On fitting ideals of second order universal modules of derivatives

Let k be an algebraically closed field with characteristic zero, [Formula: see text] and S be polynomial algebras over k. In [ 15 ], it has been shown the first nonzero Fitting ideals of [Formula: see text] and [Formula: see text] are invertible, then so is the first nonzero Fitting ideal of [Formula: see text]. It is found that the projective dimension of [Formula: see text] is less than or equal to 1. Let [Formula: see text] and [Formula: see text] denote second-order derivation of universal module over k. In [ 12 ], it has been shown that [Formula: see text] when [Formula: see text]. Alternatively, we show that [Formula: see text] is at most one by using properties of Fitting ideals and invertible Fitting ideals.