Buchsbaumness in local rings possessing constant first Hilbert coefficients of parameters

Abstract

Let $\left(A,\mathfrak{m}\right)$ be a Noetherian local ring with $d=dimA\ge 2$. Then, if $A$ is a Buchsbaum ring, the first Hilbert coefficients ${\mathrm{e}}_Q^1\left(A\right)$ of $A$ for parameter ideals $Q$ are constant and equal to $-\sum _{i=1}^{d-1}\left(\genfrac{}{}{0ex}{}{d-2}{i-1}\right)h^i\left(A\right)$, where $h^i\left(A\right)$ denotes the length of the ith local cohomology module ${\mathrm{H}}_{\mathfrak{m}}^i\left(A\right)$ of $A$ with respect to the maximal ideal $\mathfrak{m}$. This paper studies the question of whether the converse of the assertion holds true, and proves that $A$ is a Buchsbaum ring if $A$ is unmixed and the values ${\mathrm{e}}_Q^1\left(A\right)$ are constant, which are independent of the choice of parameter ideals $Q$ in $A$. Hence, a conjecture raised by [GhGHOPV] is settled affirmatively.