Artin's Criteria for Algebraicity

Artin's representability criteria provide a systematic set of conditions under which a functor (or stack) is algebraic. These criteria reduce the problem of proving algebraicity to verifiable deformation-theoretic and set-theoretic conditions, and they are the workhorse behind almost all existence results for moduli spaces and moduli stacks.

1. Artin's Axioms

TheoremArtin's Criteria for Algebraic Stacks (1974)

Let $S$ be an excellent scheme and $\mathcal{X}$ a category fibered in groupoids over $(\mathsf{Sch}/S)_{\mathrm{fppf}}$ . Then $\mathcal{X}$ is an algebraic stack locally of finite presentation over $S$ if and only if the following conditions hold:

Stack condition: $\mathcal{X}$ is a stack for the fppf (or etale) topology.
Limit preservation: $\mathcal{X}$ is locally of finite presentation, i.e., for any filtered inverse system $\lbrace T_i \rbrace$ of affine $S$ -schemes with limit $T = \varprojlim T_i$ : $\varinjlim \mathcal{X}(T_i) \xrightarrow{\sim} \mathcal{X}(T)$
Deformation theory (Schlessinger's conditions): For any local Artinian $S$ -algebra $A$ and small extension $A' \to A$ , the natural functor satisfies certain formal properties (existence of a cotangent complex).
Effectivity of formal deformations (Grothendieck's existence): Formal deformations are effective, i.e., compatible systems of deformations over $A/\mathfrak{m}^n$ arise from algebraic deformations.
Openness of versality: If $\hat{\xi}$ is a formal versal deformation at a point $x \in \mathcal{X}$ , then the locus where it remains versal is open.

2. Detailed Conditions

DefinitionSmall Extension

A small extension in $\mathsf{Art}_k$ is a surjection $A' \to A$ of Artinian local $k$ -algebras with kernel $I$ satisfying $\mathfrak{m}_{A'} \cdot I = 0$ . The kernel $I$ is then a $k$ -vector space. The fundamental small extension is $k[\varepsilon]/(\varepsilon^2) \to k$ .

DefinitionDeformation Situation

A deformation situation consists of a diagram: $0 \to M \to A' \to A \to 0$ where $A' \to A$ is a small extension of Artinian local rings and $M = \ker(A' \to A)$ is an $A/\mathfrak{m}_A \cong k$ -module. Given $\xi \in \mathcal{X}(A)$ , we ask:

Does $\xi$ lift to $\xi' \in \mathcal{X}(A')$ ?
If so, what is the set of lifts?

TheoremSchlessinger-Type Conditions

For a functor $\mathcal{X}$ satisfying Artin's axioms, at each point $\xi_0 \in \mathcal{X}(k)$ , there exist $k$ -vector spaces $T_i$ (for $i = -1, 0, 1$ ) and functorial maps such that for each deformation situation:

There is an obstruction class $\mathrm{ob}(\xi, A' \to A) \in T_1 \otimes_k M$ whose vanishing is necessary and sufficient for the existence of a lift.
When the obstruction vanishes, the set of lifts is a torsor under $T_0 \otimes_k M$ .
The automorphisms of any lift form a group isomorphic to $T_{-1} \otimes_k M$ .

These $T_i$ are the cohomology groups of the cotangent complex restricted to $\xi_0$ .

3. Examples of Verifying Artin's Criteria

ExampleModuli of Curves $\mathcal{M}_g$

To show $\mathcal{M}_g$ is algebraic via Artin's criteria:

Stack: Descent for flat proper morphisms with connected fibers is effective in the fppf topology.
Finite presentation: A smooth curve of genus $g$ over $\varprojlim T_i$ spreads out to some $T_i$ (standard limit arguments for finitely presented morphisms).
Deformation theory: $T_{-1} = H^0(C, T_C) = 0$ (for $g \geq 2$ ), $T_0 = H^1(C, T_C) \cong k^{3g-3}$ , $T_1 = H^2(C, T_C) = 0$ . Schlessinger's conditions hold.
Effectivity: By Grothendieck's formal GAGA, a compatible system of curves over $\mathrm{Spec}\, k[[t]]/t^n$ algebraizes to a curve over $\mathrm{Spec}\, k[[t]]$ .
Openness of versality: Follows from the theory of the Hilbert scheme and openness of smoothness.

ExampleModuli of Vector Bundles $\mathrm{Bun}_n(C)$

For a smooth projective curve $C$ , the stack $\mathrm{Bun}_n(C)$ of rank- $n$ vector bundles satisfies Artin's criteria:

Stack: Vector bundles satisfy fppf descent.
Finite presentation: Coherent sheaves over limits spread out.
Deformation theory: For a bundle $\mathcal{E}$ : $T_{-1} = \mathrm{End}(\mathcal{E})$ , $T_0 = \mathrm{Ext}^1(\mathcal{E}, \mathcal{E}) = H^1(C, \mathcal{E}nd(\mathcal{E}))$ , $T_1 = H^2(C, \mathcal{E}nd(\mathcal{E})) = 0$ (since $\dim C = 1$ ). Unobstructed!
Effectivity: Formal GAGA for coherent sheaves.
Openness: Standard argument.

Hence $\mathrm{Bun}_n(C)$ is a smooth algebraic stack of dimension $n^2(g-1)$ (when non-empty).

ExamplePicard Stack

The Picard stack $\mathcal{P}ic_{X/S}$ parametrizing line bundles on $X/S$ :

$T_{-1} = H^0(X, \mathcal{O}_X) = k$ (scalars, corresponding to $\mathbb{G}_m$ -automorphisms).
$T_0 = H^1(X, \mathcal{O}_X)$ (first-order deformations of a line bundle).
$T_1 = H^2(X, \mathcal{O}_X)$ (obstructions).

By Artin's criteria, $\mathcal{P}ic_{X/S}$ is an algebraic stack. The rigidification (killing the $\mathbb{G}_m$ -automorphisms) gives the Picard scheme $\mathrm{Pic}_{X/S}$ , which is an algebraic space (or scheme under mild assumptions).

ExampleModuli of Stable Maps

The Kontsevich moduli stack $\overline{\mathcal{M}}_{g,n}(X, \beta)$ of stable maps from $n$ -pointed genus- $g$ curves to a target variety $X$ representing a class $\beta \in H_2(X, \mathbb{Z})$ . Artin's criteria verify algebraicity:

Deformation theory: $T_0 = H^0(C, f^*T_X)$ balanced against $T_1 = H^1(C, f^*T_X)$ .
The expected dimension is $\dim X (1-g) + \int_\beta c_1(T_X) + 3g - 3 + n$ .
Effectivity follows from GAGA.

4. The Effectivity Condition

DefinitionFormal Object

A formal object of $\mathcal{X}$ over a complete local ring $\hat{R} = \varprojlim R/\mathfrak{m}^n$ is a compatible system $\lbrace \xi_n \in \mathcal{X}(R/\mathfrak{m}^n) \rbrace_{n \geq 1}$ with $\xi_n|_{R/\mathfrak{m}^{n-1}} \cong \xi_{n-1}$ .

TheoremGrothendieck Existence Theorem

If $f: X \to \mathrm{Spec}\, R$ is a proper morphism (with $R$ a complete local Noetherian ring), then the functor $\mathsf{Coh}(X) \to \varprojlim \mathsf{Coh}(X_n)$ is an equivalence of categories, where $X_n = X \times_{\mathrm{Spec}\, R} \mathrm{Spec}\, R/\mathfrak{m}^n$ .

ExampleEffectivity for Curves

A compatible system of curves $\lbrace C_n \to \mathrm{Spec}\, k[t]/(t^n) \rbrace$ algebraizes to a curve $\mathcal{C} \to \mathrm{Spec}\, k[[t]]$ by Grothendieck's existence theorem applied to the embedding $C_n \hookrightarrow \mathbb{P}^N_{k[t]/(t^n)}$ via the $n$ -canonical embedding.

ExampleNon-Effectivity for Non-Proper Situations

Consider formal deformations of the affine line $\mathbb{A}^1$ . The formal deformation $\mathrm{Spec}\, k[[t]][x]/(x^2 - t)$ over $k[[t]]$ gives a family of smooth affine curves. But formal deformations of non-proper schemes need not algebraize in general. The effectivity condition in Artin's criteria handles this subtlety.

5. Openness of Versality

DefinitionVersal Deformation

A deformation $\xi \in \mathcal{X}(\mathrm{Spec}\, R)$ is versal at a closed point $s \in \mathrm{Spec}\, R$ if for every deformation situation, every deformation over $A$ lifts (not necessarily uniquely) to $A'$ via a map from $R$ . It is miniversal (or semi-universal) if the induced map on tangent spaces is an isomorphism.

TheoremOpenness of Versality

If a morphism $U \to \mathcal{X}$ is versal at a point $u \in U$ , then it is versal in a Zariski open neighborhood of $u$ . This is the condition that ensures one can construct a smooth atlas from local versal deformations.

ExampleVersality for Curves

Given a versal family of curves $\mathcal{C} \to B$ over a smooth base $B$ at a point $b_0 \in B$ , the versality is open: there exists an open $U \ni b_0$ in $B$ such that $\mathcal{C}_U \to U$ is a versal family for all fibers simultaneously. The map $U \to \mathcal{M}_g$ is then smooth, providing a smooth atlas.

6. Artin's Criteria for Algebraic Spaces

TheoremArtin's Criteria for Algebraic Spaces

A functor $F: (\mathsf{Sch}/S)^{\mathrm{op}} \to \mathsf{Sets}$ is an algebraic space locally of finite presentation over $S$ if:

$F$ is a sheaf for the etale topology.
$F$ is locally of finite presentation.
The deformation theory conditions hold (with $T_{-1} = 0$ , since there are no automorphisms for functors valued in sets).
Formal deformations are effective.
Openness of versality.

ExamplePicard Scheme via Artin's Criteria

The Picard functor $\mathrm{Pic}_{X/S}$ (after rigidification) satisfies Artin's criteria for algebraic spaces. The tangent space is $H^1(X, \mathcal{O}_X)$ , obstructions lie in $H^2(X, \mathcal{O}_X)$ , and there are no automorphisms after rigidification.

ExampleHilbert Scheme via Artin (Alternative Proof)

One can re-prove Grothendieck's representability of the Hilbert functor using Artin's criteria:

The tangent space at $[Z]$ is $\mathrm{Hom}(\mathcal{I}_Z, \mathcal{O}_Z)$ .
Obstructions lie in $\mathrm{Ext}^1(\mathcal{I}_Z, \mathcal{O}_Z)$ .
No automorphisms: $T_{-1} = \mathrm{Hom}(\mathcal{O}_Z, \mathcal{O}_Z) / k = 0$ (after subtracting the identity).
Effectivity by Grothendieck's existence theorem.

7. Comparison: DM vs. Artin Stacks

RemarkWhen is a Stack DM?

An algebraic (Artin) stack $\mathcal{X}$ is Deligne-Mumford if and only if the diagonal $\Delta: \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is unramified. Equivalently:

The automorphism group scheme $\mathrm{Aut}_{\mathcal{X}}(\xi)$ is etale (hence finite, if proper) for every object $\xi$ .
In the deformation theory: $T_{-1} = 0$ (no infinitesimal automorphisms).

If additionally $\mathrm{Aut}_{\mathcal{X}}(\xi)$ is always trivial, then $\mathcal{X}$ is an algebraic space.

ExampleDM vs. Artin Comparison Table

| Stack | $T_{-1}$ | Type | |-------|----------|------| | $\mathcal{M}_g$ ( $g \geq 2$ ) | $H^0(C, T_C) = 0$ | Deligne-Mumford | | $\mathcal{M}_{1,1}$ | $H^0(E, T_E) = 0$ (using marked point) | Deligne-Mumford | | $\mathcal{M}_0 = B\mathrm{PGL}_2$ | $\mathrm{Lie}(\mathrm{PGL}_2) \neq 0$ | Artin, not DM | | $\mathrm{Bun}_n(C)$ | $\mathrm{End}(\mathcal{E}) \neq 0$ | Artin, not DM | | $\mathrm{Hilb}^P_{X/S}$ | 0 (subschemes) | Scheme | | $\mathcal{P}ic_{X/S}$ | $H^0(\mathcal{O}_X) = k$ ( $\mathbb{G}_m$ auts) | Artin, not DM |

ExampleApplying Artin's Criteria to Quotient Stacks

For a quotient stack $[X/G]$ where $G$ is a smooth group scheme acting on a finite-type $S$ -scheme $X$ :

The stack condition holds because $G$ -torsors satisfy descent.
Finite presentation follows from finite type of $X$ and $G$ .
Deformation theory: $T_{-1}$ is the Lie algebra of the stabilizer, $T_0$ is the normal space to the orbit, $T_1$ is related to $H^1$ of the normal sheaf.
Effectivity and openness follow from the algebraicity of $X$ and $G$ .

Hence $[X/G]$ is always an algebraic stack when $G$ is smooth and $X$ is of finite type over $S$ .