Invariants of FP-Projective Dimensions Under Cleft Extensions

Abstract

Let H be a finite dimensional Hopf algebra and A be an algebra over a fixed field k. Firstly, it is proved that the left FP-projective dimension is invariant under cleft extensions when H is semisimple and A is left coherent. Secondly, using (co)induction functors, we study the relations between FP-projective dimensions in A # H-Mod and the counterparts in AH-Mod. Finally, we characterize the FP-projective preenvelopes (resp., precovers) under H*-extensions and cleft extensions, respectively.