Derived categories and the analytic approach to general reciprocity laws. II
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.
Kubota's \(n\)-Hilbert reciprocity formalism
linked exact triples of derived sheaf categories
scientific article; zbMATH DE number 5282507
scientific article