Goal: For each closed point $x_0 \in \mathcal{X}$ (i.e., an object $\xi_0 \in \mathcal{X}(k)$ for a field $k$), construct a complete Noetherian local ring $(\hat{A}, \mathfrak{m})$ with residue field $k$ and a formally versal deformation $\hat{\xi} \in \mathcal{X}(\hat{A})$ lifting $\xi_0$.

Construction via Schlessinger's theorem: By conditions (3) and (6), the deformation functor $D_{\xi_0} : \mathrm{Art}_k \to \mathrm{Groupoids}, \quad A \mapsto \mathcal{X}(A) \times_{\mathcal{X}(k)} \{\xi_0\}$ (deformations of $\xi_0$ over Artinian local $k$-algebras with residue field $k$) satisfies Schlessinger's conditions:

(S1): For any surjection $A'' \to A$ and any map $A' \to A$, the natural map $D_{\xi_0}(A' \times_A A'') \to D_{\xi_0}(A') \times_{D_{\xi_0}(A)} D_{\xi_0}(A'')$ is essentially surjective.

(S2): When $A = k$, the above map is an equivalence.

These conditions, combined with the finite-dimensionality of the tangent space $T^1 = D_{\xi_0}(k[\epsilon])$ (guaranteed by condition (6) -- the tangent space is controlled by a coherent sheaf), allow us to apply Schlessinger's theorem (extended to the groupoid setting by Rim) to conclude:

There exists a complete Noetherian local ring $\hat{A} \cong k[[t_1, \ldots, t_n]]/I$ (with $n = \dim_k T^1$) and a formally versal element $\hat{\xi} \in \mathcal{X}(\hat{A})$, meaning: for every Artinian local $k$-algebra $B$ and deformation $\eta \in D_{\xi_0}(B)$, there exists a (not necessarily unique) local homomorphism $\hat{A} \to B$ pulling $\hat{\xi}$ back to $\eta$.

Obstruction analysis: The ideal $I$ is determined by the obstruction space $T^2$. More precisely, the hull $\hat{A}$ is constructed inductively: starting from $A_1 = k[T^1]/(T^1)^2$, one extends $A_n$ to $A_{n+1}$ by lifting along the surjection $A_{n+1} \to A_n$ with kernel annihilated by $\mathfrak{m}$. The obstruction to lifting lies in $T^2 \otimes \ker(A_{n+1} \to A_n)$. The elements of $I$ are precisely the obstructions encountered in this inductive construction.

When $T^2 = 0$ (the unobstructed case), $\hat{A} \cong k[[t_1, \ldots, t_n]]$ is a power series ring, and the versal deformation space is smooth.

Goal: Promote the formal versal deformation $\hat{\xi} \in \mathcal{X}(\hat{A})$ to an algebraic deformation $\xi \in \mathcal{X}(A)$ for some ring $A$ of finite type over $S$, with $\hat{A}$ as the completion of $A$ at a point.

Using effectivity (condition 4): Write $\hat{A} = \varprojlim A_n$ where $A_n = \hat{A}/\mathfrak{m}^{n+1}$. The formal deformation $\hat{\xi}$ is a compatible system $\{\xi_n \in \mathcal{X}(A_n)\}$. By the effectivity condition, this system is induced by a single object $\hat{\xi} \in \mathcal{X}(\hat{A})$.

Artin approximation: Artin's approximation theorem (which requires the excellence of $S$) states: given $\hat{\xi} \in \mathcal{X}(\hat{A})$, there exists an $S$-algebra $A$ of finite type, a point $s \in \operatorname{Spec} A$ with $\hat{\mathcal{O}}_{\operatorname{Spec} A, s} \cong \hat{A}$, and an object $\xi \in \mathcal{X}(A)$ such that $\xi$ and $\hat{\xi}$ agree to any prescribed finite order at $s$.

More precisely, for any $N \geq 1$, there exists $A$ of finite type and $\xi \in \mathcal{X}(A)$ with: $\xi \otimes_A A/\mathfrak{m}_s^N \cong \hat{\xi} \otimes_{\hat{A}} \hat{A}/\mathfrak{m}^N.$

By choosing $N$ sufficiently large (depending on the obstruction theory), we can ensure that $\xi$ retains the essential properties of $\hat{\xi}$, in particular the formal versality.

Result: We obtain a scheme $U_0 = \operatorname{Spec} A$ (or an open neighborhood of $s$ in $\operatorname{Spec} A$) of finite type over $S$, together with a morphism $\phi_0 : U_0 \to \mathcal{X}$ that is formally smooth at the point $s$.

Goal: Show that the morphism $\phi_0 : U_0 \to \mathcal{X}$ constructed in Step 2, which is formally smooth at the point $s$, is actually smooth in a Zariski neighborhood of $s$.

Using openness of versality (condition 5): This condition states precisely that the locus of points in $U_0$ where $\phi_0$ is formally smooth is a Zariski open subset. By construction, this locus contains $s$, so it contains an open neighborhood $U \ni s$.

Why this is non-trivial: Formally smooth at a point means the infinitesimal lifting property holds at that point. This is an infinitesimal condition. Openness of versality promotes this to a Zariski-open condition, bridging the formal and algebraic worlds.

Verification of openness: In practice, openness of versality follows from the coherence of the deformation and obstruction sheaves (condition 6). The formal smoothness of $\phi_0$ at a point $t \in U_0$ is equivalent to the surjectivity of a map between coherent sheaves at $t$, and surjectivity of a map of coherent sheaves is an open condition.

Result: There is an open subscheme $U \subset U_0$ containing $s$ such that $\phi = \phi_0|_U : U \to \mathcal{X}$ is formally smooth, hence smooth (since $\mathcal{X}$ and $U$ are locally of finite presentation, formal smoothness equals smoothness).

Goal: Show that by performing the construction of Steps 1--3 for all closed points, we can patch together the local smooth morphisms into a global smooth surjection $U \to \mathcal{X}$.

For each closed point $x_0 \in |\mathcal{X}|$: Steps 1--3 produce an affine scheme $U_{x_0}$ of finite type over $S$ and a smooth morphism $\phi_{x_0} : U_{x_0} \to \mathcal{X}$ whose image contains $x_0$.

Surjectivity: The images $\phi_{x_0}(U_{x_0})$ cover all closed points of $|\mathcal{X}|$. By the limit preservation condition (2), the stack $\mathcal{X}$ is locally of finite presentation, so its points are determined by finite-type points (closed points of fibers). Hence the collection $\{\phi_{x_0}\}$ covers all of $|\mathcal{X}|$.

Taking the disjoint union: Define $U = \coprod_{x_0} U_{x_0}$ (over a set of closed points covering all of $|\mathcal{X}|$). The induced morphism $\phi : U \to \mathcal{X}$ is smooth (each component is smooth) and surjective (the images cover $|\mathcal{X}|$).

Finiteness: By quasi-compactness of $\mathcal{X}$ (or by restricting to a finite cover if $\mathcal{X}$ is quasi-compact), we can take $U$ to be a finite disjoint union.

Conclusion: The morphism $\phi : U \to \mathcal{X}$ is a smooth surjection from a scheme, i.e., $\phi$ is a smooth atlas. Combined with the representability of the diagonal (which follows from the Schlessinger-type conditions on isomorphism functors), this proves $\mathcal{X}$ is an algebraic stack. $\square$

The proof that $\Delta : \mathcal{X} \to \mathcal{X} \times_S \mathcal{X}$ is representable uses the same strategy applied to the isomorphism functor:

For objects $\xi, \eta \in \mathcal{X}(T)$, the sheaf $\operatorname{Isom}_T(\xi, \eta)$ must be shown to be an algebraic space. This is a functor on $(\mathrm{Sch}/T)$: $\operatorname{Isom}_T(\xi, \eta)(T') = \{\text{isomorphisms } \xi|_{T'} \xrightarrow{\sim} \eta|_{T'}\}.$

The Schlessinger-type conditions for $\mathcal{X}$ imply analogous conditions for $\operatorname{Isom}_T(\xi, \eta)$:

• The tangent space is $T^0(\xi, \eta) = \operatorname{Hom}(\xi, \eta)$ (a coherent module).
• The obstruction space is $T^1(\xi, \eta) \subseteq \operatorname{Ext}^1(\xi, \eta)$.

By Artin's representability for algebraic spaces (Theorem 4.2), $\operatorname{Isom}_T(\xi, \eta)$ is an algebraic space. This gives the representability of $\Delta$.

For the special case $\xi = \eta$: $\operatorname{Aut}_T(\xi) = \operatorname{Isom}_T(\xi, \xi)$ is an algebraic group space over $T$, whose Lie algebra is $\operatorname{End}(\xi)$.