Open Access
2011 Geometry of Noncommutative k-Algebras
Arvid Siqveland
J. Gen. Lie Theory Appl. 5: 1-12 (2011). DOI: 10.4303/jglta/G110107


Let X be a scheme over an algebraically closed field k, and let $x\in\operatorname{Spec} R\subseteq X$ be a closed point corresponding to the maximal ideal $\mathfrak{m}\subseteq R$. Then $\hat{\mathcal{O}}_{X,x}$ is isomorphic to the prorepresenting hull, or local formal moduli, of the deformation functor $\mathrm{Def}_{R/\mathfrak{m}}:\underline{\ell}\rightarrow\mathrm{Sets}$. This suffices to reconstruct $X$ up to etalé coverings. For a noncommutative $k$-algebra $A$ the simple modules are not necessarily of dimension one, and there is a geometry between them. We replace the points in the commutative situation with finite families of points in the noncommutative situation, and replace the geometry of points with the geometry of sets of points given by noncommutative deformation theory. We apply the theory to the noncommutative moduli of three-dimensional endomorphisms.


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Arvid Siqveland. "Geometry of Noncommutative k-Algebras." J. Gen. Lie Theory Appl. 5 1 - 12, 2011.


Published: 2011
First available in Project Euclid: 29 September 2011

zbMATH: 1226.14005
MathSciNet: MR2846729
Digital Object Identifier: 10.4303/jglta/G110107

Primary: 14A22 , 14D22 , 14D23 , 16L30

Rights: Copyright © 2011 Ashdin Publishing (2009-2013) / OMICS International (2014-2016)

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