Is there a Čech-like way of computing $H^\bullet(X,M^\bullet)$ or even $\mathsf{R}f_* M^\bullet$?

Question

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!

Answer 1

Is there a Cech-like way of describing the (hyper)cohomology H∙(X,M∙) or, even better, the complex Rf∗M∙ for some map f?

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 $M$.

Answer 2

At least for complexes of quasi-coherent sheaves on a semi-separated scheme, you take the Czech complex at each level and totalize the resulting bicomplex. This complex is made of acyclic sheaves so you can compute the derived funtor of $f_*$ applying it to this resolution. This is discussed (in some generality) in §2.7 from Lipman's Lecture Notes in Mathematics, 1960, 2009; also available at: Notes on derived functors and Grothendieck duality.

In the cases you mention the same idea works mostly the same by the general result mentioned above. You have to take care, probably, of some finiteness hypothesis that guarantee the acyclicity of the Czech complex in the case at hand.