Terminal orders on arithmetic surfaces
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866910658394587136 |
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| author | Chan, Daniel Ingalls, Colin |
| author_facet | Chan, Daniel Ingalls, Colin |
| contents | The local structure of terminal Brauer classes on arithmetic surfaces were classified in [CI21] generalising the classification on geometric surfaces carried out in [CI05]. Part of the interest in these classifications is that it enables the minimal model program to be applied to the noncommutative setting of orders on surfaces. In this paper, we give etale local structure theorems for terminal orders on arithemtic surfaces, at least when the degree is a prime p >5. This generalises the structure theorem given in the geometric case. They can all be explicitly constructed as algebras of matrices over symbols. From this description one sees that such terminal orders all have global dimension two, thus generalising the fact that terminal (commutative) surfaces are smooth and hence homologically regular. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_05948 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Terminal orders on arithmetic surfaces Chan, Daniel Ingalls, Colin Rings and Algebras Algebraic Geometry 16H10, 16S38 The local structure of terminal Brauer classes on arithmetic surfaces were classified in [CI21] generalising the classification on geometric surfaces carried out in [CI05]. Part of the interest in these classifications is that it enables the minimal model program to be applied to the noncommutative setting of orders on surfaces. In this paper, we give etale local structure theorems for terminal orders on arithemtic surfaces, at least when the degree is a prime p >5. This generalises the structure theorem given in the geometric case. They can all be explicitly constructed as algebras of matrices over symbols. From this description one sees that such terminal orders all have global dimension two, thus generalising the fact that terminal (commutative) surfaces are smooth and hence homologically regular. |
| title | Terminal orders on arithmetic surfaces |
| topic | Rings and Algebras Algebraic Geometry 16H10, 16S38 |
| url | https://arxiv.org/abs/2211.05948 |