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TitleOn the Hilbert function of one-dimensional local complete intersections
Author(s) Juan Elias, M.E. Rossi, G. Valla
TypeArticle in Journal
AbstractAbstract The sequences that occur as Hilbert functions of standard graded algebras A are well understood by Macaulayʼs theorem; those that occur for graded complete intersections are elementary and were known classically. However, much less is known in the local case, once the dimension of A is greater than zero, or the embedding dimension is three or more. Using an extension to the power series ring R of Gröbner bases with respect to local degree orderings, we characterize the Hilbert functions H of one-dimensional quadratic complete intersections A = R / I , I = ( f , g ) , of type ( 2 , 2 ) that is, that are quotients of the power series ring R in three variables by a regular sequence f, g whose initial forms are linearly independent and of degree 2. We also give a structure theorem up to analytic isomorphism of A for the minimal system of generators of I, given the Hilbert function. More generally, when the type of I is ( 2 , b ) we are able to give some restrictions on the Hilbert function. In this case we can also prove that the associated graded algebra of A is Cohen–Macaulay if and only if the Hilbert function of A is strictly increasing.
KeywordsOne-dimensional local rings, Hilbert functions, Complete intersections
ISSN0021-8693
URL http://www.sciencedirect.com/science/article/pii/S0021869313005620
LanguageEnglish
JournalJournal of Algebra
Volume399
Pages489 - 515
Year2014
Edition0
Translation No
Refereed No
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