11 research outputs found
Chromatic Cyclotomic Extensions
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.Comment: 48 pages. Comments are welcome
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Quantum Groups - Algebra, Analysis and Category Theory (hybrid meeting)
The meeting was devoted to discussing the state of the art of different branches of tensor categories and quantum groups, with emphasis on the exchange of ideas between the purely algebraic and operator algebraic sides of these theories
Ambidexterity and height
We introduce and study the notion of \emph{semiadditive height} for higher semiadditive -categories, which generalizes the chromatic height. We show that the higher semiadditive structure trivializes above the height and prove a form of the redshift principle, in which categorification increases the height by one. In the stable setting, we show that a higher semiadditive -category decomposes into a product according to height, and relate the notion of height to semisimplicity properties of local systems. We place the study of higher semiadditivity and stability in the general framework of smashing localizations of , which we call \emph{modes}. Using this theory, we introduce and study the universal stable -semiadditive -category of semiadditive height , and give sufficient conditions for a stable -semiadditive -category to be -semiadditive
Cartier modules and cyclotomic spectra
We construct and study a t-structure on p-typical cyclotomic spectra and
explain how to recover crystalline cohomology of smooth schemes over perfect
fields using this t-structure. Our main tool is a new approach to p-typical
cyclotomic spectra via objects we call p-typical topological Cartier modules.
Using these, we prove that the heart of the cyclotomic t-structure is the full
subcategory of derived V-complete objects in the abelian category of p-typical
Cartier modules.Comment: Final version; to appear in J. AM
Stability and Arithmetic
Stability plays a central role in arithmetic. In this article, we explain
some basic ideas and present certain constructions for such studies. There are
two aspects: namely, general Class Field Theories for Riemann surfaces using
semi-stable parabolic bundles & for p-adic number fields using what we call
semi-stable filtered (phi,N;omega)-modules; and non-abelian zeta functions for
function fields over finite fields using semi-stable bundles & for number
fields using semi-stable lattices.Comment: 121 page
Faithfulness of certain modules and residual nilpotence of groups
Let F be a non-cyclic free group and R, S its normal subgroups. We study the abelian group H∩S/[H,S], , viewed as a module over F/RS, via conjugation in F, and residual nilpotence of the group F/[R, S]. An application to the asphericity of finite presentations is given