The Stacks project

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. $\square$


Comments (0)


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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 06WT. Beware of the difference between the letter 'O' and the digit '0'.