Projectivity and flatness over the endomorphism ring of a finitely generated comodule.

Abstract

Let $k$ be a commutative ring, $H$ a bialgebra over $k$, $A$ an $H$-algebra and $\mathcal C$ an $(H,A)$-coring, i.e a left $(H,A)$-bimodule which is an $A$-coring in a compatible way. Left $(H,A)$-bimodules and their deformations play a fundamental role in the study of the differential geometry of noncommutative manifolds. Let $\Lambda$ be a right $(H,{\mathcal C})$-comodule. The $k$-module $_HEnd^{\mathcal C}(\Lambda)$ of right $(H,{\mathcal C})$-colinear maps from $\Lambda$ to $\Lambda$ is a ring. Let us assume that $\mathcal C$ is flat as a left $A$-module. If $\Lambda$ is finitely generated (finitely presented) as a right $(H,{\mathcal C})$-comodule, we give necessary and sufficient conditions for projectivity and flatness of a module over $_HEnd^{\mathcal C}(\Lambda)$. If $\mathcal C$ contains a fixed $H$-grouplike element, we can replace $\Lambda$ with $A$.