Geometric adeles and the RiemannRoch theorem for 1cycles on surfacesTools Fesenko, Ivan (2015) Geometric adeles and the RiemannRoch theorem for 1cycles on surfaces. Moscow Mathematical Journal, 15 (3). pp. 435453. ISSN 16094514
AbstractThe classical Riemann–Roch theorem for projective irreducible curves over perfect fields can be elegantly proved using adeles and their topological selfduality. 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 twodimensional 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 selfduality 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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