THE DUAL OF THE TORSION MODULE OF DIFFERENTIALS

ABSTRACT

In this paper we will establish a duality between the torsion module of differentials, , and the highest non-zero exterior power of the module of Kähler differentials, , of reduced affine hypersurfaces with only isolated singularities in , where is an algebraically closed field of characteristic zero. In an earlier paper [5] we have shown that:

However by[5] the two sides are not isomorphic as -modules. Let denote the polynomial ring in variables over , and let denote the Jacobian ideal of the hypersurface. Both -modules also have an -module structure. We will show that when considered as -module, is dual to the torsion module of differentials, .

ACKNOWLEDGMENTS

The author gratefully acknowledges partial support through NSF grant DMS 9510654 and TARP grant 003594-0030b-1997 during the course of this work.

Notes

Dr. Ruth Michler died tragically on November 1, 2000, as the result of a traffic accident in Boston, as she was waiting to cross a street. She was an Associate Professor at University of North Texas, and a Visiting Scholar at Northeastern University. Anthony Iarrobino and Marc Levine handled subsequent correspondence with the editor, and made the corrections suggested by the referee, as well as certain other minor changes, in consultation with Prof. Gerhard Michler, Ruth's father. On Ruth's behalf, we thank the referee. Correspondence concerning the article, or requests for reprints may be sent to Prof. A. Iarrobino or Prof. M. Levine, at the Mathematics Department, 567 Lake Hall, Northeastern University, Boston, MA 02115.