Generators of top cohomology
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915314835390464 |
|---|---|
| author | Kummini, Manoj Upmanyu, Mohit |
| author_facet | Kummini, Manoj Upmanyu, Mohit |
| contents | 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 $\mathrm{Tr}_f : \mathrm{H}^n(X, ω_{X/R}) \to R$ is an isomorphism when $f$ has geometrically connected fibres. We construct an exact sequence that generates $\mathrm{Ext}_X^n(\mathcal{O}_X, ω_{X/R}) = \mathrm{H}^n(X, ω_{X/R})$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04357 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generators of top cohomology Kummini, Manoj Upmanyu, Mohit Commutative Algebra 13 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 $\mathrm{Tr}_f : \mathrm{H}^n(X, ω_{X/R}) \to R$ is an isomorphism when $f$ has geometrically connected fibres. We construct an exact sequence that generates $\mathrm{Ext}_X^n(\mathcal{O}_X, ω_{X/R}) = \mathrm{H}^n(X, ω_{X/R})$ 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. |
| title | Generators of top cohomology |
| topic | Commutative Algebra 13 |
| url | https://arxiv.org/abs/2501.04357 |