The equality of Elias–Valla and the associated graded ring of maximal ideals

Abstract

Let (A,m) be a Noetherian local ring with d=dimA>0 and Q be a parameter ideal in A which forms a reduction of maximal ideal m of A. In this article, we prove the Buchsbaumness of the associated graded ring of m in a Buchsbaum local ring A satisfying the equality 2e0(m)−e1(m)+e1(Q)=v(A)−d+2 of Elias and Valla, where e0(m), e1(m), and e1(Q) denote the Hilbert coefficients of m and Q, v(A) the embedding dimension of A, respectively. Hence a conjecture raised by Corso [1] is settled affirmatively.