Proposition 94.13.1. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Let $\mathcal{X} = [U/R]$ be the quotient stack. The category of quasi-coherent modules on $\mathcal{X}$ is equivalent to the category of quasi-coherent modules on $(U, R, s, t, c)$.

Proof. Denote $\mathit{QCoh}(U, R, s, t, c)$ the category of quasi-coherent modules on the groupoid $(U, R, s, t, c)$. We will construct quasi-inverse functors

$\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \longleftrightarrow \mathit{QCoh}(U, R, s, t, c).$

According to Lemma 94.12.2 the stackification map $[U/_{\! p}R] \to [U/R]$ (see Groupoids in Spaces, Definition 76.19.1) induces an equivalence of categories of quasi-coherent sheaves. Thus it suffices to prove the lemma with $\mathcal{X} = [U/_{\! p}R]$.

Recall that an object $x = (T, u)$ of $\mathcal{X} = [U/_{\! p}R]$ is given by a scheme $T$ and a morphism $u : T \to U$. A morphism $(T, u) \to (T', u')$ is given by a pair $(f, r)$ where $f : T \to T'$ and $r : T \to R$ with $s \circ r = u$ and $t \circ r = u' \circ f$. Let us call a special morphism any morphism of the form $(f, e \circ u' \circ f) : (T, u' \circ f) \to (T', u')$. The category of $(T, u)$ with special morphisms is just the category of schemes over $U$.

Let $\mathcal{F}$ be a quasi-coherent sheaf on $\mathcal{X}$. Then we obtain for every $x = (T, u)$ a quasi-coherent sheaf $\mathcal{F}_{(T, u)} = x^*\mathcal{F}|_{T_{\acute{e}tale}}$ on $T$. Moreover, for any morphism $(f, r) : x = (T, u) \to (T', u') = x'$ we obtain a comparison isomorphism

$c_{(f, r)} : f_{small}^*\mathcal{F}_{(T', u')} \longrightarrow \mathcal{F}_{(T, u)}$

see Lemma 94.11.6. Moreover, these isomorphisms are compatible with compositions, see Lemma 94.9.3. If $U$, $R$ are schemes, then we can construct the quasi-coherent sheaf on the groupoid as follows: First the object $(U, \text{id})$ corresponds to a quasi-coherent sheaf $\mathcal{F}_{(U, \text{id})}$ on $U$. Next, the isomorphism $\alpha : t_{small}^*\mathcal{F}_{(U, \text{id})} \to s_{small}^*\mathcal{F}_{(U, \text{id})}$ comes from

1. the morphism $(R, \text{id}_ R) : (R, s) \to (R, t)$ in the category $[U/_{\! p}R]$ which produces an isomorphism $\mathcal{F}_{(R, t)} \to \mathcal{F}_{(R, s)}$,

2. the special morphism $(R, s) \to (U, \text{id})$ which produces an isomorphism $s_{small}^*\mathcal{F}_{(U, \text{id})} \to \mathcal{F}_{(R, s)}$, and

3. the special morphism $(R, t) \to (U, \text{id})$ which produces an isomorphism $t_{small}^*\mathcal{F}_{(U, \text{id})} \to \mathcal{F}_{(R, t)}$.

The cocycle condition for $\alpha$ follows from the condition that $(U, R, s, t, c)$ is groupoid, i.e., that composition is associative (details omitted).

To do this in general, i.e., when $U$ and $R$ are algebraic spaces, it suffices to explain how to associate to an algebraic space $(W, u)$ over $U$ a quasi-coherent sheaf $\mathcal{F}_{(W, u)}$ and to construct the comparison maps for morphisms between these. We set $\mathcal{F}_{(W, u)} = x^*\mathcal{F}|_{W_{\acute{e}tale}}$ where $x$ is the $1$-morphism $\mathcal{S}_ W \to \mathcal{S}_ U \to [U/_{\! p}R]$ and the comparison maps are explained in (94.10.2.3).

Conversely, suppose that $(\mathcal{G}, \alpha )$ is a quasi-coherent module on $(U, R, s, t, c)$. We are going to define a presheaf of modules $\mathcal{F}$ on $\mathcal{X}$ as follows. Given an object $(T, u)$ of $[U/_{\! p}R]$ we set

$\mathcal{F}(T, u) : = \Gamma (T, u_{small}^*\mathcal{G}).$

Given a morphism $(f, r) : (T, u) \to (T', u')$ we get a map

\begin{align*} \mathcal{F}(T', u') & = \Gamma (T', (u')_{small}^*\mathcal{G}) \\ & \to \Gamma (T, f_{small}^*(u')_{small}^*\mathcal{G}) = \Gamma (T, (u' \circ f)_{small}^*\mathcal{G}) \\ & = \Gamma (T, (t \circ r)_{small}^*\mathcal{G}) = \Gamma (T, r_{small}^*t_{small}^*\mathcal{G}) \\ & \to \Gamma (T, r_{small}^*s_{small}^*\mathcal{G}) = \Gamma (T, (s \circ r)_{small}^*\mathcal{G}) \\ & = \Gamma (T, u_{small}^*\mathcal{G}) \\ & = \mathcal{F}(T, u) \end{align*}

where the first arrow is pullback along $f$ and the second arrow is $\alpha$. Note that if $(T, r)$ is a special morphism, then this map is just pullback along $f$ as $e_{small}^*\alpha = \text{id}$ by the axioms of a sheaf of quasi-coherent modules on a groupoid. The cocycle condition implies that $\mathcal{F}$ is a presheaf of modules (details omitted). It is immediate from the definition that $\mathcal{F}$ is quasi-coherent when pulled back to $(\mathit{Sch}/T)_{fppf}$ (by the simple description of the restriction maps of $\mathcal{F}$ in case of a special morphism).

We omit the verification that the functors constructed above are quasi-inverse to each other. $\square$

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