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Geometric adeles and the Riemann–Roch theorem for 1-cycles on surfaces

FESENKO, IVAN

Authors

IVAN FESENKO



Abstract

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 geometric 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 dimension of adelic cohomology groups. Using an adelic Euler characteristic we establish an additive adelic form of the intersection pairing on the surfaces. We derive the first adelic proof of the adelic Riemann–Roch theorem, which is also direct and relatively short. Combining with the relation between adelic and Zariski cohomology groups, this also implies the Riemann–Roch theorem for surfaces.

Citation

FESENKO, I. (2015). Geometric adeles and the Riemann–Roch theorem for 1-cycles on surfaces. Moscow Mathematical Journal, 15(3), 435-453. https://doi.org/10.17323/1609-4514-2015-15-3-435-453

Journal Article Type Article
Acceptance Date Feb 1, 2015
Online Publication Date Sep 1, 2015
Publication Date Sep 1, 2015
Deposit Date Mar 8, 2018
Journal Moscow Mathematical Journal
Print ISSN 1609-3321
Electronic ISSN 1609-4514
Publisher Nezavisimyi Moskovskii Universitet
Peer Reviewed Peer Reviewed
Volume 15
Issue 3
Pages 435-453
DOI https://doi.org/10.17323/1609-4514-2015-15-3-435-453
Keywords Higher adeles, geometric adelic structure on sur- faces, higher topologies, non locally compact groups, linear topological self- duality, adelic Euler characteristic, intersection pairing, Riemann–Roch theorem
Public URL http://www.mathjournals.org/mmj/2015-015-003/2015-015-003-003.html
Publisher URL http://www.mathjournals.org/mmj/2015-015-003/2015-015-003-003.html
Related Public URLs http://www.mathjournals.org/mmj/

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