  We prove that the Tate, Beilinson and Parshin conjectures are invariant under
Homological Projective Duality (=HPD). As an application, we obtain a proof of
these celebrated conjectures (as well as of the strong form of the Tate
conjecture) in the new cases of linear sections of determinantal varieties and
complete intersections of quadrics. Furthermore, we extend the original
conjectures of Tate, Beilinson and Parshin from schemes to stacks and prove
these extended conjectures for certain low-dimensional global orbifolds.
