is a proper morphism $f:X\rightarrow S$ such that there exists $U\subset S$ open and dense for which $f$ induces an isomorphism $f^{-1}(U)\rightarrow U$; a *resolution* of $S$ is a proper modification with $X$ smooth.

2.  The *discriminant* of a proper morphism of varieties $f:X\rightarrow S$ is the minimal closed subset $D\subset S$ such that $f$ induces an isomorphism $X\setminus f^{-1}(D)\rightarrow S\setminus D$.

The notions of proper modification, resolution and discriminant extend immediately to $n$-cubical varieties and morphisms thereof. By [@GNPP Théorème I.2.6] any $n$-cubical variety $X_{{\square}}$ admits a resolution, which is constructed by separating and resolving the irreducible components of each $X_I$ and then ‘patching together’ the pieces in a way prescribed by the cubical structure of $X_{{\square}}$.

Let $f:X_{{\square}}\rightarrow S_{{\square}}$ be a proper modification (resp. resolution) of an $n$-cubical variety. A *discriminant square* (resp. *resolution square*) for $f$ is a commutative diagram

$$\xymatrixcolsep{1.5cm}\xymatrixrowsep{1.5cm}\xymatrix{
E_{{\square}} \ar[r]^j \ar[d] & X_{{\square}} \ar[d]^f\\
D_{{\square}} \ar[r]^i                  & S_{{\square}}}$$ where the horizontal maps are closed immersions and $f$ induces an isomorphism between $X_{{\square}}-j(E_{{\square}})$ and $S_{{\square}}-i(D_{{\square}})$ (i.e. $i(D_{{\square}})$ contains the discriminant of $f$).

[@PeSt Lemma 5.20] The $(n+2)$-cubical variety defined by a discriminant square for a proper modification of an $n$-cubical variety is of cohomological descent.

