# Varieties of Modules for Z-2Z x Z-2Z

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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 $kX,Y-X^2,Y^2$. 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.

Autor: Paul Levy

Fuente: https://archive.org/