中文摘要：

令H为有限维Hopf代数且A为固定域k上的代数。证明了当H半单及A/AH为H＊-Galois扩张时,A#H的余纯（copure）投射维数与A的余纯投射维数是相同的。作为应用,进一步证明了当H半单及A/AH为H＊-Galois扩张时,A是QF环当且仅当A#H是QF环。并且利用Hopf扩张下的（co）induction函子来研究A#H-模范畴及AH-模范畴之间余纯投射维数的关系。

英文摘要：

Let H be a finite dimensional Hopf algebra and A be an algebra over a fixed field k. It is proved that the left copure projective dimension of A#H and that of A is the same when H is semisimple and the extension A /AHis H＊-Galois. Moreover,it is shown that A#H is QF if and only if A is QF. Using（ co） induction functors,we study the relations between copure projective dimensions in A#H-Mod and the counterparts in AH-Mod.