Mixed multiplicities of filtrations

Cutkosky, Steven; Sarkar, Parangama; Srinivasan, Hema

doi:10.1090/tran/7745

Mixed multiplicities of filtrations
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by Steven Dale Cutkosky, Parangama Sarkar and Hema Srinivasan PDF

Trans. Amer. Math. Soc. 372 (2019), 6183-6211 Request permission

Abstract:

In this paper we define and explore properties of mixed multiplicities of (not necessarily Noetherian) filtrations of $m_R$-primary ideals in a Noetherian local ring $R$, generalizing the classical theory for $m_R$-primary ideals. We construct a real polynomial whose coefficients give the mixed multiplicities. This polynomial exists if and only if the dimension of the nilradical of the completion of $R$ is less than the dimension of $R$, which holds, for instance, if $R$ is excellent and reduced. We show that many of the classical theorems for mixed multiplicities of $m_R$-primary ideals hold for filtrations, including the famous Minkowski inequalities of Teissier, and of Rees and Sharp.

References

Shreeram Shankar Abhyankar, Resolution of singularities of embedded algebraic surfaces, Pure and Applied Mathematics, Vol. 24, Academic Press, New York-London, 1966. MR 0217069
P. B. Bhattacharya, The Hilbert function of two ideals, Proc. Cambridge Philos. Soc. 53 (1957), 568–575. MR 89835
Nicolas Bourbaki, Commutative algebra. Chapters 1–7, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1972 edition. MR 979760
V. Cossart, U. Jannsen, and S. Saito, Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes, arXiv:0905.2191 (2009).
Steven Dale Cutkosky, On unique and almost unique factorization of complete ideals. II, Invent. Math. 98 (1989), no. 1, 59–74. MR 1010155, DOI 10.1007/BF01388844
Steven Dale Cutkosky, Multiplicities associated to graded families of ideals, Algebra Number Theory 7 (2013), no. 9, 2059–2083. MR 3152008, DOI 10.2140/ant.2013.7.2059
Steven Dale Cutkosky, Asymptotic multiplicities of graded families of ideals and linear series, Adv. Math. 264 (2014), 55–113. MR 3250280, DOI 10.1016/j.aim.2014.07.004
Steven Dale Cutkosky, Asymptotic multiplicities, J. Algebra 442 (2015), 260–298. MR 3395062, DOI 10.1016/j.jalgebra.2015.03.017
Steven Dale Cutkosky, Jürgen Herzog, and Ana Reguera, Poincaré series of resolutions of surface singularities, Trans. Amer. Math. Soc. 356 (2004), no. 5, 1833–1874. MR 2031043, DOI 10.1090/S0002-9947-03-03346-4
S. D. Cutkosky and V. Srinivas, On a problem of Zariski on dimensions of linear systems, Ann. of Math. (2) 137 (1993), no. 3, 531–559. MR 1217347, DOI 10.2307/2946531
Lawrence Ein, Robert Lazarsfeld, and Karen E. Smith, Uniform approximation of Abhyankar valuation ideals in smooth function fields, Amer. J. Math. 125 (2003), no. 2, 409–440. MR 1963690
K. Goel, R. V. Gurjar, and J. K. Verma, The Minkowski (in)equality for multiplicity of ideals, arXiv:1807.06309 (2018).
Manfred Herrmann, Eero Hyry, Jürgen Ribbe, and Zhongming Tang, Reduction numbers and multiplicities of multigraded structures, J. Algebra 197 (1997), no. 2, 311–341. MR 1483767, DOI 10.1006/jabr.1997.7128
June Huh, Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs, J. Amer. Math. Soc. 25 (2012), no. 3, 907–927. MR 2904577, DOI 10.1090/S0894-0347-2012-00731-0
Nathan Jacobson, Basic algebra. I, 2nd ed., W. H. Freeman and Company, New York, 1985. MR 780184
Daniel Katz, Note on multiplicity, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1021–1026. MR 929434, DOI 10.1090/S0002-9939-1988-0929434-8
Kiumars Kaveh and A. G. Khovanskii, Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Ann. of Math. (2) 176 (2012), no. 2, 925–978. MR 2950767, DOI 10.4007/annals.2012.176.2.5
Kiumars Kaveh and Askold Khovanskii, Convex bodies and multiplicities of ideals, Proc. Steklov Inst. Math. 286 (2014), no. 1, 268–284. Reprint of Tr. Mat. Inst. Steklova 286 (2014), 291–307. MR 3482603, DOI 10.1134/S0081543814060169
Daniel Katz and J. K. Verma, Extended Rees algebras and mixed multiplicities, Math. Z. 202 (1989), no. 1, 111–128. MR 1007742, DOI 10.1007/BF01180686
S. Lang, Algebra, Springer-Verlag, New York–Berlin–Heidelberg, 2002. Revised 3rd ed.
Joseph Lipman, Desingularization of two-dimensional schemes, Ann. of Math. (2) 107 (1978), no. 1, 151–207. MR 491722, DOI 10.2307/1971141
Robert Lazarsfeld and Mircea Mustaţă, Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 5, 783–835 (English, with English and French summaries). MR 2571958, DOI 10.24033/asens.2109
Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
H. T. Muhly and M. Sakuma, Asymptotic factorization of ideals, J. London Math. Soc. 38 (1963), 341–350. MR 159837, DOI 10.1112/jlms/s1-38.1.341
Mircea Mustaţǎ, On multiplicities of graded sequences of ideals, J. Algebra 256 (2002), no. 1, 229–249. MR 1936888, DOI 10.1016/S0021-8693(02)00112-6
Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
Andrei Okounkov, Why would multiplicities be log-concave?, The orbit method in geometry and physics (Marseille, 2000) Progr. Math., vol. 213, Birkhäuser Boston, Boston, MA, 2003, pp. 329–347. MR 1995384
D. Rees, ${\mathfrak {a}}$-transforms of local rings and a theorem on multiplicities of ideals, Proc. Cambridge Philos. Soc. 57 (1961), 8–17. MR 118750, DOI 10.1017/s0305004100034800
D. Rees, Multiplicities, Hilbert functions and degree functions, Commutative algebra: Durham 1981 (Durham, 1981) London Math. Soc. Lecture Note Ser., vol. 72, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 170–178. MR 693635
D. Rees and R. Y. Sharp, On a theorem of B. Teissier on multiplicities of ideals in local rings, J. London Math. Soc. (2) 18 (1978), no. 3, 449–463. MR 518229, DOI 10.1112/jlms/s2-18.3.449
Irena Swanson, Mixed multiplicities, joint reductions and quasi-unmixed local rings, J. London Math. Soc. (2) 48 (1993), no. 1, 1–14. MR 1223888, DOI 10.1112/jlms/s2-48.1.1
Craig Huneke and Irena Swanson, Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006. MR 2266432
Bernard Teissier, Cycles évanescents, sections planes et conditions de Whitney, Singularités à Cargèse (Rencontre Singularités Géom. Anal., Inst. Études Sci., Cargèse, 1972) Astérisque, Nos. 7 et 8, Soc. Math. France, Paris, 1973, pp. 285–362 (French). MR 0374482
B. Teissier, Sur une inégalité pour les multiplicitiés, Ann. of Math. 106 (1977), 38–44. Appendix to a paper by D. Eisenbud and H. Levine.
B. Teissier, On a Minkowski-type inequality for multiplicities. II, C. P. Ramanujam—a tribute, Tata Inst. Fund. Res. Studies in Math., vol. 8, Springer, Berlin-New York, 1978, pp. 347–361. MR 541030
Ngo Viet Trung and Jugal Verma, Mixed multiplicities of ideals versus mixed volumes of polytopes, Trans. Amer. Math. Soc. 359 (2007), no. 10, 4711–4727. MR 2320648, DOI 10.1090/S0002-9947-07-04054-8
Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581