Equivalence Theorems and Hopf–Galois Extensions

Menini, Claudia; Zuccoli, Monica

doi:10.1006/jabr.1996.6994

Cited by 10 publications

(10 citation statements)

References 16 publications

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“…It turns out, that this leads to the equivalence-theorems for bimodule-categories studied in [14] or [10].…”

Section: A As An A-h-bimodulementioning

confidence: 99%

“…It was observed by Menini and Zuccoli in [10] that parts of the theory can be described by general module-theoretic methods. Refining these techniques we are able to generalize the main part of Schneiders paper to H-comodulealgebras A for a Hopf algebra H over a ground ring R.…”

Section: Introductionmentioning

confidence: 99%

“…In particular we show in 1. 6 Using results from [10] it is shown in 1.10 that for H a semiperfect Hopf algebra over a QF-ring R the category M H A can be identified with A op #T -Mod, where T is the left rational part of H * . The second section is devoted to the question when A is a (projective) generator in M H A .…”

Section: Introductionmentioning

confidence: 99%

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A-H-BIMODULES AND EQUIVALENCES

Menini¹,

Seidel

Torrecillas³

et al. 2001

Communications in Algebra

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In [6, Theorem 2.2] Doi gave a Hopf-algebraic proof of a generalization of Oberst's theorem on affine quotients of affine schemes. He considered a commutative Hopf algebra H over a field, coacting on a commutative H-comodule algebra A. If A coH denotes the subalgebra of coinvariant elements of A and β : A ⊗ A coH A −→ A ⊗ H the canonical map, he proved that the following are equivalent: Schneider generalized this result in [14, Theorem 1] to the noncommutative situation imposing as a condition the bijectivity of the antipode of the underlying Hopf algebra. Interpreting the functor of coinvariants as a Hom-functor, Menini and Zuccoli gave in [10] a moduletheoretic presentation of parts of the theory. Refining the techniques involved we are able to generalize Schneiders result to H-comodulealgebras A for a Hopf algebra H (with bijective antipode) over a commutative ring R under fairly weak assumptions.

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“…It turns out, that this leads to the equivalence-theorems for bimodule-categories studied in [14] or [10].…”

Section: A As An A-h-bimodulementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A-H-BIMODULES AND EQUIVALENCES

Menini¹,

Seidel

Torrecillas³

et al. 2001

Communications in Algebra

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“…By [16,Lemma 3.9], Im G = σ A C * A ⊗ C , and G induces an equivalence between the categories A M C and σ A C * A ⊗ C . We will freely regard an object M ∈ A M C with both these structures.…”

Section: Proof We Know From the Theorem Thatmentioning

confidence: 96%

Co-Frobenius Hopf Algebras: Integrals, Doi–Koppinen Modules and Injective Objects

1999

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We investigate Hopf algebras with non-zero integral from a coalgebraic point of view. Categories of Doi-Koppinen modules are studied in the special case where the defining coalgebra is left and right semiperfect, and several pairs of adjoint functors are constructed. As applications we give a very short proof for the uniqueness of the integrals and provide information about injective objects in the category of Doi-Koppinen modules.

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“…In Section 2 we look at the case where C is infinite dimensional. Following the methods developed in [9,11,19], we will present two characterizations of the category of Doi-Hopf modules C M(H) B . We first introduce the notion of rational (right) C * ◮<B-module, and then we will show that the category C M(H) B is isomorphic to Rat(M C * ◮<B ), the category of rational (right) C * ◮<B-modules.…”

Section: Introductionmentioning

confidence: 99%

Doi–Hopf Modules and Yetter–Drinfeld Modules for Quasi-Hopf Algebras

Bulacu

Caenepeel

Torrecillas

2006

Communications in Algebra

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For a quasi-Hopf algebra H, a left H-comodule algebra B and a right H-module coalgebra C we will characterize the category of Doi-Hopf modules C M(H) B in terms of modules. We will also show that for an Hbicomodule algebra A and an H-bimodule coalgebra C the category of generalized Yetter-Drinfeld modules A YD(H) C is isomorphic to a certain category of Doi-Hopf modules. Using this isomorphism we will transport the properties from the category of Doi-Hopf modules to the category of generalized Yetter-Drinfeld modules.1991 Mathematics Subject Classification. 16W30. He would like to thank Free University of Brussels (Belgium) and University of Almeria (Spain) for their warm hospitality.Using the first isomorphism, we will characterize A YD(H) C as a category of comodules over a coring. In Section 3.3, we will characterize the category of Yetter-Drinfeld modules as a category of modules.A 2 M(H op ⊗ H) C . It will follow from Proposition 3.4 that these two categories are isomorphic. Proposition 3.4. Let H be a quasi-bialgebra, C a left H-module coalgebra andProof. If A 1 and A 2 are twist equivalent, then there exists V ∈ A ⊗ H satisfying (1.22-1.24). Take M ∈ A 1 M(H) C ; M becomes an object in A 2 M(H) C by keeping the same left A-module structure and defining ρ ′ M : M → M ⊗ C, ρ ′ (m) = V · ρ M (m). Conversely, take M ∈ A 2 M(H) C via the structures · and ρ ′ M . Then M can be viewed as a left-right (H, A 1 , C)-Hopf module via the same left A-action · and the right C-coaction ρ M defined by ρ M : M → M ⊗ C, ρ M (m) = V −1 · ρ ′ M (m).

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Equivalence Theorems and Hopf–Galois Extensions

Cited by 10 publications

References 16 publications

A-H-BIMODULES AND EQUIVALENCES

A-H-BIMODULES AND EQUIVALENCES

Co-Frobenius Hopf Algebras: Integrals, Doi–Koppinen Modules and Injective Objects

Doi–Hopf Modules and Yetter–Drinfeld Modules for Quasi-Hopf Algebras

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