Regular differential forms and relative duality.

*(English)*Zbl 0796.14012The 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.

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.

Reviewer: U.N.Bhosle (Bombay)