Associative Schemes and Subschemes

Abstract

In the preprint arXiv:2511.07900 we proved that there exists a localizing ring $A_M$ for $A$ an associative ring with unit, and $M=\oplus_{i=1}^rM_i$ a direct sum of $r\geq 1$ simple right $A$-modules. For a homomorphism of associative rings $A\rightarrow B$ we define the contraction of a simple $B$-module to $A.$ Then we define the set of aprime right $A$-modules ${\rm aSpec} A$ to be the set of simple $A$-modules together with contractions of such. When $A$ is commutative, ${\rm aSpec} A = {\rm Spec} A$. and we define a topology on ${\rm aSpec} A$ such that when $A$ is commutative, this is the Zariski topology. In the preprint \cite{S251}, we proved that when we have a topology and a localizing subcategory, there exists a sheaf of associative rings $\mathcal O_X$ on ${\rm aSpec} A,$ agreeing with the usual sheaf of rings on ${\rm Spec} A.$ In this text, we write out this construction, and we see that we can restrict the sheaf and topology to any subset $V\subseteq{\rm aSpec}$. In particular, this proves that we can use complex varieties in real algebraic geometry, by restricting in accordance with $\mathbb R\subseteq\mathbb C.$ Thus the theory of schemes over algebraically closed fields and its associative generalization can be applied to real (algebraic) geometry.