Hopf Galois Extensions Research Articles

Hochschild cohomology on Hopf Galois extensions

For a Hopf Galois extension, A B , we construct spectral sequences connecting the Hochschild cohomologies and homologies of A and of B.

Hopf frobenius extensions of algebras

Let AlB be an H—extension for a finite Hopf algebra H. First we characterize such Hopf extensions that are Frobenius extensions. Second we will give some characterizations of Hopf Galois extensions, which extend the result of Cohen-Fishmann-Montgomery

Hopf Galois extensions, smash products, and Morita equivalence

Hopf Galois extensions, smash products, and Morita equivalence

Representation theory of Hopf galois extensions

LetH be a Hopf algebra over the fieldk andB ⊂A a right faithfully flat rightH-Galois extension. The aim of this paper is to study some questions of representation theory connected with the ring extensionB ⊂A, such as induction and restriction of simple or indecomposable modules. In particular, generalizations are given of classical results of Clifford, Green and Blattner on representations of groups and Lie algebras. The stabilizer of a leftB-module is introduced as a subcoalgebra ofH. Very often the stabilizer is a Hopf subalgebra. The special case whenA is a finite dimensional cocommutative Hopf algebra over an algebraically closed field,B is a normal Hopf subalgebra andH is the quotient Hopf algebra was studied before by Voigt using the language of finite group schemes.

Principal homogeneous spaces for arbitrary Hopf algebras

LetH be a Hopf algebra over a field with bijective antipode,A a rightH-comodule algebra,B the subalgebra ofH-coinvariant elements and can:A ⊗ B A →A ⊗H the canonical map. ThenA is a faithfully flat (as left or rightB-module) Hopf Galois extension iffA is coflat asH-comodule and can is surjective (Theorem I). This generalizes results on affine quotients of affine schemes by Oberst and Cline, Parshall and Scott to the case of non-commutative algebras. The dual of Theorem I holds and generalizes results of Gabriel on quotients of formal schemes to the case of non-cocommutative coalgebras (Theorem II). Furthermore, in the dual situation, a normal basis theorem is proved (Theorem III) generalizing results of Oberst-Schneider, Radford and Takeuchi.

P-polynomials and H-Galois extensions

Let R be a commutative ring with identity 1 and let H be a finite cyclic Hopf algebra, i.e., H is a Hopf algebra over R and has a free basis ( 1, D,..., DC’’ ) as R-module. A finite cyclic Hopf algebra H with free basis ( 1, D,..., D” ’ ) is called a derioation rtpe if the comultiplication map d of Hisdefined byd(D)=D@l+l@D: In this paper, for a finite cyclic Hopf algebra of derivation type, we determine the structure of H-Galois estrnsions of R in the sense of [9, Definition 1.41. (An H-Galois extension of R is equivalent to an H*-Hopf Galois extension of R in the sense of [ 17, p. 66, Definition], where H* is the dual Hopf algebra Hom.(H, R).) First we show that if H is a finite cyclic Hopf algebra of derivation type, then the base ring R contains the prime field GF(p) and the algebra structure of H is determined by a p-polynomial

The cohomological aspects of Hopf Galois extensions over a commutative ring

The cohomological aspects of Hopf Galois extensions over a commutative ring