Global and finitistic dimension of Hopf-Galois extensions

Let H be a Hopf algebra over a field k and A/B a right H-Galois extension. Then in this note a spectral sequence for Ext will be constructed which yields the estimate for global dimension of A in terms of the corresponding data for H and B. As an application, we obtain the Maschke-type theorems for crossed products and twisted smash products. Finally, the relationship of finitistic dimensions between A and B will be given, if H is semisimple.