Schlessinger's Theorem

Schlessinger's theorem (1968) provides necessary and sufficient conditions for a deformation functor to have a hull (pro-representable hull), and conditions for full pro-representability. It is the foundational result in formal deformation theory and underpins the local study of moduli problems.

1. Setup and Notation

DefinitionCategory of Artinian Rings

Let $k$ be a field. Denote by $\mathsf{Art}_k$ the category of local Artinian $k$ -algebras $(A, \mathfrak{m}_A)$ with residue field $A/\mathfrak{m}_A \cong k$ . Morphisms are local $k$ -algebra homomorphisms. The ring of dual numbers $k[\varepsilon] = k[\varepsilon]/(\varepsilon^2)$ is the simplest non-trivial object.

DefinitionFunctor of Artinian Rings

A functor of Artinian rings is a covariant functor $F: \mathsf{Art}_k \to \mathsf{Sets}$ with $F(k) = \lbrace * \rbrace$ (a single point, the object being deformed). Such functors arise naturally from moduli problems by restricting to infinitesimal neighborhoods of a point.

DefinitionTangent Space

The tangent space to $F$ is $t_F = F(k[\varepsilon]/(\varepsilon^2))$ If $F$ satisfies condition (H2) below, $t_F$ carries a natural $k$ -vector space structure.

2. The Fiber Product Condition

The key technical tool is the analysis of fiber products in $\mathsf{Art}_k$ .

DefinitionFiber Product Diagram

Given morphisms $A_1 \to A_0$ and $A_2 \to A_0$ in $\mathsf{Art}_k$ , the fiber product is $A_1 \times_{A_0} A_2 = \lbrace (a_1, a_2) \in A_1 \times A_2 \mid a_1 \bmod A_0 = a_2 \bmod A_0 \rbrace$ The natural map on functors is $F(A_1 \times_{A_0} A_2) \to F(A_1) \times_{F(A_0)} F(A_2)$

3. Schlessinger's Conditions

TheoremSchlessinger's Theorem (1968)

Let $F: \mathsf{Art}_k \to \mathsf{Sets}$ with $F(k) = \lbrace * \rbrace$ and $t_F$ finite-dimensional. Consider the conditions:

(H1) For every fiber product diagram where $A_2 \to A_0$ is a small surjection, the natural map $F(A_1 \times_{A_0} A_2) \to F(A_1) \times_{F(A_0)} F(A_2)$ is surjective.

(H2) The above map is a bijection when $A_0 = k$ and $A_1 = A_2 = k[\varepsilon]$ . (This gives $t_F$ a vector space structure.)

(H3) $t_F$ is finite-dimensional over $k$ .

(H4) For every small surjection $A' \to A$ , the map $F(A' \times_A A') \to F(A') \times_{F(A)} F(A')$ is a bijection.

Then:

(H1) + (H2) + (H3) $\Longrightarrow$ $F$ has a hull (miniversal formal deformation).
(H1) + (H2) + (H3) + (H4) $\Longrightarrow$ $F$ is pro-representable.

4. Hulls and Pro-Representability

DefinitionHull (Miniversal Deformation)

A hull for $F$ is a pair $(R, \hat{\xi})$ where:

$R$ is a complete local $k$ -algebra with residue field $k$ and $\mathfrak{m}_R^2$ finitely generated.
$\hat{\xi} = \lbrace \xi_n \in F(R/\mathfrak{m}_R^n) \rbrace$ is a compatible system (formal element).
The induced morphism $h_R \to F$ is smooth (i.e., surjective on all Artinian quotients with the lifting property for small extensions).
The induced map on tangent spaces $t_{h_R} = \mathfrak{m}_R / \mathfrak{m}_R^2 \to t_F$ is an isomorphism.

DefinitionPro-Representability

$F$ is pro-representable if there exists a complete local $k$ -algebra $R$ with $F(A) \cong \mathrm{Hom}_{\text{cont}}(R, A)$ for all $A \in \mathsf{Art}_k$ . This is stronger than having a hull: the morphism $h_R \to F$ is an isomorphism (not just smooth).

5. Examples of Schlessinger's Conditions

ExampleDeformations of a Smooth Variety

For a smooth projective variety $X_0/k$ , the deformation functor $\mathrm{Def}_{X_0}$ satisfies:

(H1): Follows from the deformation theory of smooth morphisms.
(H2): The tangent space $t_F = H^1(X_0, T_{X_0})$ has a natural vector space structure.
(H3): $h^1(X_0, T_{X_0}) < \infty$ since $X_0$ is proper.
(H4): Fails in general! Two deformations over $A'$ that agree modulo a small extension need not be the same deformation.

So $\mathrm{Def}_{X_0}$ has a hull but is generally not pro-representable. It is pro-representable if and only if $\mathrm{Aut}(X_0) \to \mathrm{Aut}(H^0(X_0, \mathcal{O}_{X_0}))$ is injective (roughly, when automorphisms are "rigid").

ExampleWhy (H4) Fails for Varieties with Automorphisms

Consider an elliptic curve $E$ with automorphism $[-1]$ . Given a deformation $\mathcal{E}$ over $A'$ , the pullback $[-1]^*\mathcal{E}$ is another deformation isomorphic to $\mathcal{E}$ over $A = A'/I$ (since $[-1]$ lifts to any infinitesimal thickening). But $\mathcal{E}$ and $[-1]^*\mathcal{E}$ might be distinct elements of $F(A')$ that become equal in $F(A)$ . The fiber product $F(A' \times_A A')$ must account for this, and (H4) fails because the isomorphism $[-1]$ creates multiple elements mapping to the same pair.

ExampleDeformations of a Hypersurface Singularity

Let $f \in k[[x_1, \ldots, x_n]]$ define an isolated hypersurface singularity, and let $F$ be the functor of deformations of $X_0 = \mathrm{Spec}\, k[[x_1, \ldots, x_n]]/(f)$ .

The tangent space is $T^1 = k[[x_1, \ldots, x_n]] / (f, \partial f/\partial x_1, \ldots, \partial f/\partial x_n)$ , the Tjurina algebra. For an isolated singularity, $\dim_k T^1 < \infty$ .

Condition (H4) holds because isolated singularities have trivial automorphisms in the formal category (the identity is the only automorphism fixing the singular point). So $F$ is pro-representable.

ExampleA_n Singularity

For the $A_n$ singularity $f = x^{n+1} + y^2$ (in $k[[x,y]]$ ): $T^1 = k[[x,y]]/(x^{n+1} + y^2, (n+1)x^n, 2y) = k[x]/(x^n) \cong k^n$ The versal deformation is: $F(t_1, \ldots, t_n) = x^{n+1} + t_{n-1}x^{n-1} + \cdots + t_1 x + t_0 + y^2$ The deformation ring is $R = k[[t_0, \ldots, t_{n-1}]]$ , a smooth (power series) ring. The discriminant locus where the singularity persists is defined by the discriminant of the polynomial $x^{n+1} + t_{n-1}x^{n-1} + \cdots + t_0$ .

ExampleD_4 Singularity

For $f = x^3 + y^3$ (the $D_4$ singularity in characteristic $\neq 3$ ): $T^1 = k[[x,y]]/(x^3 + y^3, 3x^2, 3y^2) = k\lbrace 1, x, y, xy \rbrace \cong k^4$ The versal deformation is: $F = x^3 + y^3 + t_3 xy + t_2 y + t_1 x + t_0$ The deformation ring $R = k[[t_0, t_1, t_2, t_3]]$ is smooth (no obstructions for curve singularities).

ExampleCone over an Elliptic Curve

Let $X_0$ be the affine cone over an elliptic curve $E \hookrightarrow \mathbb{P}^2$ (a degree-3 cone in $\mathbb{A}^3$ ). The singularity at the vertex has: $T^1 \cong \bigoplus_{n \geq 1} H^1(E, \mathcal{O}_E(n))$ which is 1-dimensional (only $n = 1$ contributes, giving $H^1(E, \mathcal{O}_E(1)) \cong H^1(E, \mathcal{O}_E) \cong k$ ).

The obstruction space $T^2 = 0$ (the singularity is a complete intersection), so the deformation is unobstructed and $R = k[[t]]$ .

6. The Tangent-Obstruction Exact Sequence

TheoremExact Sequence for Fiber Products

For a functor $F$ satisfying (H1) and (H2), and a small extension $0 \to I \to A' \to A \to 0$ , there is an exact sequence: $0 \to t_F \otimes_k I \to F(A') \to F(A) \xrightarrow{\mathrm{ob}} \mathrm{Ob}(F) \otimes_k I$ where:

$t_F \otimes I$ acts freely and transitively on the fibers of $F(A') \to F(A)$ (when nonempty).
$\mathrm{ob}$ is the obstruction map.
$\mathrm{Ob}(F) = T^2$ is the obstruction space.

The deformation ring has the form $R = k[[t_1, \ldots, t_m]]/(f_1, \ldots, f_r)$ where $m = \dim t_F$ and $r \leq \dim \mathrm{Ob}(F)$ .

ExampleObstructed Deformation: Mumford's Example

Mumford constructed a smooth projective surface $S$ in $\mathbb{P}^4$ (a regular surface of degree 15 with $p_g = 3$ ) whose deformation functor has: $\dim t_F = h^1(S, T_S) = 0, \quad \dim \mathrm{Ob}(F) = h^2(S, T_S) \neq 0$ The surface is rigid (no deformations at all), yet the obstruction space is nonzero. This shows that the obstruction space can be "bigger than necessary."

7. Groupoid-Valued Version

TheoremSchlessinger for Groupoid-Valued Functors

For a functor $F: \mathsf{Art}_k \to \mathsf{Groupoids}$ (relevant for stacks), the analogous conditions are:

(H1'): The natural functor $F(A_1 \times_{A_0} A_2) \to F(A_1) \times_{F(A_0)} F(A_2)$ is essentially surjective when $A_2 \to A_0$ is surjective.
(H2'): It is an equivalence when $A_0 = k$ .
(H3'): $t_F$ and $t_F^{-1} = \mathrm{Aut}_F(k[\varepsilon])$ are finite-dimensional.

Under these conditions, $F$ admits a versal formal object (hull in the groupoid sense).

ExampleDeformation Groupoid of $\mathcal{M}_g$ at $[C]$

The deformation groupoid $F_{[C]}$ at a curve $[C] \in \mathcal{M}_g$ :

$t_F = H^1(C, T_C) \cong k^{3g-3}$
$t_F^{-1} = H^0(C, T_C) = 0$ for $g \geq 2$
$\mathrm{Ob}(F) = H^2(C, T_C) = 0$

Since $t_F^{-1} = 0$ , condition (H4') holds and the groupoid functor is pro-representable (by $k[[t_1, \ldots, t_{3g-3}]]$ ). The stack structure at $[C]$ is the quotient $[\mathrm{Spf}\, k[[t_1, \ldots, t_{3g-3}]] / \mathrm{Aut}(C)]$ .

ExampleDeformation Groupoid of $\mathrm{Bun}_n$ at $[\mathcal{E}]$

For a vector bundle $\mathcal{E}$ on a curve $C$ of genus $g$ :

$t_F = H^1(C, \mathcal{E}nd(\mathcal{E})) \cong k^{n^2(g-1) + \dim \mathrm{End}(\mathcal{E})}$ (by Riemann-Roch)
$t_F^{-1} = H^0(C, \mathcal{E}nd(\mathcal{E})) = \mathrm{End}(\mathcal{E})$ (nontrivial! contains at least scalars)
$\mathrm{Ob}(F) = H^2(C, \mathcal{E}nd(\mathcal{E})) = 0$ (curve has dimension 1)

The nonvanishing of $t_F^{-1}$ means (H4') fails: $F$ has a hull but is not pro-representable as a groupoid functor. This reflects the Artin stack (not DM) nature of $\mathrm{Bun}_n$ .

8. Relation to Artin's Criteria

RemarkSchlessinger to Artin

Schlessinger's theorem works at the formal level (complete local rings). To go from formal to algebraic, one needs:

Schlessinger's conditions $\Rightarrow$ existence of a formal hull.
Grothendieck's existence theorem $\Rightarrow$ effectivity (formal $\Rightarrow$ algebraic).
Openness of versality $\Rightarrow$ the versal family covers an open neighborhood.

Together, these give Artin's criteria. Schlessinger provides the infinitesimal foundation, and Artin's additional conditions bridge to the algebraic world.

ExampleSummary of the Logical Chain

$\text{Moduli Problem} \xrightarrow{\text{Schlessinger}} \text{Formal Hull} \xrightarrow{\text{Effectivity}} \text{Local Algebraic Family} \xrightarrow{\text{Openness}} \text{Smooth Atlas} \xrightarrow{\text{Artin}} \text{Algebraic Stack}$