Proposition 96.14.3. 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.**
We will construct quasi-inverse functors

where $\mathit{QCoh}(U, R, s, t, c)$ denotes the category of quasi-coherent modules on the groupoid $(U, R, s, t, c)$.

Let $\mathcal{F}$ be an object of $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$. Denote $\mathcal{U}$, $\mathcal{R}$ the categories fibred in groupoids corresponding to $U$ and $R$. Denote $x$ the (defining) object of $\mathcal{X}$ over $U$. Recall that we have a $2$-commutative diagram

See Groupoids in Spaces, Lemma 78.20.3. By Lemma 96.3.3 the $2$-arrow inherent in the diagram induces an isomorphism $\alpha : t^*x^*\mathcal{F} \to s^*x^*\mathcal{F}$ which satisfies the cocycle condition over $\mathcal{R} \times _{s, \mathcal{U}, t} \mathcal{R}$; this is a consequence of Groupoids in Spaces, Lemma 78.23.1. Thus if we set $\mathcal{G} = x^*\mathcal{F}|_{U_{\acute{e}tale}}$ then the equivalence of categories in Lemma 96.14.2 (used several times compatibly with pullbacks) gives an isomorphism $\alpha : t_{small}^*\mathcal{G} \to s_{small}^*\mathcal{G}$ satisfying the cocycle condition on $R \times _{s, U, t} R$, i.e., $(\mathcal{G}, \alpha )$ is an object of $\mathit{QCoh}(U, R, s, t, c)$. The rule $\mathcal{F} \mapsto (\mathcal{G}, \alpha )$ is our functor from left to right.

Construction of the functor in the other direction. Let $(\mathcal{G}, \alpha )$ be an object of $\mathit{QCoh}(U, R, s, t, c)$. According to Lemma 96.13.2 the stackification map $[U/_{\! p}R] \to [U/R]$ (see Groupoids in Spaces, Definition 78.20.1) induces an equivalence of categories of quasi-coherent sheaves. Thus it suffices to construct a quasi-coherent module $\mathcal{F}$ on $[U/_{\! p}R]$.

Recall that an object $x = (T, u)$ of $[U/_{\! p}R]$ 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 $U$.

With this notation in place, given an object $(T, u)$ of $[U/_{\! p}R]$, we set

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

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). We see that the restriction of $\mathcal{F}$ to $(\mathit{Sch}/T)_{fppf}$ is quasi-coherent by the simple description of the restriction maps of $\mathcal{F}$ in case of a special morphism. Hence $\mathcal{F}$ is a sheaf on $[U/_{\! p}R]$ and quasi-coherent (Lemma 96.11.3).

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

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (1)

Comment #8800 by ZL on

There are also: