Presentation of associated graded rings of Cohen-Macaulay local rings

Abstract:

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