Aka Parshin-Kato theory.
Kato ICM 1990: Generalized class field theory.
arXiv:1204.4520 Higher adeles and non-abelian Riemann-Roch from arXiv Front: math.KT by T. Chinburg, G. Pappas, M. J. Taylor 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.
Functional analysis on the eve of the 21st century Contains article by Kapranov: Analogies between the Langlands Correspondence and Topological QFT. Follow-up here at MO.
arXiv:1002.2698 A refinement of the Parshin symbol for surfaces from arXiv Front: math.AG by Ivan Horozov On an algebraic curve there are Tate symbols, which satisfy Weil reciprocity law. The analogues in higher dimensions are the Parshin symbols, which satisfy Kato-Parshin reciprocity laws. We give a refinement of the Parshin symbol for surfaces, using iterated integrals in the sense of Chen. The product of the refined symbol over the cyclic permutations of the functions recovers the Parshin symbol. Also, we construct a logarithmic version of the Parshin symbol. We prove reciprocity laws for both the refined symbol and a logarithm of the Parshin symbol.
arXiv:1006.4721 Formal completion of a category along a subcategory from arXiv Front: math.CT by Alexander I. Efimov Following an idea of Kontsevich, we introduce and study the notion of formal completion of a compactly generated (by a set of objects) enhanced triangulated category along a full thick essentially small triangulated subcategory
In particular, we prove (answering a question of Kontsevich) that using categorical formal completion, one can obtain ordinary formal completions of Noetherian schemes along closed subschemes. Moreover, we show that Beilinson-Parshin adeles can be also obtained using categorical formal completion.
nLab page on Parshin theory