The Stacks project

Lemma 85.15.4. Let $\mathcal{C}$ be a site. For $K$ in $\text{SR}(\mathcal{C})$ the functor $j^{-1}$ sends injective abelian sheaves to injective abelian sheaves. Similarly, the functor $j^{-1}$ sends K-injective complexes of abelian sheaves to K-injective complexes of abelian sheaves.

Proof. The first statement is the natural generalization of Cohomology on Sites, Lemma 21.7.1 to semi-representable objects. In fact, it follows from this lemma by the product decomposition of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K)$ and the description of the functor $j^{-1}$ given above. The second statement is the natural generalization of Cohomology on Sites, Lemma 21.20.1 and follows from it by the product decomposition of the topos.

Alternative: since $j$ induces a localization of topoi by Lemma 85.15.3 part (1) it also follows immediately from Cohomology on Sites, Lemmas 21.7.1 and 21.20.1 by enlarging the site; compare with the proof of Cohomology on Sites, Lemma 21.13.3 in the case of injective sheaves. $\square$

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