \]

where $\mathfrak {X}$ is a Noetherian formal algebraic space over $S$, there exists a proper morphism $f : X' \to X$ which “realizes the contraction”. By this we mean that there exists an identification $\mathfrak {X} = X_{/T}$ such that $\mathfrak {f} = f_{/T'} : X'_{/T'} \to X_{/T}$ where $T = f(T')$ and moreover $f$ is an isomorphism over $X \setminus T$. The proof proceeds by defining a functor $F$ on the category of locally Noetherian schemes over $S$ and proving Artin's axioms for $F$. Amusingly, in this application of Artin's axioms, openness of versality is not the hardest thing to prove, instead the proof that $F$ is limit preserving requires a lot of work and preliminary results.

Comments (2)

Comment #4232 by Jeroen van der Meer on

Aesthetic nitpick: this is the only chapter whose title isn't capitalized.

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