The Stacks project

Lemma 105.6.8. In Remark 105.6.1 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 105.6.1 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 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

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

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 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
\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 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

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

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 78.5 and the desired result follows from More on Groupoids in Spaces, Lemma 78.5.2. $\square$

[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 75.8.1. 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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