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

Abstract

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.

18 pages. One mathematical correction: number of irreducible components of centralizer (hence of restricted nilpotent commuting variety) corrected, see Lemma 2.1, Prop. 2.3, Lemma 3.1