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Geometric and analytic structures on the higher adèles. (English) Zbl 1405.11148

Summary: The adèles of a scheme have local components – these are topological higher local fields. The topology plays a large role since A. Yekutieli showed in [An explicit construction of the Grothendieck residue complex. With an appendix by Pramathanath Sastry. Paris: Société Mathématique de France (1992; Zbl 0788.14011)] 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.

MSC:

11S31 Class field theory; \(p\)-adic formal groups
14G22 Rigid analytic geometry
11R56 Adèle rings and groups
13J10 Complete rings, completion
18E10 Abelian categories, Grothendieck categories

Citations:

Zbl 0788.14011
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