 
Summary: GUIDO'S BOOK OF CONJECTURES 1
1. THE CHROMATIC REDSHIFT IN ALGEBRAIC K THEORY
Christian Ausoni and John Rognes
The algebraic K theory of the sphere spectrum S is of interest in geometric
topology, by Waldhausen's stable parametrized hcobordism theorem [WJR]
(ca. 1979). We wish to understand KS like we understand KZ, via Galois
descent. As a building block, the algebraic K theory of the Bousfield
localization LK(n)S of S with respect to the nth Morava Ktheory K(n)
might be more accessible. John has developed a theory of Galois extensions for
Salgebras, and in this framework he has stated extensions of the Lichtenbaum
Quillen conjectures. Their precise formulation is distilled from the clues
provided by our computations of the algebraic K theory of topological K 
theory and related spectra, and it is to be expected that they will keep maturing
in a cask of skepticism for a few years. Writing XhG
for the homotopy fixed
point spectrum of a finite group G acting on a spectrum X, we recall:
DEFINITION 1.1 ([Ro]). A map A B of commutative Salgebras is
a K(n)local GGalois extension if G acts on B through commutative A
algebra maps, and the canonical maps A BhG
and B A B G B are
