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
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.
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