Localization in Associative Rings and Associative Schemes

Abstract: We start with the argument for doing associative algebraic geometry: We need schemes of associative algebras to parametrize (find moduli of) noncommutative objects.

Let $A$ be a commutative ring. Then we can define the sheaf of rings on $X=Spec ~A$ by letting $O_X(U)=im A\subseteq %\underset{m\in U\text{ maximal}} \prod A_m,$ and to generalize this to rings that are not necessarily commutative, we need a replacement for the local rings $A_m.$ We change our view: The interesting point about $A_m$ is not that it is local, but rather that it is locally representing, i.e. that in the category of pointed rings, $mor( m\subset A,-)$ is represented by $A_m.$

Let $A$ be an Associative (not necessarily commutative) ring. and let $M$ be a simple right $A$-module. We prove that in the category of pointed associative rings there is a pointed associative ring $A_M$ representing $mor((A,M),-).$ Moreover, we prove that for any set of $r>0$ simple modules $M=\{M_i\}_{i=1}^r,$ the categorical product $A_M=\prod_{i=1}^r A_{M_i}$ exists. (When $A$ is noncommutative, this is certainly not the Cartesian product). Given this, we can define $aspec ~A$ as the set of simple right $A$-modules, together with the contractions of such, and we give $X=aspec ~A$ the topology generated by $\{D(f)\}_{f\in A},$ defined in such a way that if $A$ is commutative, this is the ordinary Zariski topology. Then $O_X(U)=im A\subseteq%\underset{M\in\simp A\cap U} \prod A_M$ is a sheaf, and an associative scheme is a ringed space $X$ covered by affine open sets.

We end by defining A Noncommutative Geometry. Let $Y=\mathbb R^3\times\mathbb R^3=\{(\text{observer},\text{observed})\}.$ We let $\mathbb U$ be the noncommutative blowup of $\Delta\subseteq\mathbb R^3\times\mathbb R^3$ which is adding a tangent direction to each $(x,x)\in\Delta.$ Choose a Riemannian metric on $\mathbb R^3.$ Then the maximal velocity is the length of the tangent vector on one side of the diagonal, and we also get an opposite tangent vector on the dark side of the diagonal.

Everything in this lecture are Turing computable, and so everything can be computed by infinitesimally deformation theory. See O.A. Laudal's book [2] for the study of this model.

Bibliography

1. E. Eriksen, O. A. Laudal, A. Siqveland, Noncommutative Deformation Theory. Monographs and Research Notes in Mathematics. CRC Press, Boka Raton, FL, 2017

2. O. A. Laudal, Mathematical models in science, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2021

3. A. Siqveland, Associative Algebraic Geometry, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2023 ISBN: 977-1-80061-354-6

4. A. Siqveland, Associative Schemes,\\ doi.org/10.48550/arXiv.2302.13843, 2024

5. A. Siqveland, Countably Generated Matrix Algebras,\\ doi.org/10.48550/arXiv.2408.01034, 2024

6. A. Siqveland, Shemes of Associative Algebras,\\ doi.org/10.48550/arXiv.2410.17703, 2024

7. A. Siqveland, Associative Local Function Rings,\\ doi.org/10.48550/arXiv.2410.16819, 2024

8. A. Siqveland, Categorical Construction of Schemes,\\ arxiv.org/abs/2511.03433, 2025

9. A. Siqveland, Schemes of Object in abelian Categories,\\ arxiv.org/abs/2511.04191, 2025

10. A. Siqveland, Localization in Associative Rings,\\ arxiv.org/abs/2511.07900, 2025

11. A. Siqveland, Associative Schemes and Subschemes,\\ arxiv.org/abs/2511.09176, 2025

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

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