Chromatic Cyclotomic Extensions

Abstract

We construct Galois extensions of the T(n)-local sphere, lifting all finite abelian Galois extensions of the K(n)-local sphere. This is achieved by realizing them as higher semiadditive analogues of cyclotomic extensions. Combining this with a general form of Kummer theory, we lift certain elements from the K(n)-local Picard group to the T(n)-local Picard group.

arXiv comment53 pages. Edited in response to comments of the referee. Fixed typos, some arguments expanded and made more precise, added details in section 5.2. Accepted to GnT

References in corpus (13)

• Universality of multiplicative infinite loop space machines
• Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map
• The Picard group of topological modular forms via descent theory
• On Hopkins' Picard groups for the prime 3 and chromatic level 2
• Topological resolutions in K(2)-local homotopy theory at the prime 2
• The linearity of traces in monoidal categories and bicategories
• A higher chromatic analogue of the image of J
• The telescope conjecture at height 2 and the tmf resolution
• Stably dualizable groups
• Purity in chromatically localized algebraic $K$-theory
• The Morava E-theory of Eilenberg-Mac Lane spaces
• Roots of unity in $K(n)$-local rings
• Adjoining roots in homotopy theory

Cited by in corpus (5)

• Higher semiadditive Grothendieck-Witt theory and the $K(1)$-local sphere
• Characters and transfer maps via categorified traces
• On the parametrized Tate construction
• On the Non-Existence of J-Homomorphisms of Higher Height
• The inverse limit topology and profinite descent on Picard groups in $K(n)$-local homotopy theory