The Stacks project

Lemma 105.6.4. The category $p : \mathcal{C} \to W_{spaces, {\acute{e}tale}}$ constructed in Remark 105.6.1 is a stack in groupoids.

Proof. By Lemma 105.6.3 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 105.6.2) 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 64.10.5). 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 $U$. $\square$


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