Journal of Commutative Algebra

Hilbert functions of multigraded algebras, mixed multi- plicities of ideals and their applications

N.V. Trung and J.K. Verma

Full-text: Open access

Article information

Source
J. Commut. Algebra, Volume 2, Number 4 (2010), 515-565.

Dates
First available in Project Euclid: 13 December 2010

Permanent link to this document
https://projecteuclid.org/euclid.jca/1292249710

Digital Object Identifier
doi:10.1216/JCA-2010-2-4-515

Mathematical Reviews number (MathSciNet)
MR2753721

Zentralblatt MATH identifier
1237.13048

Subjects
Primary: 13H15: Multiplicity theory and related topics [See also 14C17]

Keywords
Hilbert function joint reductions mixed volumes Rees algebra fiber cone associated graded ring Milnor number

Citation

Trung, N.V.; Verma, J.K. Hilbert functions of multigraded algebras, mixed multi- plicities of ideals and their applications. J. Commut. Algebra 2 (2010), no. 4, 515--565. doi:10.1216/JCA-2010-2-4-515. https://projecteuclid.org/euclid.jca/1292249710


Export citation

References

  • R. Achilles and M. Manaresi, Multiplicity for ideals of maximal analytic spread and intersection theory, J. Math. Kyoto Univ. 33 (1993), 1029-1046.
  • ––––, Multiplicities of a bigraded ring and intersection theory, Math. Ann. 309 (1997), 573-591.
  • ––––, Generalized Samuel multiplicities and applications, Rend. Sem. Mat., Univ. Politec. Torino 64 (2006), 345-372.
  • R. Achilles and S. Rams, Intersection numbers, Segre numbers and generalized Samuel multiplicities, Arch. Math. (Basel) 77 (2001), 391-398.
  • D.N. Bernstein, The number of roots of a system of equations, Funk. Anal. Pril. 9 (1975), 1-4 (in Russian).
  • P.B. Bhattacharya, The Hilbert function of two ideals, Proc. Cambridge Philos. Soc. 53 (1957), 568-575.
  • T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, Chelsea, New York, 1971.
  • J. Briançon and J.-P. Speder, La trivialité topologique n'implique pas les conditions de Whitney, C.R. Acad. Sci. Paris 280 (1975), A365-A367 (in French).
  • ––––, Les conditions de Whitney impliquent $\mu^*$ constant Ann. Inst. Fourier (Grenoble) 26 (1976), 153-163 (in French).
  • M. Brodmann and J. Rung, Local cohomology and the connectedness dimension in algebraic varieties, Comm. Math. Helv. 61 (1986), 481-490.
  • C. Ciuperca, A numerical characterization of the $S_2$-ification of a Rees algebra, J. Pure Appl. Algebra 178 (2003), 25-48.
  • A. Conca, J. Herzog, N.V. Trung and G. Valla, Diagonal subalgebras of bigraded algebras and embeddings of blow-ups of projective space, American J. Math. 119 (1997), 859-901.
  • N.T. Cuong, P. Schenzel and N.V. Trung, Über verallgemeinerte Cohen-Macaulay-Moduln, Math. Nachr. 85 (1978), 57-73.
  • S.D. Cutkosky and J. Herzog, Cohen-Macaulay coordinate rings of blowup schemes, Comment. Math. Helv. 72 (1997), 605-617.
  • C. D'Cruz, Multigraded Rees algebras of $\mm$-primary ideals in rings of dimension greater than one, J. Pure Appl. Algebra 155 (2000), 131-137.
  • ––––, A formula for the multiplicity of the multigraded extended Rees algebra, Commun. Algebra 31 (2003), 2573-2585.
  • E.C. Dade, Multiplicity and monoidal transforms, thesis, Princeton, 1960.
  • H. Flenner and M. Manaresi, A numerical characterization of reduction ideals, Math. Z. 238 (2001), 205-214.
  • W. Fulton, Introduction to toric varieties, Ann. Math. Stud. 131, Princeton University Press, 1993.
  • T. Gaffney and R. Gassler, Segre numbers and hypersurface singularities, J. Algebraic Geom. 8 (1999), 695-736.
  • I.M. Gelfand, M. Kapranov and A. Zelevinsky, Discriminants, resultants and multidimensional determinants, Birkhäuser, Boston, 1994.
  • M. Herrmann, E. Hyry, J. Ribbe and Z. Tang, Reduction numbers and multiplicities of mutligraded structures, J. Algebra 197 (1997), 311-341.
  • J. Herzog, T. Puthenpurakal and J.K. Verma, Hilbert polynomials and powers of ideals, Math. Proc. Cambridge Philos. Soc. 145 (2008), 623-642.
  • J. Herzog, N.V. Trung and B. Ulrich, On the multiplicity of blow-up rings of ideals generated by $d$-sequences, J. Pure Appl. Algebra 80 (1992), 273-297.
  • D. Hilbert, Über die Theorie der algebraischen Formen, Math. Ann. 36, 473-534.
  • N.D. Hoang, On mixed multiplicities of homogeneous ideals, Beiträge Algebra Geom. 42 (2001), 463-473.
  • ––––, Mixed multiplicities of ideals and of Rees algebras associated with rational normal curves, Acta Math. Vietnam 31 (2006), 61-75.
  • N.D. Hoang and N.V. Trung, Hilbert polynomials of non-standard bigraded algebras, Math. Z. 245 (2003), 309-334.
  • C. Huneke, The theory of $d$-sequences and powers of ideals, Adv. Math. 46 (1982), 249-297.
  • C. Huneke, and J.D. Sally, Birational extensions in dimension two and integrally closed ideals, J. Algebra 115 (1988), 481-500.
  • A.V. Jayanthan, Tony Puthenpurakal and J.K. Verma, On fiber cones of $\mm$-primary ideals, Canad. J. Math. 59 (2007), 109-126.
  • A.V. Jayanthan and J.K. Verma, Grothendieck-Serre formula and bigraded Cohen-Macaulay Rees algebras, J. Algebra 254 (2002), 1-20.
  • D. Katz, Note on multiplicity, Proc. Amer. Math. Soc. 104 (1988), 1021-1026.
  • D. Katz and J.K. Verma, Extended Rees algebras and mixed multiplicities, Math. Z. 202 (1989), 111-128.
  • ––––, On the multiplicity of blowups associated to almost complete intersection space curves, Comm. Algebra 22 (1994), 721-734.
  • A.G. Khovanski, Newton polytopes and toric varieties, Functional Anal. Appl. 11 (1977), 289-298.
  • E. Lasker, Zur Theorie der Moduln und Ideale, Math. Ann. 60 (1905), 20-116.
  • J. Milnor, Singular points of complex hypersurfaces, Ann. Math. Stud. 61, Princeton Univ. Press, Princeton, N.J., 1968.
  • D.G. Northcott, The Hilbert function of the tensor product of two multigraded modules, Mathematika 10 (1963), 43-57.
  • D.G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), 145-158.
  • L. O'Carroll, On two theorems concerning reductions in local rings, J. Math. Kyoto Univ. 27 (1987), 61-67.
  • K.N. Raghavan and A. Simis, Multiplicities of blowups of homogeneous quadratic sequences, J. Algebra 175 (1995), 537-567.
  • K.N. Raghavan and J.K. Verma, Mixed Hilbert coefficients of homogeneous $d$-sequences and quadratic sequences, J. Algebra 195 (1997), 211-232.
  • D. Rees, $\cal A$-transforms of local rings and a theorem on multiplicities of ideals, Proc. Cambridge Philos. Soc. 57 (1961), 8-17.
  • ––––, Multiplicities, Hilbert functions and degree functions, in Commutative algebra, Durham 1981, London Math. Soc. Lecture Note Ser. 72 (1982), 170-178.
  • ––––, The general extension of a local ring and mixed multiplicities, in Algebra, algebraic topology and their interactions, %1983, Lecture Notes Math., 1986.
  • ––––, Generalizations of reductions and mixed multiplicities, J. London Math. Soc. 29 (1984), 397-414.
  • D. Rees and R.Y. Sharp, On a theorem of B. Teissier on multiplicities of ideals in local rings, J. London Math. Soc. 18 (1978), 449-463.
  • P. Roberts, Multiplicities and Chern classes, in Commutative algebra: Syzygies, multiplicities, and birational algebra, Contemp. Math. 159 (1994), 333-350.
  • ––––, Recent developments on Serre's multiplicity conjectures: Gabber's proof of the nonnegativity conjecture, L'Enseignment Mathématique 44 (1998), 305-324.
  • ––––, Intersection multiplicities and Hilbert polynomials, Michigan Math. J. 48 (2000), 517-530.
  • G. Schiffels, Graduierte Ringe und Moduln, Bonn. Math. Schr. 11 (1960), xi+122 pp.
  • A. Simis, N.V. Trung and G. Valla, The diagonal subalgebras of a blow-up ring, J. Pure Appl. Algebra 125 (1998), 305-328.
  • R.P. Stanley, Combinatorics and commutative algebra, Second edition, Birkhäuser, Boston, 1996.
  • J. Stückrad and W. Vogel, An algebraic approach to intersection theory, The curves seminar at Queens, Vol. II (Kingston, Ont., 1981/1982), Exp. No. A, 32 pp., Queen's Papers Pure Appl. Math. 61, Queen's Univ., Kingston, ON, 1982.
  • I. Swanson, Mixed multiplicities, joint reductions and quasi-unmixed local rings, J. London Math. Soc. 48 (1993), 1-14.
  • I. Swanson and C. Huneke, Integral closure of ideals, rings and modules, Cambridge University Press, Cambridge, 2006.
  • B. Teissier, Cycles évanscents, sections planes et conditions de Whitney, Singularities à Cargèse, Astèrisque 7-8 (1973), 285-362.
  • ––––, Introduction to equisingularity problems, Proc. Sympos. Pure Math. 29 (1975), 593-632.
  • B. Teissier, Sur une inégalité à la Minkowski pour les multiplicités, (Appendix to a paper by D. Eisenbud and H.I. Levine) Ann. Math. 106 (1977), 38-44.
  • ––––, On a Minkowski-type inequality for Multiplicities-II, C.P. Ramanujam-A tribute, Tata Inst. Fund. Res. Studies Math. 8 (1978).
  • ––––, Du théorème de l'index de Hodge aux inégalités isopérimétriques, C.R. Acad. Sci. Paris 288 (1979), A287-A289.
  • Le Dung Trang and C.P. Ramanujam, The invariance of Milnor's number implies the invariance of topological type, Amer. J. Math. 98 (1976), 67-78.
  • N.V. Trung, Filter-regular sequences and multiplicity of blow-up rings of ideals of the principal class, J. Math. Kyoto Univ. 33 (1993), 665-683.
  • ––––, The Castelnuovo regularity of the Rees algebra and the associated graded ring, Trans. Amer. Math. Soc. 350 (1998), 2813-2832.
  • ––––, Positivity of mixed multiplicities, Math. Ann. 319 (2001), 33-63.
  • N.V. Trung and J.K. Verma, Mixed multiplicities of ideals versus mixed volumes of polytopes, Trans. Amer. Math. Soc. 359 (2007), 4711-4727.
  • P. Tworzewski, Intersection theory in complex analytic geometry, Ann. Polon. Math. 62 (1995), 177-191.
  • J.K. Verma, A criterion for joint reductions in dimension two, (unpublished) 1987.
  • ––––, Rees algebras and mixed multiplicities, Proc. Amer. Math. Soc. 104 (1988), 1036-1044.
  • ––––, Rees algebras of parameter ideals, J. Pure Appl. Algebra 61 (1989), 99-106.
  • ––––, Joint reductions of complete ideals, Nagoya Math. J. 118 (1990), 155-163.
  • ––––, Joint reduction of Rees algebras, Math. Proc. Cambridge Philos. Soc. 109 (1991), 335-342.
  • ––––, Multigraded Rees algebras and mixed multiplicities, J. Pure and Appl. Algebra 77 (1992), 219-228.
  • J.K. Verma, D. Katz and S. Mandal, Hilbert functions of bigraded algebras, Commutative Algebra (Trieste, 1992), World Sci. Publ., River Edge, NJ, %(1994), 291-302.
  • D.Q. Viet, Mixed multiplicities of arbitrary ideals in local rings, Comm. Algebra 28 (2000), 3803-3821.
  • ––––, On the mixed multiplicity and the multiplicity of blow-up rings of equimultiple ideals, J. Pure Appl. Algebra 183 (2003), 313-327.
  • W. Vogel, Lectures on results on Bezout's theorem, Notes by D.P. Patil, Tata Institute Fund. Research Lectures on Math. Physics 74, Tata Inst. Fund. Res. Bombay, 1984.
  • B.L. van der Waerden, On Hilbert series of composition of ideals and generalisation of a theorem of Bezout, Proc. K. Akad. Wet. Amst. 3 (1928), 749-770.
  • Ken-ichi Yoshida, Cohen-Macaulay symmetric algebras with maximal embedding dimension of almost complete intersection ideals, Comm. Algebra 23 (1995), 653-687.
  • O. Zariski, Some open questions in the theory of singularities, Bull. Amer. Math. Soc. 77 (1971), 481-491.