Noncommutative Gelfand Duality: the algebraic case
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929724620537856 |
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| author | Bambozzi, Federico Capoferri, Matteo Murro, Simone |
| author_facet | Bambozzi, Federico Capoferri, Matteo Murro, Simone |
| contents | The goal of this paper is to define a notion of non-commutative Gelfand duality. Using techniques from derived algebraic geometry, we show that the category of rings is anti-equivalent to a subcategory of pre-ringed sites, inspired by Grothendieck's work on commutative rings. Our notion of spectrum, although formally reminiscent of the Grothendieck spectrum, is new. Remarkably, an appropriately refined relative version of our spectrum agrees with the Grothendieck spectrum for finitely generated commutative algebras over the complex numbers, among others. This work aims to represent the starting point for a rigorous study of geometric properties of quantum spacetimes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11816 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Noncommutative Gelfand Duality: the algebraic case Bambozzi, Federico Capoferri, Matteo Murro, Simone Algebraic Geometry Mathematical Physics Category Theory Quantum Algebra Primary: 14A22, Secondary 14A30, 16E35, 18F20 The goal of this paper is to define a notion of non-commutative Gelfand duality. Using techniques from derived algebraic geometry, we show that the category of rings is anti-equivalent to a subcategory of pre-ringed sites, inspired by Grothendieck's work on commutative rings. Our notion of spectrum, although formally reminiscent of the Grothendieck spectrum, is new. Remarkably, an appropriately refined relative version of our spectrum agrees with the Grothendieck spectrum for finitely generated commutative algebras over the complex numbers, among others. This work aims to represent the starting point for a rigorous study of geometric properties of quantum spacetimes. |
| title | Noncommutative Gelfand Duality: the algebraic case |
| topic | Algebraic Geometry Mathematical Physics Category Theory Quantum Algebra Primary: 14A22, Secondary 14A30, 16E35, 18F20 |
| url | https://arxiv.org/abs/2411.11816 |