arXiv subjects
Publications and source records attributed to T. T. Phuong.
A problem posed by Wolmer V. Vasconcelos on the variation of the first Hilbert coefficients of parameter ideals with a common integral closure in a local ring is studied. Affirmative answers are given and counterexamples are explored as well.
This gives an alternate proof of the Theorem by the authors that shows the first Hilbert coefficient of parameter ideals in an unmixed Noetherian local ring is always negative unless the ring is Cohen--Macaulay.
The conjecture of Wolmer Vasconcelos on the vanishing of the first Hilbert coefficient $e_1(Q)$ is solved affirmatively, where $Q$ is a parameter ideal in a Noetherian local ring. Basic properties of the rings for which $e_1(Q)$ vanishes are derived. The invariance of $e_1(Q)$ for parameter ideals $Q$ and its relationship to Buchsbaum rings are studied.