K-injective (also known as hoinjective) complexes of sheaves of modules

Question

Let $(X,\mathcal O_X)$ be a ringed space (if necessary, assume that it is a scheme with suitable hypotheses). Given two complexes of sheaves $\mathcal F$ and $\mathcal G$ of $\mathcal O_X$-modules, one has the complex of morphisms \begin{equation} \operatorname{Hom}^\bullet(\mathcal F, \mathcal G),\end{equation} and moreover the complex of sheaves of morphisms \begin{equation} \mathcal H om^\bullet (\mathcal F, \mathcal G), \end{equation} which is defined on any open $U \subseteq X$ by $\mathcal H om^\bullet (\mathcal F, \mathcal G)(U) = \operatorname{Hom}^\bullet (\mathcal F_{|U}, \mathcal G_{|U})$.

We give the following definition:

A complex of sheaves of $\mathcal O_X$-modules $\mathcal I$ is called K-injective (or hoinjective) if for any exact (acyclic) complex of sheaves of $\mathcal O_X$-modules $\mathcal A$, the complex $\operatorname{Hom}^\bullet(\mathcal A, \mathcal I)$ is acyclic.

This definition should be precisely the same as the one I can found in the literature, for example here. My question is if the following characterisation holds:

A complex of $\mathcal O_X$-modules $\mathcal I$ is K-injective if and only if for any acyclic complex of $\mathcal O_X$-modules $\mathcal A$, the complex of sheaves $\mathcal H om^{\bullet}(\mathcal A, \mathcal I)$ is acyclic.

One implication is trivial because $\operatorname{Hom}^\bullet(\mathcal F, \mathcal G) = \mathcal H om^\bullet (\mathcal F, \mathcal G)(X)$ by definition, but I'm confused about the other one.

Answer 1

Let $j:U\hookrightarrow X$ be the inclusion of an open subset. Then $\mathcal H om_{\mathcal O_X}^\bullet (\mathcal A, \mathcal I)(U) = \operatorname{Hom}^\bullet_{\mathcal O_U} (j^\ast\mathcal A,j^\ast\mathcal I)=\operatorname{Hom}^\bullet_{\mathcal O_X}(j_!j^\ast\mathcal A,\mathcal I)$. The functors $j_!$ and $j^\ast$ are exact, so they send acyclic complexes of sheaves to acyclic complexes of sheaves. Also $j_!$ sends $\mathcal O_U$-modules to $\mathcal O_X$-modules. So, $j_!j^\ast\mathcal A$ is an acyclic $\mathcal O_X$-module. This gives a positive answer to your question.