Frobenius扩张下的G_(C)-平坦性

G_(C)-Flatness under Frobenius Extensions

设A/S是一个环的Frobenius扩张,且S是凝聚环,C是半对偶S-模.首先,利用构造法证明相对于半对偶模的G_(C)-平坦性在环的Frobenius扩张下是保持的,即对于A-模M,M_(A)是G_(C■_(S)A)-平坦模当且仅当M_(S)是G_(C)-平坦的;其次,证明相对于半对偶模的G_(C)-平坦维数在环的Frobenius扩张下是不变的.

Let A/S be a Frobenius extension of rings with S coherent and C a semidualizing S-module.Firstly,we prove that G_(C)-flatness with respect to a semidualizing module is invariant under Frobenius extensions by using the construction method,that is,for an A-module M,M_(A) is G_(C■_(S)A)-flat module if and only if M_(S) is G_(C)-flat.Secondly,we prove that G_(C)-flat dimension with respect to a semidualizing module is preserved under Frobenius extensions.