LMAT2925 - Categorical Galois Theory

In this course we will explain Grothendieck’s version of Galois theory of separable
algebras over a field, as well as its infinitary version. We will then follow with an exposition of
the corresponding results for commutative rings.
We will then introduce George Janelidze’s categorical formalism for Galois
theory. We will first introduce descent theory and effective descent morphisms in a category.
Afterwards we will introduce the notion of Galois structure and trivial/central/normal extensions,
and define the Galois groupoid of a normal extension, before proving the fundamental theorem
of Categorical Galois Theory.
We will then present a few applications of the theory, and the links with commutator theory and Hopf formulae for homology.