
TL;DR
This paper explores the structure of the telescopic Picard group at various heights and primes, revealing new non-Abelian Galois extensions of the $K(n)$-local sphere through higher categorical methods.
Contribution
It introduces a higher categorical framework for the periodicity theorem and constructs the first non-Abelian Galois extensions of the $K(n)$-local sphere in the telescopic setting.
Findings
Identifies a subgroup of the telescopic Picard group with specific algebraic structure.
Constructs a non-Abelian Galois extension of the $K(n)$-local sphere.
Provides a new approach using higher categories to study Picard groups and Galois extensions.
Abstract
We prove that for any prime and height , the telescopic Picard group contains a subgroup of the form , where if and if is odd. Using Kummer theory, we obtain an -Galois extension of , obtaining the first example of a lift of a non-Abelian Galois extension of the -local sphere to the telescopic world, at arbitrary positive height and prime. Our proof proceeds by setting up a higher categorical framework for the periodicity theorem, utilizing the symmetries of this framework to construct Picard elements.
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Meromorphic and Entire Functions · Mathematics and Applications
