Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2511.09176 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866915612971761664 |
|---|---|
| author | Siqveland, Arvid |
| author_facet | Siqveland, Arvid |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_09176 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Associative Schemes and Subschemes Siqveland, Arvid Algebraic Geometry 14A22 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. |
| title | Associative Schemes and Subschemes |
| topic | Algebraic Geometry 14A22 |
| url | https://arxiv.org/abs/2511.09176 |