Geometric and analytic structures on the higher ad\`eles

TL;DR

This paper proves that the topologically defined endomorphisms and the geometric Beilinson-Tate operators on higher local fields of schemes are equivalent, bridging topology and geometry in higher adèles.

Contribution

It establishes the equivalence between topological and geometric endomorphisms on higher local fields, confirming Yekutieli's conjecture.

Findings

Proves the equivalence of two notions of endomorphisms on higher local fields.

Bridges the gap between topological and geometric approaches in higher adèle theory.

Confirms Yekutieli's conjecture from 1992.

Abstract

The ad\`eles 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.