If $\mathcal F^\bullet$ is a chain complex of sheaves, to compute the hypercohomology, you take the cohomology of an injective resolution of $\mathcal F^\bullet$, i.e., a chain complex of injectives $\mathcal I^\bullet$ which is quasi-isomorphic to $\mathcal F^\bullet$.


It seems to be a well-known fact that for the algebraic de Rham complex $\Omega^\bullet\_{X/S}$ you can compute this hypercohomolgy in practice by taking the total cohomology of the Cech/de Rham double complex $(\check{C}^i(X,\Omega^j), d, \check{d})$.


Does anyone know of a reference for this (somewhere in EGA?), or is there an easy proof ?



Why does the de Rham double complex compute the algebraic de Rham cohomology ?
This has nothing to do with the details of the problem; it is a generalization of [Leray's theorem](http://math.berkeley.edu/~gasparim/charclass/leray2.pdf) to hypercohomology. If $\mathcal{F}\_{\bullet}$ is a complex of sheaves, and $U\_i$ is an open cover of $X$ such that $H^q(U\_{i\_1} \cap \cdots \cap U\_{i\_r}, \mathcal{F}\_j)=0$ for $q \geq 1$ and all $(i\_1, \ldots, i\_r)$ and $j$, then the hypercohomology of $\mathcal{F}\_{\bullet}$ is computed by the cohomology of the Cech-$\mathcal{F}$ double complex.


I don't have my books on hand to find you a reference, but it isn't hard.


