the hypercohomology of isomorphic complexes (see [@GNPP Théorème III.1.3]).

The definitions of hyperresolution and de Rham complex $DR^{\bullet}_X$ given in [@GNPP] are actually different from the ones we presented here; if we denote by ${\mathcal{C}}$ the category of separated schemes of finite type over a field ${\mathbb{K}}$ of characteristic zero, we have:

\[defn:hyperrescubGNPP\] [@GNPP Définition I.3.2] If $X_{{\square}}$ is an $n$-cubical hyperresolution of $X\in{\mathcal{C}}$, with its natural augmentation $X_{{\square}}\rightarrow X$, then $X^*_{{\square}}$ has a natural augmentation ${\varepsilon}:X^*_{{\square}}\rightarrow X$ to $X$ too. The latter is an *$n$-cubical hyperresolution* of $X$.

\[defn:DeRhamComplexBis\] [@GNPP Définition III.1.10, Proposition III.1.12] If $X\in{\mathcal{C}}$ and $X^*_{{\square}}$ is an $(n+1)$-cubical hyperresolution of $X$ with its natural augmentation ${\varepsilon}:X^*_{{\square}}\rightarrow X$, the *de Rham complex* of $X$ is

$$DR^{{\bullet}}_X:=R{\varepsilon}_*\Omega^{{\square}}_{X^*_{{\square}}}.$$

Although different, the two definitions of the de Rham complex are equivalent. Indeed, pick $X\in{\mathcal{C}}$, let $X_{{\square}}$ be an $(n+1)$-cubical hyperresolution of $X$ with its natural augmentation ${\varepsilon}:X_{{\square}}\rightarrow X$, and let ${\varepsilon}:X_{{\bullet}}\rightarrow X$ be the augmented $n$-semisimplicial resolution associated to it. Let $X^*_{{\square}}$ be the augmented $(n+1)$-cubical hyperresolution of $X$ as in Definition \[defn:hyperrescubGNPP\], and denote by ${\varepsilon}:X^*_{{\square}}\rightarrow X$ its augmentation. In order to show that Definitions \[defn:DeRhamComplex\] and \[defn:DeRhamComplexBis\] are equivalent, we need to prove that

$$R{\varepsilon}_*\Omega^{{\bullet}}_{X_{{\bullet}}}\simeq R{\varepsilon}_*\Omega^{{\square}}_{X^*_{{\square}}}$$

But this 