# 2021-06-20

## 21:57

• $$K^ \mathsf{Alg} ({\mathbb{S}})$$ is of interest due to Waldhausen’s stable parameterized h-cobordism theorem.

• How to understand it: look at Galois descent on $${\mathsf{K}}({\mathbf{Z}})$$
• A first approximation: $${\mathsf{K}}^ \mathsf{Alg} (L_{K(n)} {\mathbb{S}})$$, so localize with respect to Morava K-theory

• Rognes extends Lichtenbaum-Quillen conjectures and develops a theory for Galois extensions of $${\mathbb{S}}{\hbox{-}}$$algebras

• Redshift : algebraic K-theory increases chromatic complexity by one.

#web/quick-notes #homotopy/stable-homotopy #higher-algebra/K-theory