It has been several attempts to generalize the ordinary commutative algebraic geometry to the noncommutative situation. The main problem in the direct generalization, is the lack of localization of noncommutative $k$-algebras, $k$ algebraically closed. This can only be done for Ore sets, and does not give a satisfactory solution to the problem.

In the study of flat deformations of $A$-modules when $A$ is a commutative, finitely generated $k$-algebra, one realizes that for each maximal ideal $\frak m$, putting $V=A/\frak m$, the deformation functor $\operatorname{Def}_V$ has a (unique up to nonunique isomorphism) prorepresenting hull (local formal moduli) $\hat{H}(V)$ isomorphic to the completed local ring, that is $\hat{H}(V)\cong\hat{A}_{\frak m}.$

In the general situation with $A$ not necessarily commutative, the deformation theory can be directly generalized to right (or left) $A$-modules, and we can replace the local complete rings with the hulls of the simple modules. In fact, for a finite dimensional $k$-algebra with family of simple right modules $V=\{V_i\}_{i=1}^n,$ we have the generalized Burnside theorem: $$A\cong(\hat{H}(V_i,V_j)\otimes\operatorname{Hom}_k(V_i,V_j))$$ the right hand side considered as a matrix algebra.

I will start by giving a short introduction to the deformation theory, and give an indication on how to compute the formal local moduli. Then I will clarify the abstract above. If there is any time left, an easy example will be given.