International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 54217, 27 pages
Derived Categories and the Analytic Approach to General Reciprocity
Department of Mathematics, Loyola Marymount University, Los Angeles 90045, CA, USA
Received 13 November 2006; Accepted 13 April 2007
Academic Editor: Pentti Haukkanen
Copyright © 2007 Michael C. Berg. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Building on the topological foundations constructed in Part I, 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 -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 -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
-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 -structures to three of seven specially chosen derived categories, the collapse just mentioned is enough to
yield -Hilbert reciprocity.