Cohen–Macaulayness and type of the symmetric algebra of the inversion-factor ideal

Let $R$ be the polynomial ring in the variables ${\bf x}$, and let $D_1,\ldots,D_d$ be the inversion factors associated to the basic representatives of a general linear $(d+1)\times d$ matrix. Set

$$\mathfrak{D}:=(D_1,\ldots,D_d)\subset R.$$

The symmetric algebra $\operatorname{Sym}_R(\mathfrak{D})$ is Cohen–Macaulay of dimension $d+1$ and type two.

This concerns the homological structure of the ideal generated by the inversion factors arising from the basic representatives. The preceding theorem establishes a codimension-two ideal and gives a minimal free resolution, but the asserted Cohen–Macaulayness, dimension, and type of its symmetric algebra are not established in the supplied text.