The derived category $D^{\flat}_{c}(X,R)$ of constructible sheaves of $R$-modules on $X_{et}$ is defined as the full subcategory of $D^b(X,R)$ whose cohomology sheaves are all constructible.

Clearly, given a sequence of constructible sheaves indexed by $\mathbb{Z}$ can be realized as the cohomology of a complex of sheaves consisting entirely of constructible sheaves (just put the desired cohomology sheaf in each degree and make the differential zero). But one might imagine that there is a complex of sheaves, not all of which are constructible, whose cohomology sheaves are constructible, and which is not quasi-isomorphic to a complex consisting entirely of constructible sheaves.

My question: Can such a complex of sheaves exist? If so, under what conditions might it not exist?

I asked this question in a course of N. Katz, who answered that he did not know the answer but that he knew of some results in the affirmative when asking this question for quasi-coherent or coherent sheaves in place of constructible sheaves.

formulated in adequate generalityback into the setting of usual sheaves that satisfy whatever "niceness" properties are imposed on the homologies of the complex. – user28172 Dec 5 '12 at 3:31