Base change and Grothendieck duality for Cohen-Macaulay maps
Abstract
Description
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 ${g'}^*ω_f$ and $ω_{f'}$, where $ω_f$ and $ω_{f'}$ 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.
24 pages
24 pages