Hypersheaves vs derived category of sheaves

Question

This question arose from Peter Scholze's notes on six functor formalisms, specifically lecture VII in the proof of proposition 7.1.

We fix a LCH space $X$ and consider the functor $D(\mathrm{Ab}(X)) \to \mathrm{Shv}^{\mathrm{hyp}}(X,D(\mathbb Z))$ from the derived $\infty$-category of abelian sheaves to the infinite category of $D(\mathbb Z)$-valued hypersheaves on $X$, which (at the level of complexes) sends a complex $C^*$ of sheaves to the hypersheafification of the presheaf $U \mapsto C^*(U)$.

Scholze claims

When restricted to K-injective complexes, it turns out that the hypersheafification step is unnecessary. In other words, if $U \subset X$ has a hypercover $U_\bullet \to U$, and $C \in \operatorname{Ch}(\operatorname{Ab}(X))$ is $K$-injective, then the map $C(U) \to \lim C(U_\bullet)$ in $D(\mathbb Z)$ is an isomorphism. This follows by noting that the hypercover induces a resolution of $j_! \mathbb Z$ (where $j : U \subseteq X$ is the open immersion), and taking the associated $R{\operatorname{Hom}}$ into $C$ (which vanishes by assumption that $C$ is $K$-injective)

I had a couple of questions about this argument:

1. How is this resolution of $j_! \mathbb Z$ actually constructed from the hypercover of $U$? I assume it's something along the lines of using the hypercover to create a sequence of constant sheaves on each of the $U_\bullet$, and then using the exceptional pushforward along the inclusions $U_\bullet \hookrightarrow X$ to create a resolution of $j_! \mathbb Z$. But I don't see why this should be a ‘good’ (e.g. flasque) resolution.

2. How do we finally conclude? I don't see what $R{\operatorname{Hom}}$ming into $C$ and getting $0$ tells us about our original map being an isomorphism.

Sorry for the long question! I thought that this step in the notes used some interesting techniques (resolutions from hypercovers, and using $K$-injective complexes) and I'd like to understand them better.

Answer 1

For any cover $U^{\bullet}$ and any sheaf $F$, there exists a Čech complex $j_!^{\bullet} F\rightarrow j_!F$. Now, you just take the hypercover $U^{\bullet}$ and $F=\mathbb{Z}$. Maybe you can see section 2.8 of Sheaves on manifolds of Kashiwara–Schapira. We don't need hyper condition here.

To incorporate with $C$, you need to notice that $R{\operatorname{Hom}}(j_!\mathbb{Z},C)\simeq R\Gamma(U,C) \simeq \Gamma(U,C)=C(U)$ for all open sets $U$, where the second isomorphism follows from K-injectivity. K-Injectivity also tells you the resulting sequence is actually a resolution.