Presentation of Associated Graded Rings of Cohen-Macaulay Local Rings

Let <!-- MATH $(R,\mathfrak{m})$ --> be a local ring and be an <!-- MATH $\mathfrak{m}$ --> -primary ideal such that <!-- MATH ${\dim _k}(I/I\mathfrak{m}) = l$ --> , where <!-- MATH $k = R/\mathfrak{m}$ --> . Denote the associated graded ring with respect to <!-- MATH $I, \oplus _{n = 0}^\infty {I^n}/{I^{n + 1}}$ --> , by . Then <!-- MATH ${G_I}(R) \simeq R/I[{X_1}, \ldots {X_l}]/\mathcal{L}$ --> , for some homogeneous ideal <!-- MATH $\mathcal{L}$ --> . Set <!-- MATH $M = \max {\deg _{1 \leqslant i \leqslant t}}{f_i}$ --> , where <!-- MATH $\{ {f_1}, \ldots ,{f_t}\}$ --> is a set of homogeneous elements which form a minimal basis of <!-- MATH $\mathcal{L}$ --> . The main result in this note is that if is a Cohen-Macaulay local ring of dimension 1 and if is free over , then <!-- MATH $M \leqslant r(I) + 1$ --> , where is the reduction number of . It follows that <!-- MATH $M \leqslant e(R)$ --> where is the multiplicity of .