Higher adeles and non-abelian Riemann–Roch

Abstract

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 Fröhlich theory of the Galois module structure of rings of integers of number fields to arithmetic surfaces. ContentsIntroduction §1. Beilinson-Parshin adeles on a surface §2. Equivariant adelic Chow groups §3. Lattices, determinant functors and determinant theories §4. Pushdown maps and reciprocity laws §5. Transition matrices and the first Chern class §6. Elementary structures and the second Chern class §7. Equivariant Euler characteristics and the Riemann-Roch theorem §8. The proof of the theorem; reduction to the case of P 1 Z §9. The proof of the theorem; bundles over P 1 Z §10. Appendix: Adelic Riemann-Roch for general bundles when G = 1 References

Higher Chern classes in Iwasawa theory

“…To generalize Theorem 5.2.5, we first extend the approach to Chern classes used in Subsection 2.1 to the context of non-commutative algebras which are finite over their centers. (For related work on non-commutative Chern classes, see [6]. )…”

K1 of a p-adic group ring II. The determinantal kernel SK1

“…In Section 6.3, we also show that Corollary 1.9 implies: These and other similar results play an important role in the proof of the adelic Riemann-Roch theorem of [4] and in recent work of M. Witte [29].…”

A higher-dimensional Contou-Carrère symbol: local theory

“…] be the dilogarithm. The next lemma is directly implied by formula (11). (ii) For any f ∈ L n (A) ♯ ∩ L n (A) * , there is an equality…”