Hochschild cohomology and AS-Gorenstein property of weak Hopf Galois extensions

Let $H$ be a weak Hopf algebra with bijective antipode. This paper is devoted to the AS-Gorenstein property of restricted faithfully flat weak $H$-Galois extensions $A/B$. We first of all study the cohomologies of weak $H$-Galois extensions and then establish the spectral sequence connecting the Hochschild cohomologies of $A$ and of $B$. Finally we prove that when $A$ is a noetherian aiffine PI algebra and $B$ is AS-Gorenstein, $A$ shares the AS-Gorenstein property.