Freiman ideals and the number of generators of powers of monomial ideals

Abstract

Let $\mu (I)$ denote the number of generators of a monomial ideal I. It is well known that $\mu (I^k)<\mu (I^{k+1})$ for $k\gg 0.$ In this paper we construct monomial ideals I in $F[x,y]$ such that $\mu (I^{k+1})<\mu (I^k)$ for all $k\le l,$ given any positive integer l. Also, we extend some results of Eliahou et al. by constructing monomial ideals in $R=F[x_1,\dots ,x_n]$ with $\mu (I^2)<\mu (I)$ and investigate $\mu (I^k)$ for monomial ideals in R. Furthermore, we generalize the definition of Freiman ideals given in Herzog and Zhu and extend some results with simpler proofs. In particular, we give a complete characterization of Freiman ideals of maximum height in R.

Keywords:

• Freiman ideals
• number of generators
• powers of ideals
• Ratliff-Rush closure

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13E15
• 13F20
• 05E40