nLab Hopf-Galois extension

HopfGalois extensions

Hopf–Galois extensions


Let HH be a kk-bialgebra and EE, say, a right HH-comodule algebra (i.e. a monoid in the category of right HH-comodules) with coaction ρ:EEH\rho:E\to E\otimes H.

The subalgebra U=E coHU = E^{\mathrm{co}H} of HH-coinvariants in EE consists of all uEu\in E such that ρ(u)=u1\rho(u)=u\otimes 1.

The kk-algebra extension UEU\hookrightarrow E is Hopf–Galois over HH if the natural map E UEEHE\otimes_U E\to E\otimes H given by the kk-linear extension of the formula ee(e k1)ρ(e)e\otimes e'\mapsto (e\otimes_k 1)\rho(e') is a bijection (hence a kk-module isomorphism).

A Hopf–Galois object over a kk-bialgebra HH is any Hopf-Galois extension kEk\hookrightarrow E over HH of the ground field (or ring) kk. It is a dual (and noncommutative) analogue to a torsor over a point.

Classical Galois extensions as a special case

If kU=E Gk\subset U=E^G, k,Ek,E are fields, GG a finite group and H=(k[G]) *H = (k[G])^* is the dual Hopf algebra to the group algebra of GG, then E GEE^G\hookrightarrow E is (classically) a Galois extension iff it is a HH-Hopf–Galois extension, where the coaction of HH is induced by the action of k[G]k[G], hence of GG. One uses the Dedekind lemma on independence of automorphisms to prove this equivalence. It is possible however that E GEE^G\subset E is not (classically) Galois, but it is KK-Hopf–Galois for some Hopf algebra K(k[G]) *K\neq (k[G])^*.

Role in geometry

In algebraic geometry, given an affine algebraic kk-group scheme GG, the algebra EE of regular functions over the total scheme XX of an affine GG-torsor XYX\to Y, whose base YY also happens to be affine, is a commutative HH-Hopf–Galois extension of the algebra of regular functions UU on the base YX/GY\cong X/G, where HH is the kk-Hopf algebra of global regular functions on GG. 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 HH-Hopf-Galois extension UEU\hookrightarrow E and a left HH-comodule VV the cotensor product kk-module E HVE\Box^H V is interpreted as a space of sections of the associated fiber bundle with structure group SpecHSpec H (in noncommutative sense) and fiber VV.


  • 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.