The Adelic Differential Graded Algebra for Surfaces
| dc.contributor.advisor | Ramachandran, Niranjan | en_US |
| dc.contributor.author | Kelly, Sean James | en_US |
| dc.contributor.department | Mathematics | en_US |
| dc.contributor.publisher | Digital Repository at the University of Maryland | en_US |
| dc.contributor.publisher | University of Maryland (College Park, Md.) | en_US |
| dc.date.accessioned | 2017-06-22T06:00:56Z | |
| dc.date.available | 2017-06-22T06:00:56Z | |
| dc.date.issued | 2017 | en_US |
| dc.description.abstract | 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 [6] proved Serre duality and the Riemann–Roch theorem for surfaces using a topological duality on the adeles. On the other hand, Mattuck–Tate [18] and Grothendieck [12] 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. | en_US |
| dc.identifier | https://doi.org/10.13016/M2T296 | |
| dc.identifier.uri | http://hdl.handle.net/1903/19385 | |
| dc.language.iso | en | en_US |
| dc.subject.pqcontrolled | Mathematics | en_US |
| dc.title | The Adelic Differential Graded Algebra for Surfaces | en_US |
| dc.type | Dissertation | en_US |
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