Graded Betti numbers of powers of ideals
Abstract
Using the concept of vector partition functions, we investigate the asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field. Our main results state that if the polynomial ring is equipped with a positive $\ZZ$-grading, then the Betti numbers of powers of ideals are encoded by finitely many polynomials. More precisely, in the case of $\ZZ$-grading, $\ZZ^2$ can be splitted into a finite number of regions such that each region corresponds to a polynomial that depending to the degree $(\mu, t)$, $\dim_k \left(\tor_i^S(I^t, k)_{\mu} \right)$ is equal to one of these polynomials in $(\mu, t)$. This refines, in a graded situation, the result of Kodiyalam on Betti numbers of powers of ideals. Our main statement treats the case of a power products of homogeneous ideals in a $\ZZ^d$-graded algebra, for a positive grading.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2013
- DOI:
- 10.48550/arXiv.1308.0943
- arXiv:
- arXiv:1308.0943
- Bibcode:
- 2013arXiv1308.0943B
- Keywords:
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- Mathematics - Commutative Algebra;
- 13D45;
- 13D02
- E-Print:
- 20 pages