Generators of top cohomology

Let $R$ be a commutative noetherian ring and $f:X\to \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}R$ a proper smooth morphism, of relative dimension $n$. From Hartshorne, Residues and Duality, Springer, 1966, one knows that the trace map ${\mathrm{T}\mathrm{r}}_f:{\mathrm{H}}^n(X,{\omega }_{X\mathrm{/}R})\to R$ is an isomorphism when $f$ has geometrically connected fibres. We construct an exact sequence that generates ${\mathrm{E}\mathrm{x}\mathrm{t}}_X^n({\mathcal{O}}_X,{\omega }_{X\mathrm{/}R})={\mathrm{H}}^n(X,{\omega }_{X\mathrm{/}R})$ 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 $m\ge 4$. This partially answers a question raised by Lipman.