## Illinois Journal of Mathematics

### Looking for minimal graded Betti numbers

#### Abstract

We consider $O$-sequences that occur for arithmetically Cohen-Macaulay (ACM) schemes $X$ of codimension three in ${\pp}^n$. These are Hilbert functions $\varphi$ of Artinian algebras that are quotients of the coordinate ring of $X$ by a linear system of parameters. Using suitable decompositions of $\varphi$, we determine the minimal number of generators possible in some degree $c$ for the defining ideal of any such ACM scheme having the given $O$-sequence. We apply this result to construct Artinian Gorenstein $O$-sequences $\varphi$ of codimension $3$ such that the poset of all graded Betti sequences of the Artinian Gorenstein algebras with Hilbert function $\varphi$ admits more than one minimal element. Finally, for all $3$-codimensional complete intersection $O$-sequences we obtain conditions under which the corresponding poset of graded Betti sequences has more than one minimal element.

#### Article information

Source
Illinois J. Math., Volume 49, Number 2 (2005), 453-473.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138028

Digital Object Identifier
doi:10.1215/ijm/1258138028

Mathematical Reviews number (MathSciNet)
MR2164346

Zentralblatt MATH identifier
1097.13020

Subjects
Primary: 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
Secondary: 13D02: Syzygies, resolutions, complexes

#### Citation

Ragusa, Alfio; Zappalà, Giuseppe. Looking for minimal graded Betti numbers. Illinois J. Math. 49 (2005), no. 2, 453--473. doi:10.1215/ijm/1258138028. https://projecteuclid.org/euclid.ijm/1258138028