Source: J. Gen. Lie Theory Appl., Volume 5, 12 pages.
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.