STNB2026 (39a edició)
Hopf-Galois extensions and Hopf-Galois objects
Ponents
Cornelius Greither
Resum
This is the first of two lectures in which will try to give an overview of some basic topics in Hopf Galois theory. Building on the introductory lecture by Daniel Gil Mu\~noz, we start by introducing Hopf Galois situations, where a $K$-Hopf algebra $H$ provides ``symmetries'' on a (finite-dimensional commutative) $K$-algebra $A$. This is explained in two versions, which will turn out to be essentially equivalent. First we describe $H$-Galois extensions, where $H$ acts on $A$ (this is closer to the classical Galois situation), and second we present $H$-Galois objects, where $A$ is an $H$-comodule algebra; this setup is closer in spirit to the action of algebraic groups on varieties. We will recall a very explicit description of algebras and Hopf Galois objects in the language of finite $\Gamma_K$-sets and $\Gamma_K$-groups. In other words we try to understand zerodimensional varieties and group varieties over $K$ (which is usually a number field, hence far from algebraically closed), and the actions of the latter on the former. A generalized Galois correspondence will be discussed briefly.
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