The Stacks project

Lemma 103.4.1. Let $\mathcal{X}$ be an algebraic stack. We have $Lg_!\mathbf{Z} = \mathbf{Z}$ for either $Lg_!$ as in Lemma 103.3.1 part (1) or $Lg_!$ as in Lemma 103.3.1 part (3).

Proof. We prove this for the comparison between the flat-fppf site with the fppf site; the case of the lisse-├ętale site is exactly the same. We have to show that $H^ i(Lg_!\mathbf{Z})$ is $0$ for $i \not= 0$ and that the canonical map $H^0(Lg_!\mathbf{Z}) \to \mathbf{Z}$ is an isomorphism. Let $f : \mathcal{U} \to \mathcal{X}$ be a surjective, flat morphism where $\mathcal{U}$ is a scheme such that $f$ is also locally of finite presentation. (For example, pick a presentation $U \to \mathcal{X}$ and let $\mathcal{U}$ be the algebraic stack corresponding to $U$.) By Sheaves on Stacks, Lemmas 95.19.6 and 95.19.10 it suffices to show that the pullback $f^{-1}H^ i(Lg_!\mathbf{Z})$ is $0$ for $i \not= 0$ and that the pullback $H^0(Lg_!\mathbf{Z}) \to f^{-1}\mathbf{Z}$ is an isomorphism. By Lemma 103.3.2 we find $f^{-1}Lg_!\mathbf{Z} = L(g')_!\mathbf{Z}$ where $g' : \mathop{\mathit{Sh}}\nolimits (\mathcal{U}_{flat, fppf}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{U}_{fppf})$ is the corresponding comparision morphism for $\mathcal{U}$. This reduces us to the case studied in the next paragraph.

Assume $\mathcal{X} = (\mathit{Sch}/X)_{fppf}$ for some scheme $X$. In this case the category $\mathcal{X}_{flat, fppf}$ has a final object $e$, namely $X/X$, and moreover the functor $u : \mathcal{X}_{flat, fppf} \to \mathcal{X}_{fppf}$ sends $e$ to the final object. Since $\mathbf{Z}$ is the free abelian sheaf on the final object (provided the final object exists) we find that $Lg_!\mathbf{Z} = \mathbf{Z}$ by the very construction of $Lg_!$ in Cohomology on Sites, Lemma 21.37.2. $\square$

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