Let $H$ be a $k$-bialgebra and $E$, say, a right $H$-comodule algebra (i.e. a monoid in the category of right $H$-comodules) with coaction $\rho:E\to E\otimes H$.
The subalgebra $U = E^{\mathrm{co}H}$ of $H$-coinvariants in $E$ consists of all $u\in E$ such that $\rho(u)=u\otimes 1$.
The $k$-algebra extension $U\hookrightarrow E$ is Hopf–Galois over $H$ if the natural map $E\otimes_U E\to E\otimes H$ given by the $k$-linear extension of the formula $e\otimes e'\mapsto (e\otimes_k 1)\rho(e')$ is a bijection (hence a $k$-module isomorphism).
A Hopf–Galois object over a $k$-bialgebra $H$ is any Hopf-Galois extension $k\hookrightarrow E$ over $H$ of the ground field (or ring) $k$. It is a dual (and noncommutative) analogue to a torsor over a point.
If $k\subset U=E^G$, $k,E$ are fields, $G$ a finite group and $H = (k[G])^*$ is the dual Hopf algebra to the group algebra of $G$, then $E^G\hookrightarrow E$ is (classically) a Galois extension iff it is a $H$-Hopf–Galois extension, where the coaction of $H$ is induced by the action of $k[G]$, hence of $G$. One uses the Dedekind lemma on independence of automorphisms to prove this equivalence. It is possible however that $E^G\subset E$ is not (classically) Galois, but it is $K$-Hopf–Galois for some Hopf algebra $K\neq (k[G])^*$.
In algebraic geometry, given an affine algebraic $k$-group scheme $G$, the algebra $E$ of regular functions over the total scheme $X$ of an affine $G$-torsor $X\to Y$, whose base $Y$ also happens to be affine, is a commutative $H$-Hopf–Galois extension of the algebra of regular functions $U$ on the base $Y\cong X/G$, where $H$ is the $k$-Hopf algebra of global regular functions on $G$. In algebraic topology, a generalization to spectra (with the smash product of spectra in the role of tensor product) was studied by Rognes and others (see (Rognes 08)). In noncommutative geometry, Hopf–Galois extensions are studied as affine noncommutative principal bundles, with interesting descent theorems for Hopf modules like the Schneider's theorem. Given a right $H$-Hopf-Galois extension $U\hookrightarrow E$ and a left $H$-comodule $V$ the cotensor product $k$-module $E\Box^H V$ is interpreted as a space of sections of the associated fiber bundle with structure group $Spec H$ (in noncommutative sense) and fiber $V$.
H. F. Kreimer, M. Takeuchi, Hopf algebras and Galois extensions of an algebra, Indiana Univ. Math. J. 30 (1981), 615-692 web pdf djvu
Hans-Jürgen Schneider, Principal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math. 72 (1990) 1-2, 167–195 MR92a:16047 doi
Susan Montgomery, Hopf Galois theory: a survey, Geometry and topology monographs 16 (2009) 367–400; link, doi
Discussion for ring spectra:
Last revised on September 17, 2023 at 14:20:31. See the history of this page for a list of all contributions to it.