563 research outputs found
Graded Betti numbers of powers of ideals
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 , \dim_k \left(\tor_i^S(I^t, k)_{\mu}
\right) is equal to one of these polynomials in . 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.Comment: 20 page
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