Finitistic dimension of weak Hopf-Galois extensions

Abstract

Let H be a finite dimensional weak Hopf algebra and A/B be a right faithfully flat weak H-Galois extension. Then in this note, we first show that if H is semisimple, then the finitistic dimension of A is less than or equal to that of B. Furthermore, using duality theorem, we obtain that if H and its dual H* are both semisimple, then the finitistic dimension of A is equal to that of B, which means the finitistic dimension conjecture holds for A if and only if it holds for B. Finally, as applications, we obtain these relations for the weak crossed products and weak smash products.