The Stacks project

Lemma 18.16.3. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume that

  1. $u$ is cocontinuous,

  2. $u$ is continuous, and

  3. fibre products and equalizers exist in $\mathcal{C}$ and $u$ commutes with them.

In this case the functor $g_! : \textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D})$ is exact.

Proof. Compare with Sites, Lemma 7.21.6. Assume (a), (b), and (c). We already know that $g_!$ is right exact as it is a left adjoint, see Categories, Lemma 4.24.6 and Homology, Section 12.7. We have $g_! = (g_{p!}\ )^\# $. We have to show that $g_!$ transforms injective maps of abelian sheaves into injective maps of abelian presheaves. Recall that sheafification of abelian presheaves is exact, see Lemma 18.3.2. Thus it suffices to show that $g_{p!}$ transforms injective maps of abelian presheaves into injective maps of abelian presheaves. To do this it suffices that colimits over the categories $(\mathcal{I}_ V^ u)^{opp}$ of Sites, Section 7.5 transform injective maps between diagrams into injections. This follows from Sites, Lemma 7.5.1 and Algebra, Lemma 10.8.10. $\square$

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