On Fitting Ideals of Kahler Differential Module

Let k be an algebraically closed field of characteristic zero, and R/I and S/J be algebras over k. Omega(1) (R/I) and Omega(1) (S/J) denote universal module of first order derivation over k. The main result of this paper asserts that the first nonzero Fitting ideal Omega(1) (R/I circle times(k) S/J) is an invertible ideal, if the first nonzero Fitting ideals Omega(1) (R/I) and Omega(1) (S/J) are invertible ideals. Then using this result, we conclude that the projective dimension of Omega(1) (R/I circle times(k) S/J) is less than or equal to one.