The Stacks project

An injective abelian sheaf on a simplicial site is injective on each component

Lemma 83.3.6. In Situation 83.3.3. If $\mathcal{I}$ is injective in $\textit{Ab}(\mathcal{C}_{total})$, then $\mathcal{I}_ n$ is injective in $\textit{Ab}(\mathcal{C}_ n)$. If $\mathcal{I}^\bullet $ is a K-injective complex in $\textit{Ab}(\mathcal{C}_{total})$, then $\mathcal{I}_ n^\bullet $ is K-injective in $\textit{Ab}(\mathcal{C}_ n)$.

Proof. The first statement follows from Homology, Lemma 12.29.1 and Lemma 83.3.5. The second statement from Derived Categories, Lemma 13.31.9 and Lemma 83.3.5. $\square$


Comments (2)

Comment #1774 by shom on

Suggested slogan: An injective abelian sheaf on the (2) category of all sites is just an injective abelian sheaf on each constituent site (object) on the nose.

Comment #1775 by on

I don't think that is quite what the slogan should say. The following is probably a better approximation.

Suggested slogan: An injective abelian sheaf on a simplicial site is injective on each component.


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