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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
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