Presentation of associated graded rings of Cohen-Macaulay local rings

Abstract

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