Hopf代数的半单扩张与不变子环的同调维数

Semisimple Extensions of Hopf Algebra and Homological Dimension of Invariants for Hopf Actions

设H是域k上的有限维Hopf代数,A是左H-模代数,AH是A的H-不变子环.假定A/AH是半单扩张且A是平坦的右AH-模.如果H*是unimodular,且存在c∈C(A),使t·c=1.我们证明了WD(AH)=WD(A)=WD(A#H).此外,如果A是投射的左及右AH-模,则有LD(AH)=LD(A)=LD(A#H).

Let H be a finite dimensional Hopf algebra acting on an algebra yl, and let AH denote the invariant subring of A and A#H the smash product. Assume that A/AH is a semisimple extension and A is a flat right AH-module. It is shown that if H* is unimodular and A has an element of trace one then WD(AH) = WD(A) = WD(A#H). Moreover, if A is a projective left and right AH-module, then LD(AH) = LD(A) = LD(A#H).