Sheafification in the derived category.

Question

Let ${\cal F}$ be a presheaf on a scheme $X$ which associates each abelian group ${\cal F}(U) \in {\mathrm{Ab}}$ for each open $U \hookrightarrow X$. Then we can always associate its associated sheaf ${\cal F}^{\mathrm{a}}$.

I wonder whether this generalises to complex of abelian groups. That is, suppose we have a functor

\begin{equation*} {\cal F}^{\bullet} \colon U \mapsto {\cal F}^{\bullet}(U) \in D({\mathrm{Ab}}), \end{equation*} where $D({\mathrm{Ab}})$ stands the (derived) category of complexes of Abelian groups. (I am not sure whether the derived category for complexes of Abelian groups can exist or not). Suppose further that the above functor ${\cal F}^{\bullet}$ satisfies the functoriality. I.e., there is a unique restriction morphism $r_{UV} \colon {\cal F}^{\bullet}(V) \to {\cal F}^{\bullet}(U)$ whenever an open immersion $U \subset V$ exists such that $r_{UW} = r_{UV} \circ r_{VW}$ holds.

Finally, let us denote by $D(X)$ the derived category of sheaves of Abelian groups on $X$.

Q. Is it possible to associate the unique element ${{\cal F}^{\bullet}}^{\mathrm{a}} \in D(X)$?

This question arose when I learned that the motivic complex ${\Bbb Z}(r)_X \in D(X)$ is the one which associates Bloch's higher Chow complex ${\cal Z}^r(U,\bullet) \in D({\mathrm{Ab}})$ for each open $U \hookrightarrow X$. Roughly, my question is the naive generalisation of the sheafification of a single presheaf to that of a presheaf which associates complex of Abelian groups for each open $U$.

Answer 1

Here are some facts :

• a presheaf of complexes $U\mapsto\mathcal{F}^\bullet(U)$ is the same thing as a complex of presheaves, in other words, an element of $C(PSh(X))$. In particular, it can define an object in the derived category $D(PSh(X))$.
• The functor $PSh(X)\to\mathrm{Ab}, \mathcal{F}\mapsto\mathcal{F}(U)$ is exact, so induces a functor $D(PSh(X))\to D(\mathrm{Ab})$. In other words, given a complex of presheaves up to quasi-isomorphism, it is still meaningful to take its section on an open. It follows that we have a functor $\Gamma:D(PSh(X,\mathrm{Ab}))\to PSh(X,D(\mathrm{Ab}))$
• Note however that a presheaf with value in $D(\mathrm{Ab})$ does define an element of $D(PSh(X))$. In other words, the functor $\Gamma$ in the previous point is not an equivalence.
• The associated sheaf functor $PSh(X)\to Sh(X), \mathcal{F}\mapsto \mathcal{F}^a$ is exact, so induces a functor $D(PSh(X))\to D(Sh(X))$.

But given a presheaf with values in $D(\mathrm{Ab})$, there might not exist a canonical sheafification in $D(Sh(X))$.

The point with Bloch complex is that $\mathcal{Z}^\bullet(U,n)$ is actually a presheaf with values in the categories of complexes (and not only with value in $D(\mathcal{Ab})$, so define an element in $D(PSh(X))$ and thus can be sheafified.