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

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