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Projectivity and flatness over the endomorphism ring of a finitely generated comodule. (English) Zbl 1210.16033
Summary: Let $$k$$ be a commutative ring, $$A$$ a $$k$$-algebra, $$\mathcal C$$ an $$A$$-coring that is projective as a left $$A$$-module, $$^*\mathcal C$$ the dual ring of $$\mathcal C$$ and $$\Lambda$$ a right $$\mathcal C$$-comodule that is finitely generated as a left $$^*\mathcal C$$-module. We give necessary and sufficient conditions for projectivity and flatness of a module over the endomorphism ring $$\text{End}^{\mathcal C}(\Lambda)$$. If $$\mathcal C$$ contains a grouplike element, we can replace $$\Lambda$$ with $$A$$.
##### MSC:
 16T15 Coalgebras and comodules; corings 16D40 Free, projective, and flat modules and ideals in associative algebras 16S50 Endomorphism rings; matrix rings
##### Keywords:
corings; finitely generated comodules; duals; flatness; projectivity
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