Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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MIXED MULTIPLICITIES OF MAXIMAL DEGREES

  • Received : 2017.05.23
  • Accepted : 2017.10.25
  • Published : 2018.05.01
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Abstract

The original mixed multiplicity theory considered the class of mixed multiplicities concerning the terms of highest total degree in the Hilbert polynomial. This paper defines a broader class of mixed multiplicities that concern the maximal terms in this polynomial, and gives many results, which are not only general but also more natural than many results in the original mixed multiplicity theory.

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References

  1. M. P. Brodmann and R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge Studies in Advanced Mathematics, 60, Cambridge University Press, Cambridge, 1998.
  2. R. Callejas-Bedregal and V. H. Jorge Perez, Mixed multiplicities and the minimal number of generator of modules, J. Pure Appl. Algebra 214 (2010), no. 9, 1642-1653. https://doi.org/10.1016/j.jpaa.2009.12.009
  3. R. Callejas-Bedregal and V. H. Jorge Perez, Multiplicities for arbitrary modules and reduction, Rocky Mountain J. Math. 43 (2013), no. 4, 1077-1113. https://doi.org/10.1216/RMJ-2013-43-4-1077
  4. L. V. Dinh, N. T. Manh, and T. T. H. Thanh, On some superficial sequences, Southeast Asian Bull. Math. 38 (2014), no. 6, 803-811.
  5. M. Herrmann, E. Hyry, J. Ribbe, and Z. Tang, Reduction numbers and multiplicities of multigraded structures, J. Algebra 197 (1997), no. 2, 311-341. https://doi.org/10.1006/jabr.1997.7128
  6. D. Katz and J. K. Verma, Extended Rees algebras and mixed multiplicities, Math. Z. 202 (1989), no. 1, 111-128. https://doi.org/10.1007/BF01180686
  7. D. Kirby and D. Rees, Multiplicities in graded rings. I. The general theory, in Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992), 209-267, Contemp. Math., 159, Amer. Math. Soc., Providence, RI, 1994.
  8. D. Kirby and D. Rees, Multiplicities in graded rings. II. Integral equivalence and the Buchsbaum-Rim multiplicity, Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 3, 425-445. https://doi.org/10.1017/S0305004100074326
  9. S. Kleiman and A. Thorup, Mixed Buchsbaum-Rim multiplicities, Amer. J. Math. 118 (1996), no. 3, 529-569. https://doi.org/10.1353/ajm.1996.0026
  10. N. T. Manh and D. Q. Viet, Mixed multiplicities of modules over Noetherian local rings, Tokyo J. Math. 29 (2006), no. 2, 325-345. https://doi.org/10.3836/tjm/1170348171
  11. D. Rees, Generalizations of reductions and mixed multiplicities, J. London Math. Soc. (2) 29 (1984), no. 3, 397-414.
  12. P. Roberts, Local Chern classes, multiplicities, and perfect complexes, Mem. Soc. Math. France (N.S.) No. 38 (1989), 145-161.
  13. J. Stuckrad and W. Vogel, Buchsbaum Rings and Applications, Mathematische Monographien, 21, VEB Deutscher Verlag der Wissenschaften, Berlin, 1986.
  14. I. Swanson, Mixed multiplicities, joint reductions and quasi-unmixed local rings, J. London Math. Soc. (2) 48 (1993), no. 1, 1-14.
  15. B. Teissier, Cycles evanescents, sections planes et conditions de Whitney, in Singularites a Cargese (Rencontre Singularites Geom. Anal., Inst. Etudes Sci., Cargese, 1972), 285-362. Asterisque, Nos. 7 et 8, Soc. Math. France, Paris.
  16. N. V. Trung, Reduction exponent and degree bound for the defining equations of graded rings, Proc. Amer. Math. Soc. 101 (1987), no. 2, 229-236. https://doi.org/10.1090/S0002-9939-1987-0902533-1
  17. N. V. Trung, Positivity of mixed multiplicities, Math. Ann. 319 (2001), no. 1, 33-63. https://doi.org/10.1007/PL00004429
  18. N. V. Trung and J. Verma, Mixed multiplicities of ideals versus mixed volumes of polytopes, Trans. Amer. Math. Soc. 359 (2007), no. 10, 4711-4727. https://doi.org/10.1090/S0002-9947-07-04054-8
  19. J. K. Verma, Multigraded Rees algebras and mixed multiplicities, J. Pure Appl. Algebra 77 (1992), no. 2, 219-228. https://doi.org/10.1016/0022-4049(92)90087-V
  20. D. Q. Viet, Mixed multiplicities of arbitrary ideals in local rings, Comm. Algebra 28 (2000), no. 8, 3803-3821. https://doi.org/10.1080/00927870008827059
  21. D. Q. Viet, On some properties of (FC)-sequences of ideals in local rings, Proc. Amer. Math. Soc. 131 (2003), no. 1, 45-53. https://doi.org/10.1090/S0002-9939-02-06526-7
  22. D. Q. Viet, Sequences determining mixed multiplicities and reductions of ideals, Comm. Algebra 31 (2003), no. 10, 5047-5069. https://doi.org/10.1081/AGB-120023147
  23. D. Q. Viet, Reductions and mixed multiplicities of ideals, Comm. Algebra 32 (2004), no. 11, 4159-4178. https://doi.org/10.1081/AGB-200034021
  24. D. Q. Viet, The multiplicity and the Cohen-Macaulayness of extended Rees algebras of equimultiple ideals, J. Pure Appl. Algebra 205 (2006), no. 3, 498-509. https://doi.org/10.1016/j.jpaa.2005.07.018
  25. D. Q. Viet and L. V. Dinh, On mixed multiplicities of good filtrations, Algebra Colloq. 22 (2015), no. 3, 421-436. https://doi.org/10.1142/S1005386715000371
  26. D. Q. Viet, L. V. Dinh, and T. T. H. Thanh, A note on joint reductions and mixed multiplicities, Proc. Amer. Math. Soc. 142 (2014), no. 6, 1861-1873. https://doi.org/10.1090/S0002-9939-2014-11916-2
  27. D. Q. Viet and N. T. Manh, Mixed multiplicities of multigraded modules, Forum Math. 25 (2013), no. 2, 337-361.
  28. D. Q. Viet and T. T. H. Thanh, On (FC)-sequences and mixed multiplicities of multi-graded algebras, Tokyo J. Math. 34 (2011), no. 1, 185-202. https://doi.org/10.3836/tjm/1313074450
  29. D. Q. Viet and T. T. H. Thanh, On some multiplicity and mixed multiplicity formulas, Forum Math. 26 (2014), no. 2, 413-442. https://doi.org/10.1515/form.2011.168
  30. D. Q. Viet and T. T. H. Thanh, A note on formulas transmuting mixed multiplicities, Forum Math. 26 (2014), no. 6, 1837-1851. https://doi.org/10.1515/forum-2011-0147
  31. D. Q. Viet and T. T. H. Thanh, The Euler-Poincare characteristic and mixed multiplicities, Kyushu J. Math. 69 (2015), no. 2, 393-411. https://doi.org/10.2206/kyushujm.69.393
  32. D. Q. Viet and T. T. H. Thanh, On filter-regular sequences of multi-graded modules, Tokyo J. Math. 38 (2015), no. 2, 439-457. https://doi.org/10.3836/tjm/1452806049