Let $X$ be a topological space (or a site) and let $M$ be a sheaf on $X$. If $X$ is paracompact, or if $X$ is a noetherian separated scheme and $M$ is quasi-coherent, or if $X$ is quasi-projective over an affine scheme and $M$ is an étale sheaf, we know that the Čech cohomology $\smash{\check{\mathrm{H}}}^\bullet(X,M)$ coincides with the sheaf cohomology $H^\bullet(X,M)$.
I wonder what happens when $M$ is a (possibly unbounded) complex. Is there a Čech-like way of describing the (hyper)cohomology $H^\bullet(X,M^\bullet)$ or, even better, the complex $\mathsf{R}f_* M^\bullet$ for some map $f$?
If that's necessary, an answer using hypercovers (which I know very little about) would also be interesting!
Best Answer
Yes, the Verdier hypercovering theorem allows one to compute sheaf cohomology on any site in terms of hypercovers.
Specifically, suppose $M$ is a presheaf of unbounded cochain complexes on a site $S$ and $X∈S$.
Denote by $H/X$ the category whose objects are hypercovers of $X$ and morphisms are commutative triangles.
Denote by $\def\Ch{{\sf Ch}}\def\op{{\sf op}}H/X^\op→\Ch$ the (contravariant) functor that sends a hypercover $U→X$ to the mapping chain complex $\def\Map{\mathop{\rm Map}}\Map(U,M)$ and on morphisms is given by the restriction maps.
Denote by $C(X,M)$ the colimit of the functor $H/X^\op→\Ch$. Since hypercovers can be pulled back, a morphism $X→Y$ induces a map $C(Y,M)→C(X,M)$, which turns $C(-,M)$ into a presheaf of chain complexes.
The canonical map $M→C(-,M)$ is a local quasi-isomorphism and its target satisfies the homotopy coherent descent property with respect to all hypercovers. Therefore, the cohomology of $C(-,M)$ computes the sheaf cohomology with coefficients in $M$.