The Stacks project

Example 95.17.2. Let $f : X \to S$ be a morphism of schemes. Assume that for any $T \to S$ the base change $f_ T : X_ T \to T$ has the property that the map $\mathcal{O}_ T \to f_{T, *}\mathcal{O}_{X_ T}$ is an isomorphism. (This implies that $f$ is cohomologically flat in dimension $0$ (insert future reference here) but is stronger.) Consider the Picard stack $\mathcal{P}\! \mathit{ic}_{X/S}$, see Section 95.16. The points of its inertia stack over an $S$-scheme $T$ consist of pairs $(\mathcal{L}, \alpha )$ where $\mathcal{L}$ is a line bundle on $X_ T$ and $\alpha $ is an automorphism of that line bundle. I.e., we can think of $\alpha $ as an element of $H^0(X_ T, \mathcal{O}_{X_ T})^\times = H^0(T, \mathcal{O}_ T^*)$ by our condition. Note that $H^0(T, \mathcal{O}_ T^*) = \mathbf{G}_{m, S}(T)$, see Groupoids, Example 39.5.1. Hence the inertia stack of $\mathcal{P}\! \mathit{ic}_{X/S}$ is

\[ I_{\mathcal{P}\! \mathit{ic}_{X/S}} = \mathbf{G}_{m, S} \times _ S \mathcal{P}\! \mathit{ic}_{X/S}. \]

as a stack over $(\mathit{Sch}/S)_{fppf}$.

Comments (2)

Comment #5027 by 羽山籍真 on

For the reference for cohomologically flat in degree q, see [L. Illusie, Grothendieck’s existence theorem in formal geometry, 3.10]

Comment #5261 by on

@#5027: OK, yes, but when it says in the text "insert future reference here" it always means that we will write about the topic in the Stacks project at a later moment in time. Referring to an outside reference here would not be that helpful as we are not actually discussing the topic of being cohomologically flat at all.

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  • 2 comment(s) on Section 95.17: Examples of inertia stacks

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