The Stacks project

106.6 Lifting affines

Consider a solid diagram

\[ \xymatrix{ W \ar[d] \ar@{..>}[r] & W' \ar@{..>}[d] \\ \mathcal{X} \ar[r] & \mathcal{X}' } \]

where $\mathcal{X} \subset \mathcal{X}'$ is a thickening of algebraic stacks, $W$ is an affine scheme and $W \to \mathcal{X}$ is smooth. The question we address in this section is whether we can find $W'$ and the dotted arrows so that the square is cartesian and $W' \to \mathcal{X}'$ is smooth. We do not know the answer in general, but if $\mathcal{X} \subset \mathcal{X}'$ is a first order thickening we will prove the answer is yes.

To study this problem we introduce the following category.

Remark 106.6.1 (Category of lifts). Consider a diagram

\[ \xymatrix{ W \ar[d]_ x \\ \mathcal{X} \ar[r] & \mathcal{X}' } \]

where $\mathcal{X} \subset \mathcal{X}'$ is a thickening of algebraic stacks, $W$ is an algebraic space, and $W \to \mathcal{X}$ is smooth. We will construct a category $\mathcal{C}$ and a functor

\[ p : \mathcal{C} \longrightarrow W_{spaces, {\acute{e}tale}} \]

(see Properties of Spaces, Definition 66.18.2 for notation) as follows. An object of $\mathcal{C}$ will be a system $(U, U', a, i, x', \alpha )$ which forms a commutative diagram
\begin{equation} \label{stacks-more-morphisms-equation-object} \vcenter { \xymatrix{ U \ar[d]_ a \ar[r]_ i & U' \ar[dd]^{x'} \\ W \ar[d]_ x & \\ \mathcal{X} \ar[r] & \mathcal{X}' } } \end{equation}

with commutativity witnessed by the $2$-morphism $\alpha : x \circ a \to x' \circ i$ such that $U$ and $U'$ are algebraic spaces, $a : U \to W$ is étale, $x' : U' \to \mathcal{X}'$ is smooth, and such that $U = \mathcal{X} \times _{\mathcal{X}'} U'$. In particular $U \subset U'$ is a thickening. A morphism

\[ (U, U', a, i, x', \alpha ) \to (V, V', b, j, y', \beta ) \]

is given by $(f, f', \gamma )$ where $f : U \to V$ is a morphism over $W$, $f' : U' \to V'$ is a morphism whose restriction to $U$ gives $f$, and $\gamma : x' \circ f' \to y'$ is a $2$-morphism witnessing the commutativity in right triangle of the diagram below
\begin{equation} \label{stacks-more-morphisms-equation-morphism} \vcenter { \xymatrix{ & V \ar[ld]_ f \ar[ldd]^ b \ar[rr]_ j & & V' \ar[ld]_{f'} \ar[lddd]^{y'} \\ U \ar[d]_ a \ar[rr]_ i & & U' \ar[dd]_{x'} \\ W \ar[d]_ x & \\ \mathcal{X} \ar[rr] & & \mathcal{X}' } } \end{equation}

Finally, we require that $\gamma $ is compatible with $\alpha $ and $\beta $: in the calculus of $2$-categories of Categories, Sections 4.28 and 4.29 this reads

\[ \beta = (\gamma \star \text{id}_ j) \circ (\alpha \star \text{id}_ f) \]

(more succinctly: $\beta = j^*\gamma \circ f^*\alpha $). Another formulation is that objects are commutative diagrams ( with some additional properties and morphisms are commutative diagrams ( in the category $\textit{Spaces}/\mathcal{X}'$ introduced in Properties of Stacks, Remark 100.3.7. This makes it clear that $\mathcal{C}$ is a category and that the rule $p : \mathcal{C} \to W_{spaces, {\acute{e}tale}}$ sending $(U, U', a, i, x', \alpha )$ to $a : U \to W$ is a functor.

Proof. Namely $f : V \to U$ is étale as a morphism in $W_{spaces, {\acute{e}tale}}$ and we can apply Lemma 106.5.2 because $U' \to \mathcal{X}'$ and $V' \to \mathcal{X}'$ are smooth and $U = \mathcal{X} \times _{\mathcal{X}'} U'$ and $V = \mathcal{X} \times _{\mathcal{X}'} V'$. $\square$

Lemma 106.6.3. The category $p : \mathcal{C} \to W_{spaces, {\acute{e}tale}}$ constructed in Remark 106.6.1 is fibred in groupoids.

Proof. We claim the fibre categories of $p$ are groupoids. If $(f, f', \gamma ')$ as in ( is a morphism such that $f : U \to V$ is an isomorphism, then $f'$ is an isomorphism by Lemma 106.5.2 and hence $(f, f', \gamma ')$ is an isomorphism.

Consider a morphism $f : V \to U$ in $W_{spaces, {\acute{e}tale}}$ and an object $\xi = (U, U', a, i, x', \alpha )$ of $\mathcal{C}$ over $U$. We are going to construct the “pullback” $f^*\xi $ over $V$. Namely, set $b = a \circ f$. Let $f' : V' \to U'$ be the étale morphism whose restriction to $V$ is $f$ (More on Morphisms of Spaces, Lemma 76.8.2). Denote $j : V \to V'$ the corresponding thickening. Let $y' = x' \circ f'$ and $\gamma = \text{id} : x' \circ f' \to y'$. Set

\[ \beta = \alpha \star \text{id}_ f : x \circ b = x \circ a \circ f \to x' \circ i \circ f = x' \circ f' \circ j = y' \circ j \]

It is clear that $(f, f', \gamma ) : (V, V', b, j, y', \beta ) \to (U, U', a, i, x', \alpha )$ is a morphism as in ( The morphisms $(f, f', \gamma )$ so constructed are strongly cartesian (Categories, Definition 4.33.1). We omit the detailed proof, but essentially the reason is that given a morphism $(g, g', \epsilon ) : (Y, Y', c, k, z', \delta ) \to (U, U', a, i, x', \alpha )$ in $\mathcal{C}$ such that $g$ factors as $g = f \circ h$ for some $h : Y \to V$, then we get a unique factorization $g' = f' \circ h'$ from More on Morphisms of Spaces, Lemma 76.8.2 and after that one can produce the necessary $\zeta $ such that $(h, h', \zeta ) : (Y, Y', c, k, z', \delta ) \to (V, V', b, j, y', \beta )$ is a morphism of $\mathcal{C}$ with $(g, g', \epsilon ) = (f, f', \gamma ) \circ (h, h', \zeta )$.

Therefore $p : \mathcal{C} \to W_{\acute{e}tale}$ is a fibred category (Categories, Definition 4.33.5). Combined with the fact that the fibre categories are groupoids seen above we conclude that $p : \mathcal{C} \to W_{\acute{e}tale}$ is fibred in groupoids by Categories, Lemma 4.35.2. $\square$

Lemma 106.6.4. The category $p : \mathcal{C} \to W_{spaces, {\acute{e}tale}}$ constructed in Remark 106.6.1 is a stack in groupoids.

Proof. By Lemma 106.6.3 we see the first condition of Stacks, Definition 8.5.1 holds. As is customary we check descent of objects and we leave it to the reader to check descent of morphisms. Thus suppose we have $a : U \to W$ in $W_{spaces, {\acute{e}tale}}$, a covering $\{ U_ k \to U\} _{k \in K}$ in $W_{spaces, {\acute{e}tale}}$, objects $\xi _ k = (U_ k, U'_ k, a_ k, i_ k, x'_ k, \alpha _ k)$ of $\mathcal{C}$ over $U_ k$, and morphisms

\[ \varphi _{kk'} = (f_{kk'}, f'_{kk'}, \gamma _{kk'}) : \xi _ k|_{U_ k \times _ U U_{k'}} \to \xi _{k'}|_{U_ k \times _ U U_{k'}} \]

between restrictions satisfying the cocycle condition. In order to prove effectivity we may first refine the covering. Hence we may assume each $U_ k$ is a scheme (even an affine scheme if you like). Let us write

\[ \xi _ k|_{U_ k \times _ U U_{k'}} = (U_ k \times _ U U_{k'}, U'_{kk'}, a_{kk'}, x'_{kk'}, \alpha _{kk'}) \]

Then we get an étale (by Lemma 106.6.2) morphism $s_{kk'} : U'_{kk'} \to U'_ k$ as the second component of the morphism $\xi _ k|_{U_ k \times _ U U_{k'}} \to \xi _ k$ of $\mathcal{C}$. Similarly we obtain an étale morphism $t_{kk'} : U'_{kk'} \to U'_{k'}$ by looking at the second component of the composition

\[ \xi _ k|_{U_ k \times _ U U_{k'}} \xrightarrow {\varphi _{kk'}} \xi _{k'}|_{U_ k \times _ U U_{k'}} \to \xi _{k'} \]

We claim that

\[ j : \coprod \nolimits _{(k, k') \in K \times K} U'_{kk'} \xrightarrow {(\coprod s_{kk'}, \coprod t_{kk'})} (\coprod \nolimits _{k \in K} U'_ k) \times (\coprod \nolimits _{k \in K} U'_ k) \]

is an étale equivalence relation. First, we have already seen that the components $s, t$ of the displayed morphism are étale. The base change of the morphism $j$ by $(\coprod U_ k) \times (\coprod U_ k) \to (\coprod U'_ k) \times (\coprod U'_ k)$ is a monomorphism because it is the map

\[ \coprod \nolimits _{(k, k') \in K \times K} U_ k \times _ U U_{k'} \longrightarrow (\coprod \nolimits _{k \in K} U_ k) \times (\coprod \nolimits _{k \in K} U_ k) \]

Hence $j$ is a monomorphism by More on Morphisms, Lemma 37.3.4. Finally, symmetry of the relation $j$ comes from the fact that $\varphi _{kk'}^{-1}$ is the “flip” of $\varphi _{k'k}$ (see Stacks, Remarks 8.3.2) and transitivity comes from the cocycle condition (details omitted). Thus the quotient of $\coprod U'_ k$ by $j$ is an algebraic space $U'$ (Spaces, Theorem 65.10.5). Above we have already shown that there is a thickening $i : U \to U'$ as we saw that the restriction of $j$ on $\coprod U_ k$ gives $(\coprod U_ k) \times _ U (\coprod U_ k)$. Finally, if we temporarily view the $1$-morphisms $x'_ k : U'_ k \to \mathcal{X}'$ as objects of the stack $\mathcal{X}'$ over $U'_ k$ then we see that these come endowed with a descent datum with respect to the étale covering $\{ U'_ k \to U'\} $ given by the third component $\gamma _{kk'}$ of the morphisms $\varphi _{kk'}$ in $\mathcal{C}$. Since $\mathcal{X}'$ is a stack this descent datum is effective and translating back we obtain a $1$-morphism $x' : U' \to \mathcal{X}'$ such that the compositions $U'_ k \to U' \to \mathcal{X}'$ come equipped with isomorphisms to $x'_ k$ compatible with $\gamma _{kk'}$. This means that the morphisms $\alpha _ k : x \circ a_ k \to x'_ k \circ i_ k$ glue to a morphism $\alpha : x \circ a \to x' \circ i$. Then $\xi = (U, U', a, i, x', \alpha )$ is the desired object over $U$. $\square$

Lemma 106.6.5. Let $\mathcal{X} \subset \mathcal{X}'$ be a thickening of algebraic stacks. Let $W$ be an algebraic space and let $W \to \mathcal{X}$ be a smooth morphism. There exists an étale covering $\{ W_ i \to W\} _{i \in I}$ and for each $i$ a cartesian diagram

\[ \xymatrix{ W_ i \ar[r] \ar[d] & W_ i' \ar[d] \\ \mathcal{X} \ar[r] & \mathcal{X}' } \]

with $W_ i' \to \mathcal{X}'$ smooth.

Proof. Choose a scheme $U'$ and a surjective smooth morphism $U' \to \mathcal{X}'$. As usual we set $U = \mathcal{X} \times _{\mathcal{X}'} U'$. Then $U \to \mathcal{X}$ is a surjective smooth morphism. Therefore the base change

\[ V = W \times _{\mathcal{X}} U \longrightarrow W \]

is a surjective smooth morphism of algebraic spaces. By Topologies on Spaces, Lemma 73.4.4 we can find an étale covering $\{ W_ i \to W\} $ such that $W_ i \to W$ factors through $V \to W$. After covering $W_ i$ by affines (Properties of Spaces, Lemma 66.6.1) we may assume each $W_ i$ is affine. We may and do replace $W$ by $W_ i$ which reduces us to the situation discussed in the next paragraph.

Assume $W$ is affine and the given morphism $W \to \mathcal{X}$ factors through $U$. Picture

\[ W \xrightarrow {i} U \to \mathcal{X} \]

Since $W$ and $U$ are smooth over $\mathcal{X}$ we see that $i$ is locally of finite type (Morphisms of Stacks, Lemma 101.17.8). After replacing $U$ by $\mathbf{A}^ n_ U$ we may assume that $i$ is an immersion, see Morphisms, Lemma 29.39.2. By Morphisms of Stacks, Lemma 101.44.4 the morphism $i$ is a local complete intersection. Hence $i$ is a Koszul-regular immersion (as defined in Divisors, Definition 31.21.1) by More on Morphisms, Lemma 37.62.3.

We may still replace $W$ by an affine open covering. For every point $w \in W$ we can choose an affine open $U'_ w \subset U'$ such that if $U_ w \subset U$ is the corresponding affine open, then $w \in i^{-1}(U_ w)$ and $i^{-1}(U_ w) \to U_ w$ is a closed immersion cut out by a Koszul-regular sequence $f_1, \ldots , f_ r \in \Gamma (U_ w, \mathcal{O}_{U_ w})$. This follows from the definition of Koszul-regular immersions and Divisors, Lemma 31.20.7. Set $W_ w = i^{-1}(U_ w)$; this is an affine open neighbourhood of $w \in W$. Choose lifts $f'_1, \ldots , f'_ r \in \Gamma (U'_ w, \mathcal{O}_{U'_ w})$ of $f_1, \ldots , f_ r$. This is possible as $U_ w \to U'_ w$ is a closed immersion of affine schemes. Let $W'_ w \subset U'_ w$ be the closed subscheme cut out by $f'_1, \ldots , f'_ r$. We claim that $W'_ w \to \mathcal{X}'$ is smooth. The claim finishes the proof as $W_ w = \mathcal{X} \times _{\mathcal{X}'} W'_ w$ by construction.

To check the claim it suffices to check that the base change $W'_ w \times _{\mathcal{X}'} X' \to X'$ is smooth for every affine scheme $X'$ smooth over $\mathcal{X}'$. Choose an étale morphism

\[ Y' \to U'_ w \times _{\mathcal{X}'} X' \]

with $Y'$ affine. Because $U'_ w \times _{\mathcal{X}'} X'$ is covered by the images of such morphisms, it is enough to show that the closed subscheme $Z'$ of $Y'$ cut out by $f'_1, \ldots , f'_ r$ is smooth over $X'$. Picture

\[ \xymatrix{ Z' \ar[r] \ar[d] & Y' \ar[d] \\ W'_ w \times _{\mathcal{X}'} X' \ar[d] \ar[r] & U'_ w \times _{\mathcal{X}'} X' \ar[d] \ar[r] & X' \\ W'_ w = V(f'_1, \ldots , f'_ r) \ar[r] & U'_ w } \]

Set $X = \mathcal{X} \times _{\mathcal{X}'} X'$, $Y = X \times _{X'} Y' = \mathcal{X} \times _{\mathcal{X}'} Y'$, and $Z = Y \times _{Y'} Z' = X \times _{X'} Z' = \mathcal{X} \times _{\mathcal{X}'} Z'$. Then $(Z \subset Z') \to (Y \subset Y') \subset (X \subset X')$ are (cartesian) morphisms of thickenings of affine schemes and we are given that $Z \to X$ and $Y' \to X'$ are smooth. Finally, the sequence of functions $f'_1, \ldots , f'_ r$ map to a Koszul-regular sequence in $\Gamma (Y', \mathcal{O}_{Y'})$ by More on Algebra, Lemma 15.30.5 because $Y' \to U'_ w$ is smooth and hence flat. By More on Algebra, Lemma 15.31.6 (and the fact that Koszul-regular sequences are quasi-regular sequences by More on Algebra, Lemmas 15.30.2, 15.30.3, and 15.30.6) we conclude that $Z' \to X'$ is smooth as desired. $\square$

Lemma 106.6.6. Let $\mathcal{X} \subset \mathcal{X}'$ be a thickening of algebraic stacks. Consider a commutative diagram

\[ \xymatrix{ W'' \ar[d]_{x''} & W \ar[l] \ar[r] \ar[d]_ x & W' \ar[d]^{x'} \\ \mathcal{X}' & \mathcal{X} \ar[l] \ar[r] & \mathcal{X}' } \]

with cartesian squares where $W', W, W''$ are algebraic spaces and the vertical arrows are smooth. Then there exist

  1. an étale covering $\{ f'_ k : W'_ k \to W'\} _{k \in K}$,

  2. étale morphisms $f''_ k : W'_ k \to W''$, and

  3. $2$-morphisms $\gamma _ k : x'' \circ f''_ k \to x' \circ f'_ k$

such that (a) $(f'_ k)^{-1}(W) = (f''_ k)^{-1}(W)$, (b) $f'_ k|_{(f'_ k)^{-1}(W)} = f''_ k|_{(f''_ k)^{-1}(W)}$, and (c) pulling back $\gamma _ k$ to the closed subscheme of (a) agrees with the $2$-morphism given by the commutativity of the initial diagram over $W$.

Proof. Denote $i : W \to W'$ and $i'' : W \to W''$ the given thickenings. The commutativity of the diagram in the statement of the lemma means there is a $2$-morphism $\delta : x' \circ i' \to x'' \circ i''$ This is the $2$-morphism referred to in part (c) of the statement. Consider the algebraic space

\[ I' = W' \times _{x', \mathcal{X}', x''} W'' \]

with projections $p' : I' \to W'$ and $q' : I' \to W''$. Observe that there is a “universal” $2$-morphism $\gamma : x' \circ p' \to x'' \circ q'$ (we will use this later). The choice of $\delta $ defines a morphism

\[ \xymatrix{ W \ar[rr]_\delta & & I' \ar[ld]^{p'} \ar[rd]_{q'} \\ & W' & & W'' } \]

such that the compositions $W \to I' \to W'$ and $W \to I' \to W''$ are $i : W \to W'$ and $i' : W \to W''$. Since $x''$ is smooth, the morphism $p' : I' \to W'$ is smooth as a base change of $x''$.

Suppose we can find an étale covering $\{ f'_ k : W'_ k \to W'\} $ and morphisms $\delta _ k : W'_ k \to I'$ such that the restriction of $\delta _ k$ to $W_ k = (f'_ k)^{-1}$ is equal to $\delta \circ f_ k$ where $f_ k = f'_ k|_{W_ k}$. Picture

\[ \xymatrix{ W_ k \ar[r]^{f_ k} \ar[d] & W \ar[r]^\delta & I' \ar[d]^{p'} \\ W'_ k \ar[rr]^{f'_ k} \ar[rru]^{\delta _ k} & & W' } \]

In other words, we want to be able to extend the given section $\delta : W \to I'$ of $p'$ to a section over $W'$ after possibly replacing $W'$ by an étale covering.

If this is true, then we can set $f''_ k = q' \circ \delta _ k$ and $\gamma _ k = \gamma \star \text{id}_{\delta _ k}$ (more succinctly $\gamma _ k = \delta _ k^*\gamma $). Namely, the only thing left to show at this is that the morphism $f''_ k$ is étale. By construction the morphism $x' \circ p'$ is $2$-isomorphic to $x'' \circ q'$. Hence $x'' \circ f''_ k$ is $2$-isomorphic to $x' \circ f'_ k$. We conclude that the composition

\[ W'_ k \xrightarrow {f''_ k} W'' \xrightarrow {x''} \mathcal{X}' \]

is smooth because $x' \circ f'_ k$ is so. As $f_ k$ is étale we conclude $f''_ k$ is étale by Lemma 106.5.2.

If the thickening is a first order thickening, then we can choose any étale covering $\{ W'_ k \to W'\} $ with $W_ k'$ affine. Namely, since $p'$ is smooth we see that $p'$ is formally smooth by the infinitesimal lifting criterion (More on Morphisms of Spaces, Lemma 76.19.6). As $W_ k$ is affine and as $W_ k \to W'_ k$ is a first order thickening (as a base change of $\mathcal{X} \to \mathcal{X}'$, see Lemma 106.3.4) we get $\delta _ k$ as desired.

In the general case the existence of the covering and the morphisms $\delta _ k$ follows from More on Morphisms of Spaces, Lemma 76.19.7. $\square$

Lemma 106.6.7. The category $p : \mathcal{C} \to W_{spaces, {\acute{e}tale}}$ constructed in Remark 106.6.1 is a gerbe.

Proof. In Lemma 106.6.4 we have seen that it is a stack in groupoids. Thus it remains to check conditions (2) and (3) of Stacks, Definition 8.11.1. Condition (2) follows from Lemma 106.6.5. Condition (3) follows from Lemma 106.6.6. $\square$

Lemma 106.6.8. In Remark 106.6.1 assume $\mathcal{X} \subset \mathcal{X}'$ is a first order thickening. Then

  1. the automorphism sheaves of objects of the gerbe $p : \mathcal{C} \to W_{spaces, {\acute{e}tale}}$ constructed in Remark 106.6.1 are abelian, and

  2. the sheaf of groups $\mathcal{G}$ constructed in Stacks, Lemma 8.11.8 is a quasi-coherent $\mathcal{O}_ W$-module.

Proof. We will prove both statements at the same time. Namely, given an object $\xi = (U, U', a, i, x', \alpha )$ we will endow $\mathit{Aut}(\xi )$ with the structure of a quasi-coherent $\mathcal{O}_ U$-module on $U_{spaces, {\acute{e}tale}}$ and we will show that this structure is compatible with pullbacks. This will be sufficient by glueing of sheaves (Sites, Section 7.26) and the construction of $\mathcal{G}$ in the proof of Stacks, Lemma 8.11.8 as the glueing of the automorphism sheaves $\mathit{Aut}(\xi )$ and the fact that it suffices to check a module is quasi-coherent after going to an étale covering (Properties of Spaces, Lemma 66.29.6).

We will describe the sheaf $\mathit{Aut}(\xi )$ using the same method as used in the proof of Lemma 106.6.6. Consider the algebraic space

\[ I' = U' \times _{x', \mathcal{X}', x'} U' \]

with projections $p' : I' \to U'$ and $q' : I' \to U'$. Over $I'$ there is a universal $2$-morphism $\gamma : x' \circ p' \to x' \circ q'$. The identity $x' \to x'$ defines a diagonal morphism

\[ \xymatrix{ U' \ar[rr]_{\Delta '} & & I' \ar[ld]^{p'} \ar[rd]_{q'} \\ & U' & & U' } \]

such that the compositions $U' \to I' \to U'$ and $U' \to I' \to U'$ are the identity morphisms. We will denote the base change of $U', I', p', q', \Delta '$ to $\mathcal{X}$ by $U, I, p, q, \Delta $. Since $W' \to \mathcal{X}'$ is smooth, we see that $p' : I' \to U'$ is smooth as a base change.

A section of $\mathit{Aut}(\xi )$ over $U$ is a morphism $\delta ' : U' \to I'$ such that $\delta '|_ U = \Delta $ and such that $p' \circ \delta ' = \text{id}_{U'}$. To be explicit, $(\text{id}_ U, q' \circ \delta ', (\delta ')^*\gamma ) : \xi \to \xi $ is a formula for the corresponding automorphism. More generally, if $f : V \to U$ is an étale morphism, then there is a thickening $j : V \to V'$ and an étale morphism $f' : V' \to U'$ whose restriction to $V$ is $f$ and $f^*\xi $ corresponds to $(V, V', a \circ f, j, x' \circ f', f^*\alpha )$, see proof of Lemma 106.6.3. a section of $\mathit{Aut}(\xi )$ over $V$ is a morphism $\delta ' : V' \to I'$ such that $\delta '|_ V = \Delta \circ f$ and $p' \circ \delta ' = f'$1.

We conclude that $\mathit{Aut}(\xi )$ as a sheaf of sets agrees with the sheaf defined in More on Morphisms of Spaces, Remark 76.17.7 for the thickenings $(U \subset U')$ and $(I \subset I')$ over $(U \subset U')$ via $\text{id}_{U'}$ and $p'$. The diagonal $\Delta '$ is a section of this sheaf and by acting on this section using More on Morphisms of Spaces, Lemma 76.17.5 we get an isomorphism
\begin{equation} \label{stacks-more-morphisms-equation-isomorphism} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\Delta ^*\Omega _{I/U}, \mathcal{C}_{U/U'}) \longrightarrow \mathit{Aut}(\xi ) \end{equation}

on $U_{spaces, {\acute{e}tale}}$. There three things left to check

  1. the construction of ( commutes with étale localization,

  2. $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\Delta ^*\Omega _{I/U}, \mathcal{C}_{U/U'})$ is a quasi-coherent module on $U$,

  3. the composition in $\mathit{Aut}(\xi )$ corresponds to addition of sections in this quasi-coherent module.

We will check these in order.

To see (1) we have to show that if $f : V \to U$ is étale, then ( constructed using $\xi $ over $U$, restricts to the map (

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ V}( \Delta _ V^*\Omega _{V \times _\mathcal {X} V/V}, \mathcal{C}_{V/V'}) \to \mathit{Aut}(\xi |_ V) \]

constructed using $\xi |_ V$ over $V$ on $V_{spaces, {\acute{e}tale}}$. This follows from the discussion in the footnote above and More on Morphisms of Spaces, Lemma 76.17.8.

Proof of (2). Since $p'$ is smooth, the morphism $I \to U$ is smooth, and hence the relative module of differentials $\Omega _{I/U}$ is finite locally free (More on Morphisms of Spaces, Lemma 76.7.16). On the other hand, $\mathcal{C}_{U/U'}$ is quasi-coherent (More on Morphisms of Spaces, Definition 76.5.1). By Properties of Spaces, Lemma 66.29.7 we conclude.

Proof of (3). There exists a morphism $c' : I' \times _{p', U', q'} I' \to I'$ such that $(U', I', p', q', c')$ is a groupoid in algebraic spaces with identity $\Delta '$. See Algebraic Stacks, Lemma 94.16.1 for example. Composition in $\mathit{Aut}(\xi )$ is induced by the morphism $c'$ as follows. Suppose we have two morphisms

\[ \delta '_1, \delta '_2 : U' \longrightarrow I' \]

corresponding to sections of $\mathit{Aut}(\xi )$ over $U$ as above, in other words, we have $\delta '_ i|U = \Delta _ U$ and $p' \circ \delta '_ i = \text{id}_{U'}$. Then the composition in $\mathit{Aut}(\xi )$ is

\[ \delta '_1 \circ \delta '_2 = c'(\delta '_1 \circ q' \circ \delta '_2, \delta '_2) \]

We omit the detailed verification2. Thus we are in the situation described in More on Groupoids in Spaces, Section 79.5 and the desired result follows from More on Groupoids in Spaces, Lemma 79.5.2. $\square$


Proposition 106.6.9 (Emerton). Let $\mathcal{X} \subset \mathcal{X}'$ be a first order thickening of algebraic stacks. Let $W$ be an affine scheme and let $W \to \mathcal{X}$ be a smooth morphism. Then there exists a cartesian diagram

\[ \xymatrix{ W \ar[d] \ar[r] & W' \ar[d] \\ \mathcal{X} \ar[r] & \mathcal{X}' } \]

with $W' \to \mathcal{X}'$ smooth and $W'$ affine.

Proof. Consider the category $p : \mathcal{C} \to W_{spaces, {\acute{e}tale}}$ introduced in Remark 106.6.1. The proposition states that there exists an object of $\mathcal{C}$ lying over $W$. Namely, if we have such an object $(W, W', a, i, y', \alpha )$ then $W = \mathcal{X} \times _{\mathcal{X}'} W'$. Hence $W \to W'$ is a thickening of algebraic spaces so $W'$ is affine by More on Morphisms of Spaces, Lemma 76.9.5 and More on Morphisms, Lemma 37.2.3.

Lemma 106.6.7 tells us $\mathcal{C}$ is a gerbe over $W_{spaces, {\acute{e}tale}}$. This means we can étale locally find a solution and these local solutions are étale locally isomorphic; this part does not require the assumption that the thickening is first order. By Lemma 106.6.8 the automorphism sheaves of objects of our gerbe are abelian and fit together to form a quasi-coherent module $\mathcal{G}$ on $W_{spaces, {\acute{e}tale}}$. We will verify conditions (1) and (2) of Cohomology on Sites, Lemma 21.11.1 to conclude the existence of an object of $\mathcal{C}$ lying over $W$. Condition (1) is true: the étale coverings $\{ W_ i \to W\} $ with each $W_ i$ affine are cofinal in the collection of all coverings. For such a covering $W_ i$ and $W_ i \times _ W W_ j$ are affine and $H^1(W_ i, \mathcal{G})$ and $H^1(W_ i \times _ W W_ j, \mathcal{G})$ are zero: the cohomology of a quasi-coherent module over an affine algebraic space is zero for example by Cohomology of Spaces, Proposition 69.7.2. Finally, condition (2) is that $H^2(W, \mathcal{G}) = 0$ for our quasi-coherent sheaf $\mathcal{G}$ which again follows from Cohomology of Spaces, Proposition 69.7.2. This finishes the proof. $\square$

[1] A formula for the corresponding automorphism is $(\text{id}_ V, h', (\delta ')^*\gamma )$. Here $h' : V' \to V'$ is the unique (iso)morphism such that $h'|_ V = \text{id}_ V$ and such that
\[ \xymatrix{ V' \ar[r]_{h'} \ar[rd]_{q' \circ \delta '} & V' \ar[d]^{f'} \\ & U' } \]
commutes. Uniqueness and existence of $h'$ by topological invariance of the étale site, see More on Morphisms of Spaces, Theorem 76.8.1. The reader may feel we should instead look at morphisms $\delta '' : V' \to V' \times _{\mathcal{X}'} V'$ with $\delta '' \circ j = \Delta _{V'/\mathcal{X}'}$ and $\text{pr}_1 \circ \delta '' = \text{id}_{V'}$. This would be fine too: as $V' \times _{\mathcal{X}'} V' \to I'$ is étale, the same topological invariance tells us that sending $\delta ''$ to $\delta ' = (V' \times _{\mathcal{X}'} V' \to I') \circ \delta ''$ is a bijection between the two sets of morphisms.
[2] The reader can see immediately that it is necessary to precompose $\delta '_1$ by $q' \circ \delta '_2$ to get a well defined $U'$-valued point of the fibre product $I' \times _{p', U', q'} I'$.

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