Derived categories and the analytic approach to general reciprocity laws. II
type: Item
instance of: scholarly article
title: Derived categories and the analytic approach to general reciprocity laws. II
zbMATH Open document ID: 1135.11063
DOI: 10.1155/2007/54217
author: Michael Berg
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2008-06-03T00:00:00Z
full work available at URL: https://eudml.org/doc/53985
review text: Summary: Building on the topological foundations constructed in Part I [Int. J. Math. Math. Sci. 2005, No. 13, 2133--2158 (2005; Zbl 1081.11074)], we now go on to address the homological algebra preparatory to the projected final arithmetical phase of our attack on the analytic proof of general reciprocity for a number field. In the present work, we develop two algebraic frameworks corresponding to two interpretations of Kubota's \(n\)-Hilbert reciprocity formalism, presented in a quasi-dualized topological form in Part I, delineating two sheaf-theoretic routes toward resolving the aforementioned (open) problem. The first approach centers on factoring sheaf morphisms eventually to yield a splitting homomorphism for Kubota's \(n\)-fold cover of the adelized special linear group over the base field. The second approach employs linked exact triples of derived sheaf categories and the yoga of gluing \(t\)-structures to evolve a means of establishing the vacuity of certain vertices in diagrams of underlying topological spaces from Part I. Upon assigning properly designed \(t\)-structures to three of seven specially chosen derived categories, the collapse just mentioned is enough to yield \(n\)-Hilbert reciprocity.
Mathematics Subject Classification ID: 11R47
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11R56
zbMATH DE Number: 5282507
zbMATH Keywords: Kubota's \(n\)-Hilbert reciprocity formalism
zbMATH Keywords: linked exact triples of derived sheaf categories
MaRDI profile type: Publication
OpenAlex ID: W4252162205
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description: scientific article; zbMATH DE number 5282507