## Journal of Generalized Lie Theory and Applications

### Geometry of Noncommutative k-Algebras

Arvid Siqveland

#### Abstract

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.

#### Article information

Source
J. Gen. Lie Theory Appl., Volume 5 (2011), Article ID G110107, 12 pages.

Dates
First available in Project Euclid: 29 September 2011

https://projecteuclid.org/euclid.jglta/1317309041

Digital Object Identifier
doi:10.4303/jglta/G110107

Mathematical Reviews number (MathSciNet)
MR2846729

Zentralblatt MATH identifier
1226.14005

#### Citation

Siqveland, Arvid. Geometry of Noncommutative k-Algebras. J. Gen. Lie Theory Appl. 5 (2011), Article ID G110107, 12 pages. doi:10.4303/jglta/G110107. https://projecteuclid.org/euclid.jglta/1317309041

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