The Structure of Corings. Induction Functors, Maschke-type Theorem, and Frobenius and Galois-type Properties
نویسنده
چکیده
Given a ring A and an A-coring C we study when the forgetful functor from the category of right C-comodules to the category of right A-modules and its right adjoint − ⊗A C are separable. We then proceed to study when the induction functor − ⊗A C is also the left adjoint of the forgetful functor. This question is closely related to the problem when A → AHom(C, A) is a Frobenius extension. We introduce the notion of a Galois coring and analyse when the tensor functor over the subring of A fixed under the coaction of C is an equivalence. We also comment on possible dualisation of the notion of a coring.
منابع مشابه
Quasi-co-frobenius Corings as Galois Comodules
We compare several quasi-Frobenius-type properties for corings that appeared recently in literature and provide several new characterizations for each of these properties. By applying the theory of Galois comodules with a firm coinvariant ring, we can characterize a locally quasi-Frobenius (quasi-co-Frobenius) coring as a locally projective generator in its category of comodules.
متن کاملar X iv : m at h / 06 12 66 2 v 2 [ m at h . R A ] 3 S ep 2 00 8 QUASI - FROBENIUS FUNCTORS . APPLICATIONS
We investigate functors between abelian categories having a left adjoint and a right adjoint that are similar (these functors are called quasi-Frobenius functors). We introduce the notion of a quasi-Frobenius bimodule and give a characterization of these bimodules in terms of quasi-Frobenius functors. Some applications to corings and graded rings are presented. In particular, the concept of qua...
متن کاملFrobenius Properties and Maschke-type Theorems for Entwined Modules
Entwined modules arose from the coalgebra-Galois theory. They are a generalisation of unified Doi-Hopf modules. In this paper, Frobenius properties and Maschke-type theorems, known for Doi-Hopf modules are extended to the case of entwined modules.
متن کاملQuasi-Frobenius functors with application to corings
Müller generalized in [12] the notion of a Frobenius extension to left (right) quasi-Frobenius extension and proved the endomorphism ring theorem for these extensions. Recently, Guo observed in [9] that for a ring homomorphism φ : R → S, the restriction of scalars functor has to induction functor S ⊗R − : RM → SM as right ”quasi” adjoint if and only if φ is a left quasi-Frobenius extension. In ...
متن کاملCofrobenius Corings and Adjoint Functors
We study co-Frobenius and more generally quasi-co-Frobenius corings over arbitrary base rings and over PF base rings in particular. We generalize some results about co-Frobenius and quasi-co-Frobenius coalgebras to the case of noncommutative base rings and give several new characterizations for co-Frobenius and more generally quasi-co-Frobenius corings, some of them are new even in the coalgebr...
متن کامل