Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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GRADED BETTI NUMBERS OF GOOD FILTRATIONS

  • Lamei, Kamran (School of Mathematics Statistics and Computer Science University of Tehran) ;
  • Yassemi, Siamak (School of Mathematics Statistics and Computer Science University of Tehran)
  • Received : 2019.10.11
  • Accepted : 2020.07.09
  • Published : 2020.09.30
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Abstract

The asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field has recently been reviewed. We extend quasi-polynomial behavior of graded Betti numbers of powers of homogenous ideals to ℤ-graded algebra over Noetherian local ring. Furthermore our main result treats the Betti table of filtrations which is finite or integral over the Rees algebra.

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References

  1. A. Bagheri, M. Chardin, and H. T. Ha, The eventual shape of Betti tables of powers of ideals, Math. Res. Lett. 20 (2013), no. 6, 1033-1046. https://doi.org/10.4310/MRL.2013.v20.n6.a3
  2. A. Bagheri and K. Lamei, Graded Betti numbers of powers of ideals, J. Commut. Algebra 12 (2020), no. 2, 153-169. https://doi.org/10.1216/jca.2020.12.153
  3. A. Barvinok and J. E. Pommersheim, An algorithmic theory of lattice points in polyhedra, in New perspectives in algebraic combinatorics (Berkeley, CA, 1996-97), 91-147, Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999.
  4. N. Bourbaki. Commutative Algebra, Addison Wesley, Reading, 1972.
  5. M. Brion, Points entiers dans les polyedres convexes, Ann. Sci. Ecole Norm. Sup. (4) 21 (1988), no. 4, 653-663. https://doi.org/10.24033/asens.1572
  6. M. Brion and M. Vergne, Residue formulae, vector partition functions and lattice points in rational polytopes, J. Amer. Math. Soc. 10 (1997), no. 4, 797-833. https://doi.org/10.1090/S0894-0347-97-00242-7
  7. J. A. De Loera, R. Hemmecke, J. Tauzer, and R. Yoshida, Effective lattice point counting in rational convex polytopes, J. Symbolic Comput. 38 (2004), no. 4, 1273-1302. https://doi.org/10.1016/j.jsc.2003.04.003
  8. L. T. Hoa and S. Zarzuela, Reduction number and a-invariant of good filtrations, Comm. Algebra 22 (1994), no. 14, 5635-5656. https://doi.org/10.1080/00927879408825151
  9. V. Kodiyalam, Homological invariants of powers of an ideal, Proc. Amer. Math. Soc. 118 (1993), no. 3, 757-764. https://doi.org/10.2307/2160118
  10. E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, Graduate Texts in Mathematics, 227, Springer-Verlag, New York, 2005.
  11. J. S. Okon and L. J. Ratliff, Jr., Reductions of filtrations, Pacific J. Math. 144 (1990), no. 1, 137-154. http://projecteuclid.org/euclid.pjm/1102645830 https://doi.org/10.2140/pjm.1990.144.137
  12. L. J. Ratliff, Jr., and D. E. Rush, Two notes on reductions of ideals, Indiana Univ. Math. J. 27 (1978), no. 6, 929-934. https://doi.org/10.1512/iumj.1978.27.27062
  13. D. Rees, A note on analytically unramified local rings, J. London Math. Soc. 36 (1961), 24-28. https://doi.org/10.1112/jlms/s1-36.1.24
  14. R. P. Stanley, Combinatorial reciprocity theorems, in Combinatorics, Part 2: Graph theory; foundations, partitions and combinatorial geometry (Proc. Adv. Study Inst., Breukelen, 1974), 107-118. Math. Centre Tracts, 56, Math. Centrum, Amsterdam, 1974.
  15. B. Sturmfels, On vector partition functions, J. Combin. Theory Ser. A 72 (1995), no. 2, 302-309. https://doi.org/10.1016/0097-3165(95)90067-5