On the Hilbert coefficients, depth of associated graded rings and reduction numbers

Let (R,𝔪) be a d-dimensional Cohen–Macaulay local ring, I be an 𝔪-primary ideal of R and let J=(x1,…,xd) be a minimal reduction of I. We show that if, for i=0 or 1, Jd−1=(x1,…,xd−1) and ∑n=1∞λ(In+1∩Jd−1)∕(J“In∩Jd−1)=i, then depthG(I)≥d−i−1. Moreover, we prove that if e2(I)= ∑n=2∞(n−1)λ(In∕J“In−1)−2, or if e2(I)= ∑n=2∞(n−1)λ(In∕J“In−1)−3 and I is integrally closed, then e1(I)= ∑n=1∞λ(In∕J“In−1)−1, where the integers ei are the Hilbert coefficients of I. In addition, if J is a minimal reduction of I then we prove that the reduction number rJ(I) is independent of J.