The Stacks project

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

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

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

\[ \xymatrix{ \mathcal{R} \ar[r]_ s \ar[d]_ t & \mathcal{U} \ar[d]^ x \\ \mathcal{U} \ar[r]^ x & \mathcal{X} } \]

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

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

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). 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. $\square$

Comments (1)

Comment #8800 by ZL on

Typo: the th line from the below "Note that if is a special ..." should be "Note that if is a special ..."

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