# Moscow Mathematical Journal

Volume 15, Issue 3, July–September 2015 pp. 435–453.

Geometric Adeles and the Riemann–Roch Theorem for 1-Cycles on Surfaces

**Authors**:
Ivan Fesenko

**Author institution:**School of Mathematical Sciences University of Nottingham, Nottingham NG7 2RD England

**Summary: **

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 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.

2010 Math. Subj. Class. 11R56, 14A99, 14C40, 14J99, 22A99, 57N17.

**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.

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