Lemma 105.6.8. In Remark 105.6.1 assume $\mathcal{X} \subset \mathcal{X}'$ is a first order thickening. Then

**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 65.29.6).

We will describe the sheaf $\mathit{Aut}(\xi )$ using the same method as used in the proof of Lemma 105.6.6. Consider the algebraic space

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

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 105.6.3. 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 75.17.7 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 75.17.5 we get an isomorphism

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

the construction of (105.6.8.1) commutes with étale localization,

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

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 (105.6.8.1) constructed using $\xi $ over $U$, restricts to the map (105.6.8.1)

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 75.17.8.

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 75.7.16). On the other hand, $\mathcal{C}_{U/U'}$ is quasi-coherent (More on Morphisms of Spaces, Definition 75.5.1). By Properties of Spaces, Lemma 65.29.7 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 93.16.1 for example. Composition in $\mathit{Aut}(\xi )$ is induced by the morphism $c'$ as follows. Suppose we have two morphisms

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

We omit the detailed verification^{2}. Thus we are in the situation described in More on Groupoids in Spaces, Section 78.5 and the desired result follows from More on Groupoids in Spaces, Lemma 78.5.2.
$\square$

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