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 (0)

There are also:

  • 2 comment(s) on Section 96.14: Quasi-coherent sheaves and presentations

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 076S. Beware of the difference between the letter 'O' and the digit '0'.