Base change and Grothendieck duality for Cohen-Macaulay maps

Details Base change and Grothendieck duality for Cohen-Macaulay maps Base change and Grothendieck duality for Cohen-Macaulay maps

2000-11-20

arxiv.org/abs/math/0011138v1

Mathematics

Algebraic Geometry

24 pages

Scientific paper

Let $f:X\to Y$ be a Cohen-Macaulay map of finite type between Noetherian schemes, and $:Y'\to Y$ a base change map, with $Y'$ Noetherian. Let $f'$ be the base change of $f$ under $g$ and $g'$ the base change of $g$ under $f$. We show that there is a canonical isomorphism between ${g'}^*\omega_f$ and $\omega_{f'}$, where $\omega_f$ and $\omega_{f'}$ are the relative dualizing sheaves. The map underlying this isomorphism is easily described when $f$ is proper, and has subtler description when $f$ is not. If $f$ is smooth we show that this map between the dualizing sheaves corresponds to the canonical identification of differential forms. Our results generalize the results of B. Conrad in two directions - wedo not need the properness assumption, and we do not need to assume that theschemes involved carry dualizing complexes. Residual complexes do not appear in this paper.

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