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Definition

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 $\rho:E\to E\otimes H$, $e\mapsto \sum e_{(0)}\otimes e_{(1)}$.

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 $e\otimes e'\mapsto (e\otimes_k 1)\rho(e') = \sum e e'_{(0)}\otimes e'_{(1)}$ 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.

Classical Galois extensions as a special case

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 $K\neq (k[G])^*$.

Role in geometry

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 descent 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 $V$.

Literature

A class of Hopf-Galois extensions admitting a cleaving map is dedicated a separate entry, cleft extension.

Early papers

• H. F. Kreimer, Mitsuhiro Takeuchi, Hopf algebras and Galois extensions of an algebra, Indiana Univ. Math. J. 30 (1981), 615-692 web pdf djvu

• Y. Doi, M. Takeuchi, Hopf-Galois extensions of algebras, the Miyashita-Ulbrich action, and Azumaya algebras, J. Algebra 121 (1989) 488–516

Schneider's descent theorem for Hopf-Galois extensions is proven in

• Hans-Jürgen Schneider, Principal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math. 72 (1990) 1-2, 167–195 MR92a:16047 doi

Surveys

• Susan Montgomery, Hopf Galois theory: a survey, Geometry and topology monographs 16 (2009) 367–400; link, doi

Newer references

• Stefaan Caenepeel, Septimiu Crivei, Andrei Marcus, Mitsuhiro Takeuchi, Morita equivalences induced by bimodules over Hopf-Galois extensions, J. Algebra 314 (2007) 267–302 pdf

• Peter Schauenburg, Hopf bimodules over Hopf-Galois extensions, Miyashita–Ulbrich actions, and monoidal center constructions, Comm. Algebra 24 (1996) 143–163 doi

• Peter Schauenburg, Hopf-Galois and bi-Galois extensions, from: “Galois theory, Hopf

  algebras, and semiabelian categories”, (G Janelidze, B Pareigis, W Tholen, editors), Fields Inst. Commun. 43, Amer. Math. Soc. (2004) 469–515 MR2075600

Categorifications and homotopifications

• Kathryn Hess, Homotopic Hopf-Galois extensions: foundations and examples, Geometry & Topology Monographs 16 (2009) 79–132 pdf arXiv:0902.3393

Discussion for ring spectra:

• John Rognes, Galois extensions of structured ring spectra/Stably dualizable groups, Memoirs of the American Mathematical Society, 192(898), 2008, (partial pdf, doi:10.1090/memo/0898)