Generators of top cohomology

TL;DR

This paper constructs explicit generators for the top cohomology of certain smooth proper morphisms over specific rings, extending classical duality results and addressing a question by Lipman.

Contribution

It provides explicit exact sequences generating top cohomology modules in new cases, including DVRs with sections and Grassmannians over integers.

Findings

Generated top cohomology modules explicitly in new cases.

Connected classical duality with explicit generators.

Partially answered Lipman's question.

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.