The Stacks project

Lemma 84.12.3. In Situation 84.3.3 let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in Remark 84.4.1.

  1. The pullback $a^{-1}\mathcal{G}$ of a sheaf of sets or abelian groups on $\mathcal{D}$ is cartesian.

  2. The pullback $a^{-1}K$ of an object $K$ of $D(\mathcal{D})$ is cartesian.

Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$ and $\mathcal{O}_\mathcal {D}$ a sheaf of rings on $\mathcal{D}$ and $a^\sharp : \mathcal{O}_\mathcal {D} \to a_*\mathcal{O}$ a morphism as in Section 84.11.

  1. The pullback $a^*\mathcal{F}$ of a sheaf of $\mathcal{O}_\mathcal {D}$-modules is cartesian.

  2. The derived pullback $La^*K$ of an object $K$ of $D(\mathcal{O}_\mathcal {D})$ is cartesian.

Proof. This follows immediately from the identities $a_ m \circ f_\varphi = a_ n$ for all $\varphi : [m] \to [n]$. See Lemma 84.4.2 and the discussion in Section 84.11. $\square$

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