Base change and Grothendieck duality for Cohen-Macaulay maps

Abstract

Let $f:X\to Y$ be a Cohen-Macaulay map of finite type between Noetherian schemes, and $:Y^{\prime }\to Y$ a base change map, with $Y^{\prime }$ Noetherian. Let $f^{\prime }$ be the base change of $f$ under $g$ and $g^{\prime }$ the base change of $g$ under $f$. We show that there is a canonical isomorphism between ${g^{\prime }}^{\ast }{\omega }_f$ and ${\omega }_{f^{\prime }}$, where ${\omega }_f$ and ${\omega }_{f^{\prime }}$ 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