The subalgebra of -coinvariants in consists of all such that .
The -algebra extension is Hopf–Galois over if the natural map given by the -linear extension of the formula is a bijection (hence a -module isomorphism).
A Hopf–Galois object over a -bialgebra is any Hopf-Galois extension over of the ground field (or ring) . It is a dual (and noncommutative) analogue to a torsor over a point.
If , are fields, a finite group and is the dual Hopf algebra to the group algebra of , then is (classically) a Galois extension iff it is a -Hopf–Galois extension, where the coaction of is induced by the action of , hence of . One uses the Dedekind lemma on independence of automorphisms to prove this equivalence. It is possible however that is not (classically) Galois, but it is -Hopf–Galois for some Hopf algebra .
In algebraic geometry, given an affine algebraic -group scheme , the algebra of regular functions over the total scheme of an affine -torsor , whose base also happens to be affine, is a commutative -Hopf–Galois extension of the algebra of regular functions on the base , where is the -Hopf algebra of global regular functions on . 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. 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 -Hopf-Galois extension and a left -comodule the cotensor product -module is interpreted as a space of sections of the associated fiber bundle with structure group (in noncommutative sense) and fiber .