The adjunction morphism for retgular differential forms and relative duality

Abstract

Let $f:X\rightarrow Y$ be a morphism of noetherian schemes, generically smooth and equidimensional of dimension $d, \iota : X^{\prime} \rightarrow X$ a closed embedding such that $f \circ \iota : X^{\prime} \rightarrow Y$ is generically smooth and equidimensional of dimension $d^{\prime}$, and $X^{\prime}, X$ and $Y$ are excellent schemes without embedded components. We exhibit a concrete morphism\[{\rm Res}_{X^{\prime}/X}: {\rm det}\,{\cal N}_{X^{\prime}/X} \otimes_{{\cal O}_{X^{\prime}}} \iota^* \omega^d_{X/Y} \rightarrow \omega^{d^{\prime}}_{X^{\prime}/Y},\] which transforms the integral of $X/Y$ into the integral of$X^{\prime}/Y$. Here ${\cal N}_{X^{\prime}/X}$ denotes the normal sheaf of $X^{\prime}/X$ and$\omega^d_{X/Y}$ resp. $\omega^{d^{\prime}}_{X^{\prime}/Y}$ denotes the sheaf of regular differential forms of $X/Y$ resp. $X^{\prime}/Y$. Using generalized fractions we provide a canonical description of residual complexes and residue pairs of Cohen-Macaulay varieties, and obtain a very explicit description of fundamental classes and their traces.