Vanishing and a counterexample for Witt divisorial sheaves

First we refine the duality theory for Witt divisorial sheaves on smooth projective varieties over a perfect field of positive characteristic. Building on previous work [Lem22], we remove the residual derived limit to obtain a cleaner isomorphism. As an application, we prove a Ramanujam-type vanishing theorem for Witt divisorial sheaves of nef and big divisors on surfaces. Finally, we show that a surface constructed by Langer [Lan16] with a divisor constructed by Cascini and Tanaka [CT18] gives a counterexample to Kawamata-Viehweg-type vanishing of Witt divisorial sheaves in dimension two.