Generators of top cohomology

Abstract

Let $R$ be a commutative noetherian ring and $f: X \to \mathrm{Spec} R$ a proper smooth morphism, of relative dimension $n$. From Hartshorne, Residues and Duality, Springer, 1966, one knows that the trace map $\mathrm{Tr}_f : \mathrm{H}^n(X, \omega_{X/R}) \to R$ is an isomorphism when $f$ has geometrically connected fibres. We construct an exact sequence that generates $\mathrm{Ext}_X^n(\mathcal{O}_X, \omega_{X/R}) = \mathrm{H}^n(X, \omega_{X/R})$ as an $R$-module in the following cases: (1) when $R$ is a DVR and $f$ has a section; (2) when $R=\mathbb{Z}$ and $X$ is the Grassmannian $G_{2,m}$ for some $m \geq 4$. This partially answers a question raised by Lipman.