The Stacks project

Lemma 105.6.6. 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 105.5.2.

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 75.19.6). 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 105.3.4) 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 75.19.7. $\square$


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