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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2410.16819 |
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| _version_ | 1866916448813711360 |
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| author | Siqveland, Arvid |
| author_facet | Siqveland, Arvid |
| contents | We prove that for an arbitrary field $k,$ a complete, associative $k^r$-algebra $\hat H$ augmented over $k^r$ has exactly $r$ maximal two-sided ideals and deserves the name $r$-pointed. If $A$ is any $k$-algebra, $M=\{M_i\}_{i=1}^r$ is a family of simple right $A$-modules with a countable $k$-basis, and there is a homomorphism $ρ_A:A\rightarrow\enm_{\hat H}(H\hat{\otimes}_{k^r}(\oplus_{i=1}^r M_i))=:\hat O(M)$ then $\hat O(M)$ is $r$-pointed and $M$ is contained in the set of right simple $\hat O(M)$-modules. Our main result is that the subalgebra generated $ρ_A(A)$ and all $ρ_A(a)^{-1}$ whenever $ρ_A(a)$ is a unit, is a natural substitute for the localization $A(M)$ of the $k$-algebra $A$ in $M$ which only exists when the Ore condition is fulfilled. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16819 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Associative Local Function Rings Siqveland, Arvid Algebraic Geometry 14a22, 16B70 We prove that for an arbitrary field $k,$ a complete, associative $k^r$-algebra $\hat H$ augmented over $k^r$ has exactly $r$ maximal two-sided ideals and deserves the name $r$-pointed. If $A$ is any $k$-algebra, $M=\{M_i\}_{i=1}^r$ is a family of simple right $A$-modules with a countable $k$-basis, and there is a homomorphism $ρ_A:A\rightarrow\enm_{\hat H}(H\hat{\otimes}_{k^r}(\oplus_{i=1}^r M_i))=:\hat O(M)$ then $\hat O(M)$ is $r$-pointed and $M$ is contained in the set of right simple $\hat O(M)$-modules. Our main result is that the subalgebra generated $ρ_A(A)$ and all $ρ_A(a)^{-1}$ whenever $ρ_A(a)$ is a unit, is a natural substitute for the localization $A(M)$ of the $k$-algebra $A$ in $M$ which only exists when the Ore condition is fulfilled. |
| title | Associative Local Function Rings |
| topic | Algebraic Geometry 14a22, 16B70 |
| url | https://arxiv.org/abs/2410.16819 |