Higher adeles and non-abelian Riemann-Roch

Abstract

We show a Riemann-Roch theorem for group ring bundles over an arithmetic surface; this is expressed using the higher adeles of Beilinson-Parshin and the tame symbol via a theory of adelic equivariant Chow groups and Chern classes. The theorem is obtained by combining a group ring coefficient version of the local Riemann-Roch formula as in Kapranov-Vasserot with results on K-groups of group rings and an explicit description of group ring bundles over P^1. Our set-up provides an extension of several aspects of the classical Fr"ohlich theory of the Galois module structure of rings of integers of number fields to arithmetic surfaces.

77 pp, various further corrections and changes

References in corpus (7)

• Infinite-dimensional vector bundles in algebraic geometry (an introduction)
• Noncommutative localisation in algebraic K-theory I
• An introduction to higher dimensional local fields and adeles
• Integral Grothendieck-Riemann-Roch theorem
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Cited by in corpus (2)

• K_1 of a p-adic group ring II. The determinantal kernel SK_1
• Adams operations and Galois structure