AI 中文总结
设k为交换环，H为k上双代数，C为(H,A)-余环，若C作为左A-模平坦，有限生成的右(H,C)-余模Λ对应的自同态环上的模具有投射性与平坦性的充要条件被给出；当C含固定H-群样元时，可将Λ替换为A。
AI 中文摘要
设$k$为交换环，$H$为$k$上的双代数，$A$为$H$-代数，${\rm C}$为$(H,A)$-余环，即兼具左$(H,A)$-双模结构且以兼容方式构成$A$-余环的对象。左$(H,A)$-双模及其变形在非交换流形的微分几何研究中具有基础作用。设$\rm\bigtriangleup$为右$(H,{\rm C})$-余模，从$\rm\bigtriangleup$到自身的右$(H,{\rm C})$-余线性映射构成的$k$-模$_HEnd^{\rm C}(\rm\bigtriangleup)$是一个环。假设${\rm C}$作为左$A$-模是平坦的，若$\rm\bigtriangleup$作为右$(H,{\rm C})$-余模是有限生成（有限表现）的，本文给出$_HEnd^{\rm C}(\rm\bigtriangleup)$上的模具有投射性与平坦性的充要条件；若${\rm C}$包含一个固定的$H$-群样元，则可将$\rm\bigtriangleup$替换为$A$。
英文摘要
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 $Λ$ be a right $(H,{\mathcal C})$-comodule. The $k$-module $_HEnd^{\mathcal C}(Λ)$ of right $(H,{\mathcal C})$-colinear maps from $Λ$ to $Λ$ is a ring. Let us assume that $\mathcal C$ is flat as a left $A$-module. If $Λ$ 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 $_HEnd^{\mathcal C}(Λ)$. If $\mathcal C$ contains a fixed $H$-grouplike element, we can replace $Λ$ with $A$.
