Hilbert functions of d-regular ideals

Abstract

In the present paper, we characterize all possible Hilbert functions of graded ideals in a polynomial ring whose regularity is smaller than or equal to $d$, where $d$ is a positive integer. In addition, we prove the following result which is a generalization of Bigatti, Hulett and Pardue's result: Let $p \geq 0$ and $d>0$ be integers. If the base field is a field of characteristic 0 and there is a graded ideal $I$ whose projective dimension $\mathrm{proj\ dim}(I)$ is smaller than or equal to $p$ and whose regularity $\mathrm{reg}(I)$ is smaller than or equal to $d$, then there exists a monomial ideal $L$ having the maximal graded Betti numbers among graded ideals $J$ which have the same Hilbert function as $I$ and which satisfy $\mathrm{proj dim}(J) \leq p$ and $\mathrm{reg}(J) \leq d$. We also prove the same fact for squarefree monomial ideals. The main methods for proofs are generic initial ideals and combinatorics on strongly stable ideals.

33 pages, minor changes, to appear in J. Algebra