The sequences that occur as Hilbert functions of standard graded algebras $A$ are well understood by Macaulay's 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 Gr\"{o}bner 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 two. 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.
On the Hilbert function of one-dimensional local complete intersections
ROSSI, MARIA EVELINA;
2014-01-01
Abstract
The sequences that occur as Hilbert functions of standard graded algebras $A$ are well understood by Macaulay's 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 Gr\"{o}bner 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 two. 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.
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