Student AG Seminar: Galois descent and the Picard group of K-theory

The Picard group of a ring is a powerful invariant worth computing. Galois descent is a powerful tool for passing information from a larger and more understandable ring to a more complicated and often more interesting ring. In this talk, we will use Galois descent to compute the derived Picard group by passing to a homotopical context. We will introduce a useful generalization of the Picard group and discuss how it interacts with the Galois theory of rings and ring spectra. Finally, we will discuss a famous Galois extension involving topological K-theory to compute the Picard group of real K-theory, \$\operatorname{Pic}(KO)\$.