Geometric adeles and the Riemann-Roch theorem for 1-cycles on surfaces
The classical Riemann–Roch theorem for projective irreducible curves over perfect fields can be elegantly proved using adeles and their topological self-duality. This was known already to E. Artin and K. Iwasawa and can be viewed as a relation between adelic geometry and algebraic geometry in dimens...
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| Format: | Article |
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Independent University of Moscow
2015
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| Online Access: | https://eprints.nottingham.ac.uk/30393/ |
| _version_ | 1848793975757996032 |
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| author | Fesenko, Ivan |
| author_facet | Fesenko, Ivan |
| author_sort | Fesenko, Ivan |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | The classical Riemann–Roch theorem for projective irreducible curves over perfect fields can be elegantly proved using adeles and their topological self-duality. This was known already to E. Artin and K. Iwasawa and can be viewed as a relation between adelic geometry and algebraic geometry in dimension one. In this paper we study geo- metric two-dimensional adelic objects, endowed with appropriate higher topology, on algebraic proper smooth irreducible surfaces over perfect fields. We establish several new results about adelic objects and prove topological self-duality of the geometric adeles and the discreteness of the function field. We apply this to give a direct proof of finite dimen- sion of adelic cohomology groups. Using an adelic Euler characteristic we establish an additive adelic form of the intersection pairing on the surfaces. We derive a direct and relatively short proof of the adelic Riemann–Roch theorem. Combining with the relation between adelic and Zariski cohomology groups, this also implies the Riemann–Roch theorem for surfaces. |
| first_indexed | 2025-11-14T19:08:50Z |
| format | Article |
| id | nottingham-30393 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:08:50Z |
| publishDate | 2015 |
| publisher | Independent University of Moscow |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-303932020-05-04T20:06:53Z https://eprints.nottingham.ac.uk/30393/ Geometric adeles and the Riemann-Roch theorem for 1-cycles on surfaces Fesenko, Ivan The classical Riemann–Roch theorem for projective irreducible curves over perfect fields can be elegantly proved using adeles and their topological self-duality. This was known already to E. Artin and K. Iwasawa and can be viewed as a relation between adelic geometry and algebraic geometry in dimension one. In this paper we study geo- metric two-dimensional adelic objects, endowed with appropriate higher topology, on algebraic proper smooth irreducible surfaces over perfect fields. We establish several new results about adelic objects and prove topological self-duality of the geometric adeles and the discreteness of the function field. We apply this to give a direct proof of finite dimen- sion of adelic cohomology groups. Using an adelic Euler characteristic we establish an additive adelic form of the intersection pairing on the surfaces. We derive a direct and relatively short proof of the adelic Riemann–Roch theorem. Combining with the relation between adelic and Zariski cohomology groups, this also implies the Riemann–Roch theorem for surfaces. Independent University of Moscow 2015-10 Article PeerReviewed Fesenko, Ivan (2015) Geometric adeles and the Riemann-Roch theorem for 1-cycles on surfaces. Moscow Mathematical Journal, 15 (3). pp. 435-453. ISSN 1609-4514 Higher adeles Geometric adelic structure on surfaces Higher topologies Non locally compact groups Linear topological selfduality Adelic Euler characteristic Intersection pairing Riemann–Roch theorem http://www.mathjournals.org/mmj/2015-015-003/2015-015-003-003.html |
| spellingShingle | Higher adeles Geometric adelic structure on surfaces Higher topologies Non locally compact groups Linear topological selfduality Adelic Euler characteristic Intersection pairing Riemann–Roch theorem Fesenko, Ivan Geometric adeles and the Riemann-Roch theorem for 1-cycles on surfaces |
| title | Geometric adeles and the Riemann-Roch theorem for 1-cycles on surfaces |
| title_full | Geometric adeles and the Riemann-Roch theorem for 1-cycles on surfaces |
| title_fullStr | Geometric adeles and the Riemann-Roch theorem for 1-cycles on surfaces |
| title_full_unstemmed | Geometric adeles and the Riemann-Roch theorem for 1-cycles on surfaces |
| title_short | Geometric adeles and the Riemann-Roch theorem for 1-cycles on surfaces |
| title_sort | geometric adeles and the riemann-roch theorem for 1-cycles on surfaces |
| topic | Higher adeles Geometric adelic structure on surfaces Higher topologies Non locally compact groups Linear topological selfduality Adelic Euler characteristic Intersection pairing Riemann–Roch theorem |
| url | https://eprints.nottingham.ac.uk/30393/ https://eprints.nottingham.ac.uk/30393/ |