The adjunction morphism for retgular differential forms and relative duality
Tóm tắt
Let
$$f:X \to Y$$
be a morphism of noetherian schemes,generically smooth and equidimensional of dimension
$$d,\iota :X\prime \to X$$
a closed embedding such that
$$f \circ \iota :X\prime \to Y$$
is generically smooth and equidimensional ofdimension d
$$\prime $$
, and X
$$\prime $$
, X and Y are excellent schemes withoutembedded components. We exhibit a concrete morphism
$$Res_{X\prime /X} :det \mathcal{N}_{X\prime /X} \otimes _{\mathcal{O}_{X\prime } } \iota *\omega _{X/Y}^d \to \omega _{X\prime /Y\prime }^{d\prime } ,$$
which transforms the integral of X/Y into the integral ofX
$$\prime $$
/Y. Here
$$\mathcal{N}_{X\prime /X} $$
denotes the normal sheaf of X
$$\prime $$
/X and
$$\omega _{X/Y}^d $$
resp.
$$\omega _{X\prime /Y\prime }^{d\prime } $$
denotes the sheaf ofregular differential forms of X/Y resp. X
$$\prime $$
/Y. Usinggeneralized fractions we provide a canonical description ofresidual complexes and residue pairs of Cohen--Macaulayvarieties, and obtain a very explicit description of fundamentalclasses and their traces.
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