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

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 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.

Adelic geometry on arithmetic surfaces II: Completed adeles and idelic Arakelov intersection theory

“…Now we show that the spaces A 01 and A 02 are equal to their orthogonal complements. Compare these results with the "geometric counterpart" in [10].…”

“…The inclusions A 01 ⊆ A ⊥ 01 and A 02 ⊆ A ⊥ 02 are a direct consequence of the completed reciprocity laws, thus the self-orthogonality of A 01 and A 02 can be interpreted as "strong reciprocity laws" for arithmetic surfaces. The "strong reciprocity laws" for surfaces over a perfect field were proved in [10].…”

“…When X is an algebraic surface over a perfect field k, the algebraic and topological properties of the subspaces A * were studied in [10].…”

“…However, the explicit adelic structures introduced in [20] are not equivalent to Beilinson's, since [20] worked with objects that now are called rational adeles. The gap on the definitions was partially fixed in [21], but a complete account of 2-dimensional explicit adelic theory was given by Fesenko in [10], where he also proved an adelic Riemann-Roch theorem for an algebraic surface over a perfect field by using properties of adelic cohomology. In particular, Fesenko showed that the function field of an algebraic surface X can be seen as a discrete subspace inside the ring of 2-dimensional adeles attached to X.…”

“…The topology on A X is obtained after a process of several inductive an projective limits by starting from the local topologies on all K x,y . In [10] it is shown that A X is self-dual as k-vector space. For 2-dimensional local fields with the same structure of K x,y there is a well known theory of differential forms and residues (e.g.…”