On the Betti polynomials of certain graded ideals

ABSTRACT

Let $S=K[x_1,\dots ,x_n]$ be a polynomial ring over a field K and I be a nonzero graded ideal of S. Then, for t≫0, the Betti number ${\beta }_q(S∕I^t)$ is a polynomial in t, which is denoted by ${𝔅}_q^I\left(t\right)$. It is proved that ${𝔅}_q^I\left(t\right)$ is vanished or of degree ℓ(I)−1 provided I is a monomial ideal generated in a single degree or grade(𝔪R(I)) = codim(𝔪R(I)) where $𝔪=(x_1,\dots ,x_n)$ and R(I) is the Rees ring of I. One lower bound for the leading coefficient of ${𝔅}_q^I\left(t\right)$ is given. When I is a Borel principal monomial ideal, ${𝔅}_q^I\left(t\right)$ is calculated explicitly.

KEYWORDS:

• Betti polynomial
• Borel principal ideal
• degree

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13A30

Acknowledgments

The authors are grateful to Professor J. Herzog for his helpful conversations. The paper was carried out when the second author was visiting the University of Messina, the author would like to thank INDAM (Istituto Nazionale di Alta Matematica, F. Severi, Roma, Italy). The authors thank the anonymous referee for helpful comments.