The Étale Spectrum of Non-commutative Rings
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2025
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| author | Revista, Zen MATH, 10 |
| author_facet | Revista, Zen MATH, 10 |
| contents | This paper explores the theoretical construction and properties of the étale spectrum in the context of non-commutative rings. While the classical Zariski spectrum provides a foundational geometric insight for commutative algebra, its direct generalization to non-commutative settings is often inadequate due to the absence of prime ideals behaving as points. The étale topology, a powerful tool in commutative algebraic geometry for studying local properties and covering spaces, offers a promising avenue. We develop a framework for defining a suitable étale site for certain classes of non-commutative rings, leveraging categorical methods and the theory of non-commutative localization. The methodology involves identifying appropriate categories of non-commutative modules or representations that can serve as the underlying "objects" of the site, and then defining coverings based on faithful flatness or related concepts suitable for non-commutative structures. We investigate the resulting spectral space, its functorial properties, and its relationship to existing non-commutative spectra such as the primitive spectrum or the prime spectrum of specific non-commutative algebras. Through concrete examples, including Weyl algebras and enveloping algebras of Lie algebras, we demonstrate the applicability of this construction and highlight how the non-commutative étale spectrum can provide a richer geometric understanding of these algebras, revealing aspects not captured by other spectral theories. The discussion focuses on the implications for non-commutative algebraic geometry, representation theory, and potential connections to quantum group theory and mathematical physics. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17805966 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Étale Spectrum of Non-commutative Rings Revista, Zen MATH, 10 This paper explores the theoretical construction and properties of the étale spectrum in the context of non-commutative rings. While the classical Zariski spectrum provides a foundational geometric insight for commutative algebra, its direct generalization to non-commutative settings is often inadequate due to the absence of prime ideals behaving as points. The étale topology, a powerful tool in commutative algebraic geometry for studying local properties and covering spaces, offers a promising avenue. We develop a framework for defining a suitable étale site for certain classes of non-commutative rings, leveraging categorical methods and the theory of non-commutative localization. The methodology involves identifying appropriate categories of non-commutative modules or representations that can serve as the underlying "objects" of the site, and then defining coverings based on faithful flatness or related concepts suitable for non-commutative structures. We investigate the resulting spectral space, its functorial properties, and its relationship to existing non-commutative spectra such as the primitive spectrum or the prime spectrum of specific non-commutative algebras. Through concrete examples, including Weyl algebras and enveloping algebras of Lie algebras, we demonstrate the applicability of this construction and highlight how the non-commutative étale spectrum can provide a richer geometric understanding of these algebras, revealing aspects not captured by other spectral theories. The discussion focuses on the implications for non-commutative algebraic geometry, representation theory, and potential connections to quantum group theory and mathematical physics. |
| title | The Étale Spectrum of Non-commutative Rings |
| url | https://doi.org/10.5281/zenodo.17805966 |