Differentiably simple rings and ring extensions defined by $p$-basis
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866912061163831296 |
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| author | de Andrés, Celia del Buey Sulca, Diego Villamayor, Orlando E. |
| author_facet | de Andrés, Celia del Buey Sulca, Diego Villamayor, Orlando E. |
| contents | We review the concept of differentiably simple ring and we give a new proof of Harper's Theorem on the characterization of Noetherian differentiably simple rings in positive characteristic. We then study flat families of differentiably simple rings, or equivalently, finite flat extensions of rings which locally admit $p$-basis. These extensions are called "Galois extensions of exponent one". For such an extension $A\subset C$, we introduce an $A$-scheme, called the "Yuan scheme", which parametrizes subextensions $A\subset B\subset C$ such that $B\subset C$ is Galois of a fixed rank. So, roughly, the Yuan scheme can be thought of as a kind of Grassmannian of Galois subextensions. We finally prove that the Yuan scheme is smooth and compute the dimension of the fibers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_09125 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Differentiably simple rings and ring extensions defined by $p$-basis de Andrés, Celia del Buey Sulca, Diego Villamayor, Orlando E. Commutative Algebra Algebraic Geometry 13B05, 14M15, 14L17 We review the concept of differentiably simple ring and we give a new proof of Harper's Theorem on the characterization of Noetherian differentiably simple rings in positive characteristic. We then study flat families of differentiably simple rings, or equivalently, finite flat extensions of rings which locally admit $p$-basis. These extensions are called "Galois extensions of exponent one". For such an extension $A\subset C$, we introduce an $A$-scheme, called the "Yuan scheme", which parametrizes subextensions $A\subset B\subset C$ such that $B\subset C$ is Galois of a fixed rank. So, roughly, the Yuan scheme can be thought of as a kind of Grassmannian of Galois subextensions. We finally prove that the Yuan scheme is smooth and compute the dimension of the fibers. |
| title | Differentiably simple rings and ring extensions defined by $p$-basis |
| topic | Commutative Algebra Algebraic Geometry 13B05, 14M15, 14L17 |
| url | https://arxiv.org/abs/2211.09125 |