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Regular differential forms and relative duality. (English) Zbl 0796.14012
The paper deals with relative duality theory introduced by S. L. Kleiman [Compos. Math. 41, 39-60 (1980; Zbl 0403.14003)] and developed by E. Kunz [Manuscr. Math. 15, 91-108 (1975; Zbl 0299.14013) and Abh. Math. Semin. Univ. Hamburg 47, 42-70 (1978; Zbl 0379.14005), J. Lipman [“Dualizing sheaves, differentials and residues on algebraic varieties”, Astérisque 117 (1984; Zbl 0562.14003)] and others. It does not deal with the general Grothendieck duality theory. The authors connect here the (local) theory of regular differential forms [E. Kunz and R. Waldi, “Regular differential forms”, Contemp. Math. 79 (1988; Zbl 0658.13019)] with the (global) theory of duality.
Let $$Y$$ be an excellent noetherian scheme without components. $$D^ d_ Y =:\{Y$$-schemes $$f:X \to Y$$ without embedded components, such that $$X | Y$$ is of finite type, equidimensional of dimension $$d$$ and generically smooth}. The sheaf of regular $$d$$-differential forms $$\omega^ d_{X | Y}$$ is well-defined for $$X \in D^ d_ Y$$. The main result of the paper (the duality theorem) is that $$\{\omega^ d_{X | Y}\}_{X \in D^ d_ y}$$ is equipped with a dualising structure i.e. for $$f$$ proper, there is an $${\mathcal O}_ Y$$-linear ‘trace map’ \hbox$$I_{X | Y} : R^ df_ * \omega^ d_{X | Y} \to {\mathcal O}_ Y$$ such that the induced canonical map $$f_ * \operatorname{Hom}_{{\mathcal O}_ X} ({\mathcal F}, \omega^ d_{X | Y}) \to \operatorname{Hom}_{{\mathcal O}_ Y} (R^ df_ *{\mathcal F}, {\mathcal O}_ Y)$$ is an isomorphism. A complete description of the trace map is given in terms of local integrals and residues which can be calculated easily. The dualising structure on $$\omega^ d_{X | Y}$$ is completely characterised by the two properties:
(1) If $$f$$ is proper, then $$\omega^ d_{X | Y}$$ is compatible with a flat base change to a noetherian scheme. (A counter example in case of a non-flat base change is described.)
(2) Residue theorem: If $$Z \subseteq X$$ is a closed subscheme satisfying suitable conditions, then the trace maps $$I_{X | Y}$$, $$I_{X | Y,Z}$$ are compatible with the canonical map $$R^ d_ Zf_ * \omega^ d_{X | Y} \to R^ df_ * \omega^ d_{X | Y}$$. The generalisation of these results to the relative case when $$X,Y$$ are $$S$$- schemes and $$f:X \to Y$$ a morphism of $$S$$-schemes is done in the final section.

##### MSC:
 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 32C37 Duality theorems for analytic spaces 13N05 Modules of differentials 32C38 Sheaves of differential operators and their modules, $$D$$-modules
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