Buchsbaumness in local rings possessing constant first Hilbert coefficients of parameters

Shiro Goto; Kazuho Ozeki

doi:10.1215/00277630-2010-004

Nagoya Mathematical Journal

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

Buchsbaumness in local rings possessing constant first Hilbert coefficients of parameters

Shiro Goto and Kazuho Ozeki

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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

Permanent link to this document
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

Subjects
Primary: 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05]
Secondary: 13A30: Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 13B22: Integral closure of rings and ideals [See also 13A35]; integrally closed rings, related rings (Japanese, etc.) 13H15: Multiplicity theory and related topics [See also 14C17]

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

References

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Digital Object Identifier: doi:10.1307/mmj/1220879434
Project Euclid: euclid.mmj/1220879434

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Nagoya Mathematical Journal

Buchsbaumness in local rings possessing constant first Hilbert coefficients of parameters

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