1. Morita theory of comodules over corings
- Author
-
Joost Vercruysse and Gabriella Böhm
- Subjects
Pure mathematics ,Algebra and Number Theory ,Endomorphism ,Mathematics::Rings and Algebras ,Sigma ,Mathematics - Rings and Algebras ,Algèbre - théorie des anneaux - théorie des corps ,16D90, 16W30 ,Comodule ,Rings and Algebras (math.RA) ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Morita therapy ,FOS: Mathematics ,Canonical map ,Finitely-generated abelian group ,Equivalence (formal languages) ,Structured program theorem ,Mathematics - Abstract
By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita context induces an equivalence between appropriate subcategories of the module categories of the two rings in the Morita context. These are in fact categories of firm modules for non-unital subrings. We apply this result to various Morita contexts associated to a comodule $\Sigma$ of an $A$-coring $\cC$. This allows to extend (weak and strong) structure theorems in the literature, in particular beyond the cases when any of the coring $\cC$ or the comodule $\Sigma$ is finitely generated and projective as an $A$-module. That is, we obtain relations between the category of $\cC$-comodules and the category of firm modules for a firm ring $R$, which is an ideal of the endomorphism algebra $^\cC(\Sigma)$. For a firmly projective comodule of a coseparable coring we prove a strong structure theorem assuming only surjectivity of the canonical map., Comment: LaTeX, 35 pages. v2: Minor changes including the title, examples added in Section 2
- Published
- 2007