The Stacks project

96.14 Quasi-coherent sheaves and presentations

Let us first match quasi-coherent sheaves with our previously defined notions for schemes and algebraic spaces.

Lemma 96.14.1. Let $S$ be a scheme. Let $\mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids wich is representable by an algebraic space $F$. If $\mathcal{F}$ is in $\textit{LQCoh}(\mathcal{O}_\mathcal {X})$ then the restriction $\mathcal{F}|_{F_{\acute{e}tale}}$ ( is quasi-coherent.

Proof. Let $U$ be a scheme étale over $F$. Then $\mathcal{F}|_{U_{\acute{e}tale}} = (\mathcal{F}|_{F_{\acute{e}tale}})|_{U_{\acute{e}tale}}$. This is clear but see also Remark 96.10.2. Thus the assertion follows from the definitions. $\square$

Lemma 96.14.2. Let $S$ be a scheme. Let $\mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids wich is representable by an algebraic space $F$. The functor ( defines an equivalence

\[ \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \mathit{QCoh}(\mathcal{O}_ F),\quad \mathcal{F} \longmapsto \mathcal{F}|_{F_{\acute{e}tale}} \]

with quasi-inverse given by $\mathcal{G} \mapsto \pi _ F^*\mathcal{G}$. This equivalence is compatible with pullback for morphisms between categories fibred in groupoids representable by algebraic spaces.

Proof. By Lemma 96.11.4 we may work with the étale topology. We will use the notation and results of Lemma 96.10.1 without further mention. Recall that the restriction functor $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) \to \textit{Mod}(F_{\acute{e}tale}, \mathcal{O}_ F)$, $\mathcal{F} \mapsto \mathcal{F}|_{F_{\acute{e}tale}}$ is given by $i_ F^*$. By Lemma 96.14.1 or by Modules on Sites, Lemma 18.23.4 we see that $\mathcal{F}|_{F_{\acute{e}tale}}$ is quasi-coherent if $\mathcal{F}$ is quasi-coherent. Hence we get a functor as indicated in the statement of the lemma and we get a functor $\pi _ F^*$ in the opposite direction. Since $\pi _ F \circ i_ F = \text{id}$ we see that $i_ F^*\pi _ F^*\mathcal{G} = \mathcal{G}$.

For $\mathcal{F}$ in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ there is a canonical map $\pi _ F^*(\mathcal{F}|_{F_{\acute{e}tale}}) \to \mathcal{F}$, namely the map adjoint to the identification $\mathcal{F}|_{F_{\acute{e}tale}} = \pi _{F, *}\mathcal{F}$. We will show that this map is an isomorphism if $\mathcal{F}$ is a quasi-coherent module on $\mathcal{X}$. Choose a scheme $U$ and a surjective étale morphism $U \to F$. Denote $x : U \to \mathcal{X}$ the corresponding object of $\mathcal{X}$ over $U$. It suffices to show that $\pi _ F^*(\mathcal{F}|_{F_{\acute{e}tale}}) \to \mathcal{F}$ is an isomorphism after restricting to $\mathcal{X}_{\acute{e}tale}/x = (\mathit{Sch}/U)_{\acute{e}tale}$. Since $U \to F$ is étale, it follows from Remark 96.10.2 that

\[ \pi _ F^*(\mathcal{F}|_{F_{\acute{e}tale}})|_{\mathcal{X}_{\acute{e}tale}/x} = \pi _ U^*(\mathcal{F}|_{U_{\acute{e}tale}}) \]

and that the restriction of the map $\pi _ F^*(\mathcal{F}|_{F_{\acute{e}tale}}) \to \mathcal{F}$ to $\mathcal{X}_{\acute{e}tale}/x = (\mathit{Sch}/U)_{\acute{e}tale}$ is equal to the corresponding map $\pi _ U^*(\mathcal{F}|_{U_{\acute{e}tale}}) \to \mathcal{F}|_{(\mathit{Sch}/U)_{\acute{e}tale}}$. Since we have seen the result is true for schemes in Descent, Section 35.81 we conclude.

Compatibility with pullbacks follows from the fact that the quasi-inverse is given by $\pi _ F^*$ and the commutative diagram of ringed topoi in Lemma 96.10.3. $\square$

In Groupoids in Spaces, Definition 78.12.1 we have the defined the notion of a quasi-coherent module on an arbitrary groupoid. The following (formal) proposition tells us that we can study quasi-coherent sheaves on quotient stacks in terms of quasi-coherent modules on presentations.

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. $\square$

We finish this section with a technical lemma on maps out of quasi-coherent sheaves. It is an analogue of Schemes, Lemma 26.7.1. We will see later (Criteria for Representability, Theorem 97.17.2) that the assumptions on the groupoid imply that $\mathcal{X}$ is an algebraic stack.

Lemma 96.14.4. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Assume $s, t$ are flat and locally of finite presentation. Let $\mathcal{X} = [U/R]$ be the quotient stack. Denote $x$ the object of $\mathcal{X}$ over $U$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_\mathcal {X}$-module, and let $\mathcal{H}$ be any object of $\textit{Mod}(\mathcal{O}_\mathcal {X})$. The map

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{H}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(x^*\mathcal{F}|_{U_{\acute{e}tale}}, x^*\mathcal{H}|_{U_{\acute{e}tale}}), \quad \phi \longmapsto x^*\phi |_{U_{\acute{e}tale}} \]

is injective and its image consists of exactly those $\varphi : x^*\mathcal{F}|_{U_{\acute{e}tale}} \to x^*\mathcal{H}|_{U_{\acute{e}tale}}$ which give rise to a commutative diagram

\[ \xymatrix{ s_{small}^*(x^*\mathcal{F}|_{U_{\acute{e}tale}}) \ar[r] \ar[d]^{s_{small}^*\varphi } & (x \circ s)^*\mathcal{F}|_{R_{\acute{e}tale}} = (x \circ t)^*\mathcal{F}|_{R_{\acute{e}tale}} & t_{small}^*(x^*\mathcal{F}|_{U_{\acute{e}tale}}) \ar[l] \ar[d]_{t_{small}^*\varphi } \\ s_{small}^*(x^*\mathcal{H}|_{U_{\acute{e}tale}}) \ar[r] & (x \circ s)^*\mathcal{H}|_{R_{\acute{e}tale}} = (x \circ t)^*\mathcal{H}|_{R_{\acute{e}tale}} & t_{small}^*(x^*\mathcal{H}|_{U_{\acute{e}tale}}) \ar[l] } \]

of modules on $R_{\acute{e}tale}$ where the horizontal arrows are the comparison maps (

Proof. 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 and of fppf $\mathcal{O}$-modules. Thus it suffices to prove the lemma with $\mathcal{X} = [U/_{\! p}R]$. By Proposition 96.14.3 and its proof there exists a quasi-coherent module $(\mathcal{G}, \alpha )$ on $(U, R, s, t, c)$ such that $\mathcal{F}$ is given by the rule $\mathcal{F}(T, u) = \Gamma (T, u^*\mathcal{G})$. In particular $x^*\mathcal{F}|_{U_{\acute{e}tale}} = \mathcal{G}$ and it is clear that the map of the statement of the lemma is injective. Moreover, given a map $\varphi : \mathcal{G} \to x^*\mathcal{H}|_{U_{\acute{e}tale}}$ and given any object $y = (T, u)$ of $[U/_{\! p}R]$ we can consider the map

\[ \mathcal{F}(y) = \Gamma (T, u^*\mathcal{G}) \xrightarrow {u_{small}^*\varphi } \Gamma (T, u_{small}^*x^*\mathcal{H}|_{U_{\acute{e}tale}}) \rightarrow \Gamma (T, y^*\mathcal{H}|_{T_{\acute{e}tale}}) = \mathcal{H}(y) \]

where the second arrow is the comparison map ( for the sheaf $\mathcal{H}$. This assignment is compatible with the restriction mappings of the sheaves $\mathcal{F}$ and $\mathcal{G}$ for morphisms of $[U/_{\! p}R]$ if the cocycle condition of the lemma is satisfied. Proof omitted. Hint: the restriction maps of $\mathcal{F}$ are made explicit in terms of $(\mathcal{G}, \alpha )$ in the proof of Proposition 96.14.3. $\square$

[1] Namely, if $U$ is a scheme and $\mathcal{F}$ is quasi-coherent on $(\mathit{Sch}/U)_{\acute{e}tale}$, then $\mathcal{F} = \mathcal{H}^ a$ for some quasi-coherent module $\mathcal{H}$ on the scheme $U$ by Descent, Proposition 35.8.9. In other words, $\mathcal{F} = (\text{id}_{{\acute{e}tale},Zar})^*\mathcal{H}$ by Descent, Remark 35.8.6 with notation as in Descent, Lemma 35.8.5. Then we have $\text{id}_{{\acute{e}tale},Zar} = \pi _ U \circ \text{id}_{small,{\acute{e}tale},Zar}$ and hence we see that $\mathcal{F} = \pi _ U^*\mathcal{G}$ where $\mathcal{G} = (\text{id}_{small,{\acute{e}tale},Zar})^*\mathcal{H}$ is quasi-coherent. Then $\pi _ U^*i_ U^*\mathcal{F} = \pi _ U^*i_ U^*\pi _ U^*\mathcal{G} = \pi _ U^*\mathcal{G} = \mathcal{F}$ as desired.

Comments (1)

Comment #8657 by Sergey Guminov on

There are two instances of "wich" instead of "which" in the first two lemmas on this page.

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