The Stacks project

Example 94.17.2. 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 94.16. 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.

There are also:

  • 2 comment(s) on Section 94.17: Examples of inertia stacks

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