Spectral Sequence and Finitely Presented Dimension for Weak Hopf–Galois Extensions

Let $H$ be a weak Hopf algebra, $A$ a right weak $H$-comodule algebra, and $B$ the subalgebra of the $H$-coinvariant elements of $A$. Let $A/B$ be a right weak $H$-Galois extension. In this paper, a spectral sequence for $\operatorname{Ext}$ which yields an estimate for the global dimension of $A$ in terms of the corresponding data for $H$ and $B$ is constructed. Next, the relationship between the finitely presented dimensions of $A$ and its subalgebra $B$ are given. Further, the case in which $A$ is an $n$-Gorenstein algebra is studied.