A bound on the reduction number of a primary ideal

Abstract

Let ( A , M ) ( A,\mathcal {M}) be a local ring of positive dimension d d and let I I be an M \mathcal {M} -primary ideal. We denote the reduction number of I I by r ( I ) r(I) , which is the smallest integer r r such that I r + 1 = J I r I^{r+1}=JI^r for some reduction J J of I . I. In this paper we give an upper bound on r ( I ) r(I) in terms of numerical invariants which are related with the Hilbert coefficients of I I when A A is Cohen-Macaulay. If d = 1 d=1 , it is known that r ( I ) ≤ e ( I ) − 1 r(I) \le e(I) -1 where e ( I ) e(I) denotes the multiplicity of I . I. If d ≤ 2 , d \le 2, in Corollary 1.5 we prove r ( I ) ≤ e 1 ( I ) − e ( I ) + λ ( A / I ) + 1 r(I) \le e_1(I) - e(I) + \lambda (A/I) + 1 where e 1 ( I ) e_1(I) is the first Hilbert coefficient of I . I. From this bound several results follow. Theorem 1.3 gives an upper bound on r ( I ) r(I) in a more general setting.