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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.

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
Full Text: DOI
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