nLab Hopf-Galois extension




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, ee (0)e (1)e\mapsto \sum e_{(0)}\otimes e_{(1)}.

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


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


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

Discussion for ring spectra:

Last revised on May 14, 2024 at 17:30:27. See the history of this page for a list of all contributions to it.