Akhil Mathew (Harvard): Descent theorems in algebraic K-theory

In this talk, we will present some applications of the "transfer" to algebraic K-theory, inspired by the work of Thomason. Let A --> B be a G-Galois extension of rings, or more generally of E-infinity ring spectra in the sense of Rognes. A basic question in algebraic K-theory asks how close the map K(A) --> K(B)^hG is to being an equivalence, i.e., how close K is to satisfying Galois descent. Motivated by the classical descent theorem of Thomason, one also expects such a result after "periodic" localization. We formulate and prove a general lemma that enables one to translate rational descent statements as above into descent statements after telescopic localization. As a result, we prove various descent results in the telescopically localized K-theory, TC, etc. of ring spectra, and verify several cases of a conjecture of Ausoni-Rognes. This is joint work with Dustin Clausen, Niko Naumann, and Justin Noel.