Algebra/Topology Seminar 04122020

Speaker: Shachar Carmeli

Title: Cyclotomic Extensions in Chromatic Homotopy Theory

Abstract: In classical algebra, cyclotomic extensions are an essential source of Galois extensions for fields and rings.  

In Chromatic homotopy theory, the p-complete sphere spectrum is studied through various localizations, known as the K(n)-local spheres. Closely related are the telescopic localizations, known as the T(n)-local spheres. It is an open problem whether the two families of localizations coincide. 

In my talk, I will present a generalization, based on ambidexterity, of the classical theory of cyclotomic extensions, suitable for producing non-trivial Galois extensions in the T(n)-local and K(n)-local context. This construction gives a new family of Galois extensions of the T(n)-local sphere and allows to lift the well known maximal abelian extension of the K(n)-local sphere to the T(n)-local world. 

I will then describe some applications, including a chromatic version of the Fourier transform and affine-ness results for local systems valued in T(n)-local spectra. 

This is a joint project with Tomer Schlank and Lior Yanovski.