## Galois comodules (2006)

Venue: | Commun. Algebra |

Citations: | 18 - 7 self |

### BibTeX

@ARTICLE{Wisbauer06galoiscomodules,

author = {Robert Wisbauer},

title = {Galois comodules},

journal = {Commun. Algebra},

year = {2006},

pages = {2683--2711}

}

### OpenURL

### Abstract

Generalizing the notion of Galois corings, Galois comodules were introduced as comodules P over an A-coring �� � for which PA is finitely generated and projective and the evaluation map ��� � � Hom �� � �P � ��� � ⊗S P → �� � is an isomorphism (of corings) where S = End �� � �P�. It has been observed that for such comodules the functors

### Citations

87 | The structure of corings: Inductions functors, Maschke type theorem, and Frobenius and Galois-type properties
- Brzeziński
(Show Context)
Citation Context ... the ring A and put S = End C (A). A grouplike element g ∈ C makes A a right C-comodule by the coaction ̺ A : A → A ⊗A C, a ↦→ 1 ⊗ ga. The notion of Galois corings (C, g) was introduced in Brzeziński =-=[2]-=- by requiring the canonical map, χ : A ⊗S A → C, a ⊗ a ′ ↦→ aga ′ , to be an isomorphism (of corings). It was pointed out in [13] that this can be seen as the evaluation map µC : Hom C (Ag, C) ⊗S A → ... |

59 |
Foundations of Modules and Ring Theory
- Wisbauer
- 1991
(Show Context)
Citation Context ...P ∗ � ≃ Y ⊗ B S� □ 3.3. P as Generator in M��� . Recall that P is a generator in M� if and only if the functor Hom � �P� −� �M� → MS is faithful and that faithful functors reflect epimorphisms (e.g., =-=Wisbauer, 1991-=-, 11.3). Since, for any N ∈ M� , Hom � �P� � N ��Hom � �P� Hom � �P� N� ⊗ S P� → Hom � �P� N� is an epimorphism (surjective), we conclude that �N is an epimorphism (that is surjective) in M� , provide... |

53 | On modules associated to coalgebra-Galois extensions - Brzeziński - 1999 |

47 | Comatrix corings: Galois corings, descent theory, and a structure theorem for cosemisimple corings
- Kaoutit, Gómez-Torrecillas
(Show Context)
Citation Context ... C, and that it implies bijectivity of µN : Hom C (Ag, N) ⊗S A → N, for every (C, A)-injective comodule N. The notion of Galois corings was extended to comodules by El Kaoutit and GómezTorrecillas in =-=[6]-=-, where to any bimodule SPA with PA finitely generated and projective, a coring P ∗ ⊗S P was associated and it was shown that the canonical map ˜µA : HomA(P, A) ⊗S P → C is a coring morphism provided ... |

24 | Modules and algebras: bimodule structure and group actions on algebras - Wisbauer - 1996 |

8 |
Static modules and equivalences, in “Interactions between Ring Theory and Representation
- Wisbauer
- 2000
(Show Context)
Citation Context ...tic: they all share the property that they are generators in their respective categories provided they are flat over their endomorphism rings. Hence the presentation is partly motivated by the papers =-=[11, 12]-=- on tilting and static modules. Some results from [3] and [5] are obtained in a more general setting (e.g., 5.10, 5.8). Relative injectivity of comodules is of special interest in the context of our i... |

7 |
Hopf-Galois extensions
- Schauenburg
- 1996
(Show Context)
Citation Context ...�A�-injective and for a B-strongly ���A�-injective comodule M and any X ∈ M B, X ⊗ B M is ���A�injective. For coalgebras B-strongly ���A�-injective comodules are named Bequivariantly �-injective (see =-=Schauenburg and Schneider, 2005-=-, Definition 5.1). Cointegrals � making M fully ���A�-injective are said to be M-normalized in Caenepeel et al. (to appear, Proposition 5.1). The fact that under projectivity conditions comodule prope... |

6 | Infinite comatrix corings - Kaoutit, Gómez-Torrecillas |

5 | Modules and Algebras : Bimodule Structure and - Wisbauer - 1996 |

5 |
On Galois corings. [In:] Hopf algebras in non-commutative geometry and physics
- Wisbauer
- 2003
(Show Context)
Citation Context ... ⊗ ga. The notion of Galois corings (C, g) was introduced in Brzeziński [2] by requiring the canonical map, χ : A ⊗S A → C, a ⊗ a ′ ↦→ aga ′ , to be an isomorphism (of corings). It was pointed out in =-=[13]-=- that this can be seen as the evaluation map µC : Hom C (Ag, C) ⊗S A → C, and that it implies bijectivity of µN : Hom C (Ag, N) ⊗S A → N, for every (C, A)-injective comodule N. The notion of Galois co... |

5 |
Modules with chain conditions for finite matrix subgroups
- Zimmermann
- 1997
(Show Context)
Citation Context ...essentially means that all P-injective modules in σ[P] are P-static and - if P is a balanced bimodule - this can be seen as descending chain condition on certain matrix subgroups of P (see [11, 5.4], =-=[14]-=-). In all these cases the functor Hom C (P, −) induces equivalences between the P-static classes and the corresponding adstatic classes. Properties of these classes correspond to properties of the mod... |

2 |
Galois theory for comatrix corings: Descent theory, Morita theory, Frobenius and separability properties
- Vercruysse, J
- 2004
(Show Context)
Citation Context ...omodules and pointed out their relevance for descent theory, vector bundles, and non-commutative geometry. Related questions are, for example, also considered by Caenepeel, De Groot and Vercruysse in =-=[5]-=-. In this note we concentrate on comodule properties and we want to free the notion from the condition that PA has to be finitely generated and projective. This is done by taking the above mentioned i... |

1 | theory for comatrix corings: Descent theory, Morita theory, Frobenius and separability properties. arXiv:math.RA/0406436 - LNS |

1 | Tilting in module categories
- Wisbauer
- 1998
(Show Context)
Citation Context ...st case was handled in 3.4 for A� flat. In module categories, the second case describes an important property of self-tilting modules; for those, an additional projectivity condition is required (see =-=Wisbauer, 1998-=-, 4.2; Wisbauer, 2000, 4.4). The third case generalizes tilting modules (see Wisbauer, 2000, 4.3). For a module P, the corresponding property (4) essentially means that all P-injective modules in ��P�... |

1 | Static modules and equivalences
- Wisbauer
- 2000
(Show Context)
Citation Context ...r Hom � �P� −�. Now assume A� to be flat. Then K = Ke � is a comodule, and we have the commutative diagram with exact rows where �K is surjective and the central vertical map is an isomorphism (e.g., =-=Wisbauer, 2000-=-, Corollary 3.4). By diagram lemmata, this implies that �N is injective (hence an isomorphism).ON GALOIS COMODULES 2693 3.4. Properties of Generators. Assume A� to be flat. (1) If P generates the (fi... |

1 | On Galois Corings, Hopf algebras - Wisbauer - 2004 |

1 |
theory for comatrix corings: Descent theory, Morita theory, Frobenius and separability properties
- Soc
- 2003
(Show Context)
Citation Context ... the infinite comatrix corings, as introduced by El Kaoutit and Gómez Torrecillas in [7], find a natural application (Section 6). 2 Preliminaries Throughout we will essentially follow the notation in =-=[4]-=-. For convenience we recall some basic notions. 2.1. Corings. Let A be an associative ring with unit and C an A-coring with coproduct and counit ∆ : C → C ⊗A C, ε : C → A. Associated to this there are... |