Associative Schemes and Subschemes
Arvid Siqveland
TL;DR
The work extends scheme theory to associative rings by introducing aprime modules and the aSpec space, along with a canonical sheaf of associative rings that specializes to the usual structure sheaf on commutative spectra. It builds localizing rings $A_M$ and proves a universal property that permits affine localization, yielding an equivalence between associative rings and affine associative schemes, while also enabling the construction of induced subschemes and restriction to subsets. It then analyzes algebraic closures and $k$-points, establishing criteria for simple modules under base change and linking real points to closed points of complexified affine spaces through root-points $X^{< abla inite}$. Overall, the framework provides a path to apply complex-geometry methods to real algebraic geometry via associative schemes and their module-theoretic moduli, enabling a moduli perspective for noncommutative objects.
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.