Geometric and analytic structures on the higher adèles

Abstract

The adèles of a scheme have local components - these are topological higher local fields. The topology plays a large role since Yekutieli showed in 1992 that there can be an abundance of inequivalent topologies on a higher local field and no canonical way to pick one. Using the datum of a topology, one can isolate a special class of continuous endomorphisms. Quite differently, one can bypass topology entirely and single out special endomorphisms (global Beilinson-Tate operators) from the geometry of the scheme. Yekutieli's "Conjecture 0.12" proposes that these two notions agree. We prove this.

arXiv commentThe final version of this preprint was published as: Braunling, O., Groechenig, M. & Wolfson, J. Res Math Sci (2016) 3: 22. https://doi.org/10.1186/s40687-016-0064-y