Regular differentials and equidimensional scheme-maps.

*(English)*Zbl 0812.14011Let \(S\) be a scheme, and consider \(S\)-schemes \(Z\) such that \(g : Z \to S\) is proper and equidimensional of relative dimension \(d\); under additional hypotheses one can identify the relative dualizing sheaf \(g^ ! {\mathcal O}_ S\) with the sheaf \(\omega_ Z\) of regular \(d\)-forms [cf. E. Kunz and R. Waldi, “Regular differential forms”, Contemp. Math. 79 (1988; Zbl 0658.13019)]. Now let \(f:X \to Y\) be a suitable restricted morphism of such schemes; by a result of S. L. Kleiman [Compos. Math. 41, 39-60 (1980; Zbl 0423.32006)] there is a canonical map \(\eta : f^* \omega_ Y \otimes \omega_ f \to \omega_ X = f^ ! \omega_ Y\) where \(\omega_ f\) is the canonical dualizing sheaf for \(f\). On the other hand, R. Hübl [Manuscr. Math. 65, No. 2, 213-224 (1989; Zbl 0704.13004)] gives a rather explicit description by methods of commutative algebra of a morphism \(\varphi : f^* \omega_ Y \otimes \omega_ f \to \omega_ X\).

The bulk of the present paper consists of showing that \(\eta = \varphi\). To prove this result the author makes use of the residue formalism developed by R. Hübl and E. Kunz [J. Reine Angew. Math. 410, 53-83 (1990; Zbl 0712.14006) and ibid. 84-108 (1990; Zbl 0709.14014)]. This paper contains many results on differential forms, residues, generalizations of the residue theorem of R. Hübl and P. Sastry [Am. J. Math. 115, No. 4, 749-787 (1993; Zbl 0796.14012)] and related topics and should be a must for anybody interested in these questions. With respect to residues of differential forms, one may also consult the Habilitationsschrift of R. Hübl [“Residues of differential forms, de Rham cohomology and Chern classes” (Regensburg 1994)].

The bulk of the present paper consists of showing that \(\eta = \varphi\). To prove this result the author makes use of the residue formalism developed by R. Hübl and E. Kunz [J. Reine Angew. Math. 410, 53-83 (1990; Zbl 0712.14006) and ibid. 84-108 (1990; Zbl 0709.14014)]. This paper contains many results on differential forms, residues, generalizations of the residue theorem of R. Hübl and P. Sastry [Am. J. Math. 115, No. 4, 749-787 (1993; Zbl 0796.14012)] and related topics and should be a must for anybody interested in these questions. With respect to residues of differential forms, one may also consult the Habilitationsschrift of R. Hübl [“Residues of differential forms, de Rham cohomology and Chern classes” (Regensburg 1994)].

Reviewer: K.Kiyek (Paderborn)

##### MSC:

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

13N05 | Modules of differentials |