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.