2025/11/12 by Siqveland, Arvid
#14A22 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2511.09176
In the preprint arXiv:2511.07900 we proved that there exists a localizing ring AM for A an associative ring with unit, and M=⊕i=1rMi a direct sum of r≥ 1 simple right A-modules. For a homomorphism of associative rings A→ 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 \citeS251, we proved that when we have a topology and a localizing subcategory, there exists a sheaf of associative rings \mathcal OX 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⊆\rm aSpec. In particular, this proves that we can use complex varieties in real algebraic geometry, by restricting in accordance with \mathbb R⊆\mathbb C. Thus the theory of schemes over algebraically closed fields and its associative generalization can be applied to real (algebraic) geometry.