(Co)torsion of exterior powers of differentials over complete intersections

The main goal of this paper is to establish a generalized Lipman-Zariski result in characteristic zero for complete intersection germs whose singular locus has codimension at least three, generalizing the corresponding result of Graf for hypersurfaces. More precisely, we prove that the condition that the sheaf of reflexive Kähler differential p-forms is free implies smoothness. The proof given here rests on recognizing the torsion, as was known previously, and the cotorsion as homologies of the generalized Koszul complexes constructed by Kirby, Buchsbaum-Eisenbud, Lebelt, and Bruns-Vetter and applying certain rigidity and symmetry results based on work of Lebelt and Rodicio, yielding also a different proof of Graf’s result. To make the paper accessible to both complex analysts and algebraic geometers we include full descriptions of the necessary background.