## Nagoya Mathematical Journal

### Buchsbaumness in local rings possessing constant first Hilbert coefficients of parameters

#### Abstract

Let $(A,\mathfrak{m})$ be a Noetherian local ring with $d=\operatorname{dim}A\ge 2$. Then, if $A$ is a Buchsbaum ring, the first Hilbert coefficients $\mathrm{e}_Q^1(A)$ of $A$ for parameter ideals $Q$ are constant and equal to $-\sum_{i=1}^{d-1}\binom{d-2}{i-1}h^i(A)$, where $h^i(A)$ denotes the length of the ith local cohomology module $\mathrm{H}_{\mathfrak{m}}^i(A)$ 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 $\mathrm{e}_Q^1(A)$ are constant, which are independent of the choice of parameter ideals $Q$ in $A$. Hence, a conjecture raised by [GhGHOPV] is settled affirmatively.

#### Article information

Source
Nagoya Math. J., Volume 199 (2010), 95-105.

Dates
First available in Project Euclid: 14 September 2010

https://projecteuclid.org/euclid.nmj/1284471571

Digital Object Identifier
doi:10.1215/00277630-2010-004

Mathematical Reviews number (MathSciNet)
MR2730412

Zentralblatt MATH identifier
1210.13021

#### Citation

Goto, Shiro; Ozeki, Kazuho. Buchsbaumness in local rings possessing constant first Hilbert coefficients of parameters. Nagoya Math. J. 199 (2010), 95--105. doi:10.1215/00277630-2010-004. https://projecteuclid.org/euclid.nmj/1284471571

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