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
an étale covering \{ f'_ k : W'_ k \to W'\} _{k \in K},
étale morphisms f''_ k : W'_ k \to W'', and
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
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