The Stacks project

Lemma 98.9.5. Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be an algebraic stack over $S$. The functor ( is an equivalence.

Proof. Case I: $\mathcal{X}$ is representable (by a scheme). Say $\mathcal{X} = (\mathit{Sch}/X)_{fppf}$ for some scheme $X$ over $S$. Unwinding the definitions we have to prove the following: Given a Noetherian complete local $S$-algebra $R$ with $R/\mathfrak m$ of finite type over $S$ we have

\[ \mathop{\mathrm{Mor}}\nolimits _ S(\mathop{\mathrm{Spec}}(R), X) \longrightarrow \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Mor}}\nolimits _ S(\mathop{\mathrm{Spec}}(R/\mathfrak m^ n), X) \]

is bijective. This follows from Formal Spaces, Lemma 87.33.2.

Case II. $\mathcal{X}$ is representable by an algebraic space. Say $\mathcal{X}$ is representable by $X$. Again we have to show that

\[ \mathop{\mathrm{Mor}}\nolimits _ S(\mathop{\mathrm{Spec}}(R), X) \longrightarrow \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Mor}}\nolimits _ S(\mathop{\mathrm{Spec}}(R/\mathfrak m^ n), X) \]

is bijective for $R$ as above. This is Formal Spaces, Lemma 87.33.3.

Case III: General case of an algebraic stack. A general remark is that the left and right hand side of ( are categories fibred in groupoids over the category of affine schemes over $S$ which are spectra of Noetherian complete local rings with residue field of finite type over $S$. We will also see in the proof below that they form stacks for a certain topology on this category.

We first prove fully faithfulness. Let $R$ be a Noetherian complete local $S$-algebra with $k = R/\mathfrak m$ of finite type over $S$. Let $x, x'$ be objects of $\mathcal{X}$ over $R$. As $\mathcal{X}$ is an algebraic stack $\mathit{Isom}(x, x')$ is representable by an algebraic space $I$ over $\mathop{\mathrm{Spec}}(R)$, see Algebraic Stacks, Lemma 94.10.11. Applying Case II to $I$ over $\mathop{\mathrm{Spec}}(R)$ implies immediately that ( is fully faithful on fibre categories over $\mathop{\mathrm{Spec}}(R)$. Hence the functor is fully faithful by Categories, Lemma 4.35.9.

Essential surjectivity. Let $\xi = (R, \xi _ n, f_ n)$ be a formal object of $\mathcal{X}$. Choose a scheme $U$ over $S$ and a surjective smooth morphism $f : (\mathit{Sch}/U)_{fppf} \to \mathcal{X}$. For every $n$ consider the fibre product

\[ (\mathit{Sch}/\mathop{\mathrm{Spec}}(R/\mathfrak m^ n))_{fppf} \times _{\xi _ n, \mathcal{X}, f} (\mathit{Sch}/U)_{fppf} \]

By assumption this is representable by an algebraic space $V_ n$ surjective and smooth over $\mathop{\mathrm{Spec}}(R/\mathfrak m^ n)$. The morphisms $f_ n : \xi _ n \to \xi _{n + 1}$ induce cartesian squares

\[ \xymatrix{ V_{n + 1} \ar[d] & V_ n \ar[d] \ar[l] \\ \mathop{\mathrm{Spec}}(R/\mathfrak m^{n + 1}) & \mathop{\mathrm{Spec}}(R/\mathfrak m^ n) \ar[l] } \]

of algebraic spaces. By Spaces over Fields, Lemma 72.16.2 we can find a finite separable extension $k'/k$ and a point $v'_1 : \mathop{\mathrm{Spec}}(k') \to V_1$ over $k$. Let $R \subset R'$ be the finite ├ętale extension whose residue field extension is $k'/k$ (exists and is unique by Algebra, Lemmas 10.153.7 and 10.153.9). By the infinitesimal lifting criterion of smoothness (see More on Morphisms of Spaces, Lemma 76.19.6) applied to $V_ n \to \mathop{\mathrm{Spec}}(R/\mathfrak m^ n)$ for $n = 2, 3, 4, \ldots $ we can successively find morphisms $v'_ n : \mathop{\mathrm{Spec}}(R'/(\mathfrak m')^ n) \to V_ n$ over $\mathop{\mathrm{Spec}}(R/\mathfrak m^ n)$ fitting into commutative diagrams

\[ \xymatrix{ \mathop{\mathrm{Spec}}(R'/(\mathfrak m')^{n + 1}) \ar[d]_{v'_{n + 1}} & \mathop{\mathrm{Spec}}(R'/(\mathfrak m')^ n) \ar[d]^{v'_ n} \ar[l] \\ V_{n + 1} & V_ n \ar[l] } \]

Composing with the projection morphisms $V_ n \to U$ we obtain a compatible system of morphisms $u'_ n : \mathop{\mathrm{Spec}}(R'/(\mathfrak m')^ n) \to U$. By Case I the family $(u'_ n)$ comes from a unique morphism $u' : \mathop{\mathrm{Spec}}(R') \to U$. Denote $x'$ the object of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(R')$ we get by applying the $1$-morphism $f$ to $u'$. By construction, there exists a morphism of formal objects

\[ (07X6)(x') = (R', x'|_{\mathop{\mathrm{Spec}}(R'/(\mathfrak m')^ n)}, \ldots ) \longrightarrow (R, \xi _ n, f_ n) \]

lying over $\mathop{\mathrm{Spec}}(R') \to \mathop{\mathrm{Spec}}(R)$. Note that $R' \otimes _ R R'$ is a finite product of spectra of Noetherian complete local rings to which our current discussion applies. Denote $p_0, p_1 : \mathop{\mathrm{Spec}}(R' \otimes _ R R') \to \mathop{\mathrm{Spec}}(R')$ the two projections. By the fully faithfulness shown above there exists a canonical isomorphism $\varphi : p_0^*x' \to p_1^*x'$ because we have such isomorphisms over $\mathop{\mathrm{Spec}}((R' \otimes _ R R')/\mathfrak m^ n(R' \otimes _ R R'))$. We omit the proof that the isomorphism $\varphi $ satisfies the cocycle condition (see Stacks, Definition 8.3.1). Since $\{ \mathop{\mathrm{Spec}}(R') \to \mathop{\mathrm{Spec}}(R)\} $ is an fppf covering we conclude that $x'$ descends to an object $x$ of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(R)$. We omit the proof that $x_ n$ is the restriction of $x$ to $\mathop{\mathrm{Spec}}(R/\mathfrak m^ n)$. $\square$

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