Generators of top cohomology

IF 0.9 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2025-06-24 DOI:10.1112/blms.70132

Manoj Kummini, Mohit Upmanyu

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引用次数: 0

Abstract

Let $R$ be a commutative Noetherian ring and $f : X ⟶ Spec R$ a proper smooth morphism, of relative dimension $n$ . From Hartshorne, Residues and Duality, Springer, 1966, one knows that the trace map ${Tr}_{f} : H^{n} (X, ω_{X / R}) ⟶ R$ is an isomorphism when $f$ has geometrically connected fibres. We construct an exact sequence that generates ${Ext}_{X}^{n} (O_{X}, ω_{X / R}) = H^{n} (X, ω_{X / R})$ as an $R$ -module in the following cases:

This partially answers a question raised by Lipman.

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上上同调的生成

设R$ R$是一个交换诺瑟环，并且f: X {Spec $R} f: X \ lonightarrow \operatorname{Spec}R$是一个相对维数为n$ n$的光滑态射。从Hartshorne，残数与对偶性，b施普林格，1966，可知迹映射Tr f：H n (X, ω X / R) {R $\operatorname{Tr}_f：\operatorname{H}^n(X, \ ω _{X/R}) \ lonightarrow R$是一个同构，当f$ f$具有几何连接的纤维。我们构造一个精确的序列生成Ext X n （O X）ω X / R = H n (X，ω X/R)$ \operatorname{Ext}_X^n(\mathcal O_X, \omega _{X/R}) = \operatorname{H}^n(X, \omega _{X/R})$在以下情况下作为R$ R$ -模块：这部分地回答了利普曼提出的一个问题。

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