Generators of top cohomology

Abstract Let be a commutative Noetherian ring and a proper smooth morphism, of relative dimension . From Hartshorne, Residues and Duality , Springer, 1966, one knows that the trace map is an isomorphism when has geometrically connected fibres. We construct an exact sequence that generates as an ‐module in the following cases: when is a DVR and has a section; when and is the Grassmannian for some . This partially answers a question raised by Lipman.