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Morita theory of comodules over corings
- Source :
- Communications in Algebra, 37 (9
- Publication Year :
- 2007
-
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.<br />Comment: LaTeX, 35 pages. v2: Minor changes including the title, examples added in Section 2
- 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
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Journal :
- Communications in Algebra, 37 (9
- Accession number :
- edsair.doi.dedup.....cab0939d8cb950c5f27d18f170199f66