On Weak Crossed Products of Weak Hopf Algebras

Let H be a weak Hopf algebra in the sense of Böhm et al. (J Algebra 221:385–438, 1999) measuring an algebra A. Let A#σH be a weak crossed product with σ invertible. Then in this paper we first give some conditions for A#σH to be a weak Hopf algebra. Next the spectral sequence for Ext will be constructed which yields an estimate for the global dimension of A#σH in terms of the corresponding data for H and A. Furthermore, we will investigate when A ⊂ A#σH becomes a separable extension. Finally, we prove that if H and its dual H* are both semisimple, then the finitistic dimension of A#σH is equal to that of A.