The Stacks project

Proposition 3.3.6 of “Intro to Algebraic Stacks” by Anatoly Preygel.

Proposition 105.15.1. Let $\mathcal{X}$ be an algebraic stack with quasi-affine1 diagonal. Then $\mathcal{X}$ satisfies descent for fpqc coverings.

Proof. Our conventions are that $\mathcal{X}$ is a stack in groupoids $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ over the category of schemes over a base scheme $S$ endowed with the fppf topology. The statement means the following: given an fpqc covering $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ of schemes over $S$ the functor

\[ \mathcal{X}_ U \longrightarrow DD(\mathcal{U}) \]

is an equivalence. Here on the left we have the category of objects of $\mathcal{X}$ over $U$ and on the right we have the category of descent data in $\mathcal{X}$ relative to $\mathcal{U}$. See discussion in Stacks, Section 8.3.

Fully faithfulness. Suppose we have two objects $x, y$ of $\mathcal{X}$ over $U$. Then $I = \mathit{Isom}(x, y)$ is an algebraic space over $U$. Hence a collection of sections of $I$ over $U_ i$ whose restrictions to $U_ i \times _ U U_ j$ agree, come from a unique section over $U$ by the analogue of the proposition for algebraic spaces, see Properties of Spaces, Proposition 65.17.1. Thus our functor is fully faithful.

Essential surjectivity. Here we are given objects $x_ i$ over $U_ i$ and isomorphisms $\varphi _{ij} : \text{pr}_0^*x_ i \to \text{pr}_1^*x_ j$ over $U_ i \times _ U U_ j$ satisfying the cocyle condition over $U_ i \times _ U U_ j \times _ U U_ k$.

Let $W$ be an affine scheme and let $W \to \mathcal{X}$ be a morphism. For each $i$ we can form

\[ W_ i = U_ i \times _{x_ i, \mathcal{X}} W \]

The projection $W_ i \to U_ i$ is quasi-affine as the diagonal of $\mathcal{X}$ is quasi-affine. For each pair $i, j \in I$ the isomorphism $\varphi _{ij}$ induces an isomorphism

\[ W_ i \times _ U U_ j = (U_ i \times _ U U_ j) \times _{x_ i \circ \text{pr}_0, \mathcal{X}} W \to (U_ i \times _ U U_ j) \times _{x_ j \circ \text{pr}_1, \mathcal{X}} W = U_ i \times _ U W_ j \]

Moreover, these isomorphisms satisfy the cocycle condition over $U_ i \times _ U U_ j \times _ U U_ k$. In other words, these isomorphisms define a descent datum on the schemes $W_ i/U_ i$ relative to $\mathcal{U}$. By Descent, Lemma 35.37.1 we see that this descent datum is effective2. We conclude that there exists a quasi-affine morphism $W' \to U$ and a commutative diagram

\[ \xymatrix{ W' \ar[d] & W_ i \ar[l] \ar[d] \ar[r] & W \ar[d] \\ U & U_ i \ar[l] \ar[r]^{x_ i} & \mathcal{X} } \]

whose squares are cartesian. Since $\{ W_ i \to W'\} _{i \in I}$ is the base change of $\mathcal{U}$ by $W' \to U$ we conclude that it is an fpqc covering. Since $W$ satisfies the sheaf condition for fpqc coverings, we obtain a unique morphism $W' \to W$ such that $W_ i \to W' \to W$ is the given morphism $W_ i \to W$. In other words, we have the commutative diagrams

\[ \xymatrix{ W_ i \ar[d] \ar[r] & W' \ar[d] \ar[r] & W \ar[d] \\ U_ i \ar[r] \ar@/_1pc/[rr]_{x_ i} & U & \mathcal{X} } \]

compatible with the isomorphisms $\varphi _{ij}$ and whose square and rectangle are cartesian.

Choose a collection of affine schemes $W_\alpha $, $\alpha \in A$ and smooth morphisms $W_\alpha \to \mathcal{X}$ such that $\coprod W_\alpha \to \mathcal{X}$ is surjective. By the procedure of the preceding paragraph we produce a diagram

\[ \xymatrix{ W_{\alpha , i} \ar[d] \ar[r] & W_\alpha ' \ar[d] \ar[r] & W_\alpha \ar[d] \\ U_ i \ar[r] \ar@/_1pc/[rr]_{x_ i} & U & \mathcal{X} } \]

for each $\alpha $. Then the morphisms $W_\alpha ' \to U$ are smooth and jointly surjective.

Denote $x_\alpha $ the object of $\mathcal{X}$ over $W_\alpha '$ corresponding to $W_\alpha ' \to W_\alpha \to \mathcal{X}$. Since $\mathcal{X}$ is an fppf stack and since $\{ W_\alpha ' \to U\} $ is an fppf covering, it suffices to show that there are isomorphisms $\text{pr}_0^*x_\alpha \to \text{pr}_1^*x_\beta $ over $W_\alpha ' \times _ U W'_\beta $ satisfying the cocycle condition. However, after pulling back to $W_{\alpha , i}$ we do have such isomorphisms over $W_{\alpha , i} \times _{U_ i} W_{\beta , i} = U_ i \times _ U (W_\alpha ' \times _ U W'_\beta )$ since the pullback of $x_\alpha $ to $W_{\alpha , i}$ is isomorphic to the pullback of $x_ i$ to $W_{\alpha , i}$. Since $\{ U_ i \times _ U (W_\alpha ' \times _ U W'_\beta ) \to W_\alpha ' \times _ U W'_\beta \} _{i \in I}$ is an fpqc covering and by the aforementioned compatibility of the diagrams above with $\varphi _{ij}$ these isomorphisms descend to $W_\alpha ' \times _ U W'_\beta $ and the proof is complete. $\square$

[1] It suffices to assume ind-quasi-affine.
[2] Or use More on Groupoids, Lemma 40.15.3 in the case of ind-quasi-affine diagonal.

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