Let be a Noetherian local ring with . Then, if is a Buchsbaum ring, the first Hilbert coefficients of for parameter ideals are constant and equal to , where denotes the length of the ith local cohomology module of with respect to the maximal ideal . This paper studies the question of whether the converse of the assertion holds true, and proves that is a Buchsbaum ring if is unmixed and the values are constant, which are independent of the choice of parameter ideals in . Hence, a conjecture raised by [GhGHOPV] is settled affirmatively.
"Buchsbaumness in local rings possessing constant first Hilbert coefficients of parameters." Nagoya Math. J. 199 95 - 105, September 2010. https://doi.org/10.1215/00277630-2010-004