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

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dc.creatorLevy, Paul
dc.date2006-09-06
dc.date2007-06-11
dc.date.accessioned2026-07-07T08:04:45Z
dc.date.available2026-07-07T08:04:45Z
dc.descriptionLet $k$ 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.
dc.description18 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
dc.identifierhttps://arxiv.org/abs/math/0609180
dc.identifierhttp://arxiv.org/abs/math/0609180
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/130078
dc.subjectRings and Algebras
dc.subjectRepresentation Theory
dc.subject17B45
dc.titleVarieties of Modules for Z/2Z x Z/2Z
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