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Graded Betti numbers of powers of ideals. (English) Zbl 1451.13015
In the paper under review, the authors investigate the asymptotic behaviour of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field. More precisely, they prove that if the polynomial ring is equipped with a positive \(\mathbb{Z}^d\)-grading, then the Betti numbers of powers of ideals are encoded by finitely many polynomials. In particular, for \(d=1\) this result states that there is a decomposition of \(\mathbb{Z}^2\) in a finite number of regions such that for each region there is a polynomial in \(\mu\) and \(t\) that computes \(\dim_k (Tor_i^S(I^t,k))_\mu \). This refines, in a graded situation, the result of [V. Kodiyalam, Proc. Am. Math. Soc. 118, No. 3, 757–764 (1993; Zbl 0780.13007)] on Betti numbers of powers of ideals.
More generally, the authors investigate the case of a power products of homogeneous ideals \(I_1,\ldots,I_s\) in a \(\mathbb{Z}^d\)-graded algebra, for a positive grading.

13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13D02 Syzygies, resolutions, complexes and commutative rings
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
Full Text: DOI Euclid
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