Mathematics – Rings and Algebras
Scientific paper
2007-10-04
Comm. Algebra 37 (2009), no. 9, 3207-3247
Mathematics
Rings and Algebras
LaTeX, 35 pages. v2: Minor changes including the title, examples added in Section 2
Scientific paper
10.1080/00927870902747993
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.
Böhm Gabriella
Vercruysse Joost
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