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Hopf-Galois extension

Hopf–Galois extensions

Definition

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. In noncommutative geometry, Hopf–Galois extensions are studied as noncommutative affine torsors, with interesting theorems involving descent theorems for Hopf modules generalizing 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.

Literature

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

Revised on December 30, 2009 03:43:11 by Toby Bartels (151.213.42.84)