Varieties of Modules for Z/2Z x Z/2Z

Let $k$ be an algebraically closed field of characteristic 2. We prove that the restricted nilpotent commuting variety ${\mathcal C}$, that is the set of pairs of $(n\times n)$-matrices $(A,B)$ such that $A^2=B^2=[A,B]=0$, is equidimensional. ${\mathcal C}$ can be identified with the `variety of $n$-dimensional modules' for ${\mathbb Z}/2{\mathbb Z}\times{\mathbb Z}/2{\mathbb Z}$, or equivalently, for $k[X,Y]/(X^2,Y^2)$. On the other hand, we provide an example showing that the restricted nilpotent commuting variety is not equidimensional for fields of characteristic $>2$. We also prove that if $e^2=0$ then the set of elements of the centralizer of $e$ whose square is zero is equidimensional. Finally, we express each irreducible component of ${\mathcal C}$ as a direct sum of indecomposable components of varieties of ${\mathbb Z}/{2{\mathbb Z}}\times{\mathbb Z}/2{\mathbb Z}$-modules.