implies the following:

\[thm:iperrescubcub\] Any $n$-cubical variety $X_{{\square}}$ admits a hyperresolution by an $m$-cubical variety $Y_{{\square}}$ whose entries are $n$-cubical varieties.

\[rmk:iperrescubcub\] Assume that $X_{{\square}}=\{X_{\emptyset},X_{\{0\}},X_{\{1\}},X_{\{01\}}\}$ is a $2$-cubical variety and $Y_{{\square}}$ is an $m$-cubical hyperresolution of $X_{{\square}}$, then $Y_{{\square}}$ can be thought of as a $2$-cubical variety $Y'_{{\square}}=\{Y'_{\emptyset},Y'_{\{0\}},Y'_{\{1\}},Y'_{\{01\}}\}$ of $(m-2)$-cubical varieties; by construction, for any $I\in{\square}_2$ we have that $Y'_I$ is an $(m-2)$-cubical hyperresolution of $X_I$.

In [@GNPP] cubical hyperresolutions were used to define a cohomology theory for possibly singular algebraic varieties; namely:

\[defn:DeRhamComplex\] Let $X$ be a separated scheme of finite type over a field ${\mathbb{K}}$ of characteristic zero, and let ${\varepsilon}:X_{{\square}}\rightarrow X$ be an $(n+1)$-cubical hyperresolution of $X$ together with its natural augmentation; the *de Rham complex* and $k$-th algebraic de Rham cohomology group of $X$ are defined as

$$DR^{{\bullet}}_X:=R{\varepsilon}_*\Omega^{{\bullet}}_{X_{{\bullet}}}\hspace{1cm}H_{DR}^k(X):=\mathbb{H}^k(X,DR^{{\bullet}}_X).$$

If $V\subset X$ is a closed subset, the $k$-th algebraic de Rham cohomology group of $X$ with supports in $V$ is defined as

$$H_{DR,V}^k(X):=\mathbb{H}^k(V,R\Gamma_V DR^{{\bullet}}_X).$$

In both cases, ${\varepsilon}:X_{{\bullet}}\rightarrow X$ is the augmented $n$-semisimplicial resolution of $X$ associated to $X_{{\square}}$.

This cohomology theory coincides with the one developed by Hartshorne in [@H1] in case of an embeddable scheme $X$ over ${\mathbb{C}}$, since the cohomology groups are defined as 