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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
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