Bounds for the reduction number of primary ideal in dimension three

Abstract

Let ( R , m ) (R,\mathfrak {m}) be a Cohen-Macaulay local ring of dimension d ≥ 3 d\geq 3 and I I an m \mathfrak {m} -primary ideal of R R . Let r J ( I ) r_J(I) be the reduction number of I I with respect to a minimal reduction J J of I I . Suppose depth ⁡ G ( I ) ≥ d − 3 \operatorname {depth} G(I)\geq d-3 . We prove that r J ( I ) ≤ e 1 ( I ) − e 0 ( I ) + λ ( R / I ) + 1 + ( e 2 ( I ) − 1 ) e 2 ( I ) − e 3 ( I ) r_J(I)\leq e_1(I)-e_0(I)+\lambda (R/I)+1+(e_2(I)-1)e_2(I)-e_3(I) , where e i ( I ) e_i(I) are Hilbert coefficients. Suppose d = 3 d=3 and depth ⁡ G ( I t ) > 0 \operatorname {depth} G(I^t)>0 for some t ≥ 1 t\geq 1 . Then we prove that r J ( I ) ≤ e 1 ( I ) − e 0 ( I ) + λ ( R / I ) + t r_J(I)\leq e_1(I)-e_0(I)+\lambda (R/I)+t .