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

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

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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.

Item Type: Article
Additional Information: Copyright Independent University of Moscow 2015
Keywords: Higher adeles, Geometric adelic structure on surfaces, Higher topologies, Non locally compact groups, Linear topological selfduality, Adelic Euler characteristic, Intersection pairing, Riemann–Roch theorem
Schools/Departments: University of Nottingham, UK > Faculty of Science > School of Mathematical Sciences
Depositing User: Fesenko, Professor Ivan
Date Deposited: 07 Oct 2015 08:45
Last Modified: 04 May 2020 20:06

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