Let $B = k[x_1, \ldots, x_n]$ be a polynomial ring over a field $k$ , and let $A$ be a quotient ring of $B$ by a homogeneous ideal $J$ . Let $\frak{m}$ denote the maximal graded ideal of $A$ . Then the Rees algebra $R = A[{\frak{m}} t]$ also has a presentation as a quotient ring of the polynomial ring $k[x_1, \ldots, x_n, y_1, \ldots, y_n]$ by a homogeneous ideal $J^*$ . For instance, if $A = k[x_1, \ldots, x_n]$ , then \[ R \cong k[x_1, \ldots, x_n, y_1, \ldots, y_n]/(x_i y_j - x_j y_i\mid i,j = 1, \ldots, n). \] In this paper we want to compare the homological properties of the homogeneous ideals $J$ and $J^*$ .