Associative Schemes

Abstract

We state results from noncommutative deformation theory of modules over an associative $k$-algebra $A,$ $k$ a field, necessary for this work. We define a set of $A$-modules $\operatorname{aSpec}A$ containing the simple modules, whose elements we call spectral, 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 $\mathcal O_X$ on the topological space $X=\operatorname{aSpec}A$ giving it the structure of a pointed ringed space. In general, an associative variety $X$ is a ringed space with an open covering $\{U_i=\operatorname{aSpec}{A_i}\}_{i\in I}.$ When $A$ is a commutative $k$-algebra, $\operatorname{aSpec}A\simeq\spec A,$ and so the category $\cat{aVar}_k$ of associative varieties is an extension of the category of varieties $\cat{Var}_k,$ i.e. there exists a faithfully full functor $I:\cat{Var}_k\rightarrow\cat{aVar}_k.$ Our main result says that any associative variety $X$ is $\operatorname{aSpec}(\mathcal O_X(X))$ for the $k$-algebra $\mathcal O_X(X),$ and so any study of varieties can be reduced to the study of the associative algebra $\mathcal O_X(X).$