### Speaker

### Description

We state results from noncommutative deformation theory of modules over an associative k-algebra A necessary for this work. We define a set of A-modules containing the simple modules whose elements we call spectral, aSpec A, for which there exists a topology where the simple modules are the closed points. Applying results from deformation theory we prove that there exists a sheaf of associative rings O_X on the topological space X = aSpec A giving it the structure of a locally ringed space. In general, an associative variety X is a ringed space with an open covering {Ui = aSpec Ai}_{i∈I}. When A is a commutative k-algebra, aSpec A ≃ Spec A,

and so the category aVark of associative varieties is an extension of the category of varieties Vark , i.e. there exists a faithfully flat functor I : Vark → aVark . Our main result says that any associative variety X is aSpec(O_X(X)) for the k-algebra O_X(X), and so any study of varieties can be reduced to the study of the associative algebra O_X(X). (We make

Geometry Algebraic again.)