Julián Komáromy: Galois theory in higher algebra

Abstract: In this thesis, we introduce Galois theory in higher algebra and show that it arises naturally from a reformulation of classical Galois theory. To motivate the definition of the Galois category of an E∞-ring, we study the Galois theory of classical rings and introduce Mathew’s axiomatic Galois theory. This itself is a reformulation of Grothendieck’s original Galois theory for schemes.
Higher algebra, especially E∞-rings and modules over them, are briefly introduced. Using this language, we define (weak) finite covers of an E∞-ring, which, together with the axiomatic Galois theory introduced previously, allows us to assign a Galois group(oid) to an E∞-ring. We compare the resulting theory to Rognes’ original theory of G-Galois extensions and show that G-torsors in the category of weak finite covers are precisely the faithful G-Galois extensions. We reprove some of the key theorems of Rognes using∞-categorical language. Finally, real and complex K-theory spectra are introduced as E∞-rings and parts of Rognes’ classic proof that complexification KO → KU is a Galois extension of ring spectra are proven. This provides a non-algebraic example of a C2-Galois extension.