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Buchsbaumness in local rings possessing constant first Hilbert coefficients of parameters. (English) Zbl 1210.13021
Let $$A$$ be a Noetherian local ring with maximal ideal $$\mathfrak{m},$$ and let $$d=\dim A>0.$$ Let $$\{e_I(A)\}_{0\leq i\leq d}$$ be the Hilbert coefficients of $$A$$ with respect to an $$\mathfrak{m}$$-primary ideal $$I.$$ $$A$$ is said unmixed if $$\dim\widehat{A}/\mathfrak{p}=d,$$ for every $$\mathfrak{p}\in \text{Ass}\widehat{A}.$$ The main result of the paper states that if $$A$$ is unmixed and $$d\geq 2$$ then $$A$$ is a Buchsbaum local ring iff the first Hilbert coefficients $$e^1_Q(A)$$ are constant and independent of the choice of parameter ideals $$Q$$ in $$A.$$ Moreover the authors give a characterization of Noetherian local rings such that the set $$\{e^1_Q(A)\mid Q \text{ is a parameter ideal of }A\}$$ has only one element.

##### MSC:
 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 13B22 Integral closure of commutative rings and ideals 13H15 Multiplicity theory and related topics
##### Keywords:
Buchsbaum rings; local rings; Hilbert coefficients
Full Text:
##### References:
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