Coinduction Functor and Simple Comodules *

Consider a coring with exact rational functor, and a finitely generated and projective right comodule. We construct a functor (coinduction functor) which is right adjoint to the hom-functor represented by this comodule. Using the coinduction functor, we establish a bijective map between the set of representative classes of torsion simple right comodules and the set of representative classes of simple right modules over the endomorphism ring. A detailed application to a group-graded modules is also given. Introduction Let G be a group with neutral element e and A = ⊕x∈GAx a G-graded ring. To each element x ∈ G, one can associate the restriction functor (−)x : gr-A → ModAe from the category of G-graded right A-modules to the category of right Ae-modules. This functor sends an object M ∈ gr-A to its homogeneous component Mx. Since (−)x is right exact and commutes with direct sums, a classical result of P. Gabriel tell us that it has a right adjoint functor which we denote by Coind : ModAe → gr-A. The functor Coind x is known in the literature as coinduction functor, and was first introduced by C. Năstăsescu in [13], for x = e. In [1] G. Abrams and C. Menini defined and studied Coind in the case of semigroup-graded ring. The use of coinduction functor was crucial to study both simple and injective objects in either a group-graded or semigroup-graded modules categories, see [13, 1, 12]. ∗Research supported by grant MTM2007-61673 from the Ministerio de Educación y Ciencia of Spain, and P06-FQM-01889 from Junta de Andalućıa