The Adelic Differential Graded Algebra for Surfaces
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For any variety X/k, we consider the Beilinson–Huber adeles AX as a differ- ential graded k-algebra and examine the category Moddg of differential graded A-modules. We characterize the modules associated to certain quasi-coherent sheaves and define an adelic Chern class c(M) for modules which are graded free of rank 1. We study the intersection pairing in terms of a cup product and prove a version of the Bloch–Quillen formula that respects this cup product.
Fesenko  proved Serre duality and the Riemann–Roch theorem for surfaces using a topological duality on the adeles. On the other hand, Mattuck–Tate  and Grothendieck  provided proofs of the Riemann hypothesis for curves using the Riemann–Roch theorem for surfaces by studying the graph of the Frobenius morphism on the surface S = C × C . Therefore the combined results of Fesenko and Mattuck–Tate–Grothendieck can be said to provide an adelic proof of the Riemann hypothesis for a curve C over a finite field. We apply the results of this thesis to the adelic intersection pairing, and state a version of the Hodge index theorem which implies the Riemann hypothesis for curves.