R{\varepsilon}_*{\varepsilon}^{-1}{\mathcal{F}}^{\emptyset}].$$

From now on, we will take for ${\mathcal{C}}$ the category whose objects are reduced separated schemes of finite type over ${\mathbb{C}}$, which we will simply call varieties, and whose morphisms are morphisms of schemes; this is not fully consistent with the existing literature, in which the term ‘algebraic variety’ is usually reserved for *integral* separated schemes of finite type over some field.

1.  An augmented $n$-semisimplicial variety is of cohomological descent if this is the case for the associated augmented $n$-semisimplicial space.

2.  Let $X$ be a variety. An $n$-*semisimplicial resolution* of $X$ is an $n$-semisimplicial variety ${\varepsilon}:X_{{\bullet}}\rightarrow X$ augmented to $X$ such that all maps $X_m\rightarrow X$ are proper, all $X_m$ are smooth and ${\varepsilon}$ is of cohomological descent.

3.  An $(n+1)$-cubical variety is of cohomological descent (resp. a *cubical hyperresolution*) if the associated augmented $n$-semisimplicial variety is of cohomological descent (resp. an $n$-semisimplicial resolution). By the definition of cohomological descent and (\[eq:cono\]), $X_{{\square}}$ is of cohomological descent if and only if $C^{{\bullet}}(X_{{\square}},{\mathcal{F}}^{{\square}})$ is acyclic for any sheaf of abelian groups ${\mathcal{F}}^{{\square}}$ on $X_{{\square}}$.

We want to show that any variety $X$ admits an $m$-cubical hyperresolution $X_{{\square}}$ for some $m$.

1.  A *proper modification* of a variety $S$ 