×

zbMATH — the first resource for mathematics

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.

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] A. Bagheri, M. Chardin, and H. T. Hà, “The eventual shape of Betti tables of powers of ideals”, Math. Res. Lett. 20:6 (2013), 1033-1046. · Zbl 1307.13018
[2] A. Barvinok, A course in convexity, Graduate Studies in Mathematics 54, Amer. Math. Soc., 2002. · Zbl 1014.52001
[3] G. R. Blakley, “Combinatorial remarks on partitions of a multipartite number”, Duke Math. J. 31 (1964), 335-340. · Zbl 0122.01404
[4] M. Brion and M. Vergne, “Residue formulae, vector partition functions and lattice points in rational polytopes”, J. Amer. Math. Soc. 10:4 (1997), 797-833. · Zbl 0926.52016
[5] S. D. Cutkosky, J. Herzog, and N. V. Trung, “Asymptotic behaviour of the Castelnuovo-Mumford regularity”, Compositio Math. 118:3 (1999), 243-261. · Zbl 0974.13015
[6] D. Eisenbud, Commutative algebra: with a view toward algebraic geometry, Graduate Texts in Mathematics 150, Springer, 1995. · Zbl 0819.13001
[7] V. Kodiyalam, “Homological invariants of powers of an ideal”, Proc. Amer. Math. Soc. 118:3 (1993), 757-764. · Zbl 0780.13007
[8] V. Kodiyalam, “Asymptotic behaviour of Castelnuovo-Mumford regularity”, Proc. Amer. Math. Soc. 128:2 (2000), 407-411. · Zbl 0929.13004
[9] E. Miller and B. Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics 227, Springer, 2005. · Zbl 1090.13001
[10] D. G. Northcott and D. Rees, “Reductions of ideals in local rings”, Proc. Cambridge Philos. Soc. 50 (1954), 145-158. · Zbl 0057.02601
[11] B. Sturmfels, “On vector partition functions”, J. Combin. Theory Ser. A 72:2 (1995), 302-309. · Zbl 0837.11055
[12] G. · Zbl 0823.52002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.