  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.
