Lemma 96.14.4 (076S)—The Stacks project

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 (96.10.3.3).

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 (96.9.4.1) 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. $\square$

Comments (1)

Comment #9901 by ZL on December 22, 2024 at 20:15

I am a little confused. We know that $x\circ s$ and $x\circ t$ are connected by a $2$ -isomorphism defining $\alpha$ . Hence the two "equalities" $(x\circ s)^{\ast}F| _ {R_{\acute{e}tale}}=(x\circ t)^{\ast}F| _ {R_{\acute{e}tale}}$ and $(x\circ s)^{\ast}H| _ {R_{\acute{e}tale}}=(x\circ t)^{\ast}H| _ {R_{\acute{e}tale}}$ in the commutative diagramme are in fact the canonical isomorphisms induced from $x\circ s\rightarrow x\circ t$ ? If I understand correctly, the upper horizontal morphism (after inverting $s_ {small}^{\ast} (x^{\ast} \mathcal{F}| _ {U_ {\acute{e}tale}}) \rightarrow (x \circ s) ^* \mathcal{F}| _ {R_ {\acute{e}tale}}$ ) is exactly the $\alpha: t^{\ast}_ {small}( x^{\ast} \mathcal{F}| _ {U_ {\acute{e}tale}})\rightarrow s^{\ast}_ {small} (x^{\ast} \mathcal{F}| _ {U_ {\acute{e}tale}})$ ?

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