The Stacks project

Lemma 103.3.1. Let $\mathcal{X}$ be an algebraic stack. Notation as in Cohomology of Stacks, Lemmas 102.14.2 and 102.14.4.

  1. The functor $g_! : \textit{Ab}(\mathcal{X}_{lisse,{\acute{e}tale}}) \to \textit{Ab}(\mathcal{X}_{\acute{e}tale})$ has a left derived functor

    \[ Lg_! : D(\mathcal{X}_{lisse,{\acute{e}tale}}) \longrightarrow D(\mathcal{X}_{\acute{e}tale}) \]

    which is left adjoint to $g^{-1}$ and such that $g^{-1}Lg_! = \text{id}$.

  2. The functor $g_! : \textit{Mod}(\mathcal{X}_{lisse,{\acute{e}tale}}, \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}) \to \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_{\mathcal{X}})$ has a left derived functor

    \[ Lg_! : D(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}) \longrightarrow D(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) \]

    which is left adjoint to $g^*$ and such that $g^*Lg_! = \text{id}$.

  3. The functor $g_! : \textit{Ab}(\mathcal{X}_{flat,fppf}) \to \textit{Ab}(\mathcal{X}_{fppf})$ has a left derived functor

    \[ Lg_! : D(\mathcal{X}_{flat, fppf}) \longrightarrow D(\mathcal{X}_{fppf}) \]

    which is left adjoint to $g^{-1}$ and such that $g^{-1}Lg_! = \text{id}$.

  4. The functor $g_! : \textit{Mod}(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}}) \to \textit{Mod}(\mathcal{X}_{fppf}, \mathcal{O}_{\mathcal{X}})$ has a left derived functor

    \[ Lg_! : D(\mathcal{O}_{\mathcal{X}_{flat, fppf}}) \longrightarrow D(\mathcal{O}_\mathcal {X}) \]

    which is left adjoint to $g^*$ and such that $g^*Lg_! = \text{id}$.

Warning: It is not clear (a priori) that $Lg_!$ on modules agrees with $Lg_!$ on abelian sheaves, see Cohomology on Sites, Remark 21.37.3.

Proof. The existence of the functor $Lg_!$ and adjointness to $g^*$ is Cohomology on Sites, Lemma 21.37.2. (For the case of abelian sheaves use the constant sheaf $\mathbf{Z}$ as the structure sheaves.) Moreover, it is computed on a complex $\mathcal{H}^\bullet $ by taking a suitable left resolution $\mathcal{K}^\bullet \to \mathcal{H}^\bullet $ and applying the functor $g_!$ to $\mathcal{K}^\bullet $. Since $g^{-1}g_!\mathcal{K}^\bullet = \mathcal{K}^\bullet $ by Cohomology of Stacks, Lemmas 102.14.4 and 102.14.2 we see that the final assertion holds in each case. $\square$


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