Homological Integrals for Weak Hopf Algebras

We introduce the notion of a homological integral for an infinite-dimensional weak Hopf algebra and use the homological integral to prove several structure theorems. For example, we prove that the Artin–Schelter property and the Van den Bergh condition are equivalent for a noetherian weak Hopf algebra, and that the antipode is automatically invertible in this case. We also prove a decomposition theorem that states that any weak Hopf algebra finite over an affine center is a direct sum of Artin–Schelter Gorenstein, Cohen–Macaulay, GK dimension homogeneous weak Hopf algebras.