Abstract
We develop a theory of multiplicities and mixed multiplicities of filtrations, extending the theory for filtrations of m-primary ideals to arbitrary (not necessarily Noetherian) filtrations. The mixed multiplicities of r filtrations on an analytically unramified local ring R come from the coefficients of a suitable homogeneous polynomial in r variables of degree equal to the dimension of the ring, analogously to the classical case of the mixed multiplicities of m-primary ideals in a local ring. We prove that the Minkowski inequalities hold for arbitrary filtrations. The characterization of equality in the Minkowski inequality for m-primary ideals in a local ring by Teissier, Rees and Sharp and Katz does not extend to arbitrary filtrations, but we show that they are true in a large and important subcategory of filtrations. We define divisorial and bounded filtrations. The filtration of powers of a fixed ideal is a bounded filtration, as is a divisorial filtration. We show that in an excellent local domain, the characterization of equality in the Minkowski equality is characterized by the condition that the integral closures of suitable Rees like algebras are the same, strictly generalizing the theorem of Teissier, Rees and Sharp and Katz. We also prove that a theorem of Rees characterizing the inclusion of ideals with the same multiplicity generalizes to bounded filtrations in excellent local domains. We give a number of other applications, extending classical theorems for ideals.
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References
Bhattacharya, P.B.: The Hilbert function of two ideals. Proc. Camb. Phil. Soc. 53, 568–575 (1957)
Bourbaki, N.: Commutative Algebra, Chapters 1–7. Springer Verlag, Berlin (1989)
Böger, E.: Einige Bemerkungen zur Theorie der ganzalgebraischen Abhandigkeit von Idealen. Math. Ann. 185, 303–308 (1970)
Brodmann, M.: Asymptotic stability of \(\text{ Ass }(M/I^nM)\) Proc. Am. Math. Soc. 74, 16–18 (1979)
Bruns, W., Herzog, J.: Cohen-Macaulay rings, Cambridge studies in Advanced Mathematics 39, Cambridge University Press, Cambridge (1993) 13H10 (13-02)
Cid-Ruiz, Y., Montaño, J.: Mixed Multiplicities of Graded Families of Ideals, arXiv:2010.11862v1
Cutkosky, S.D.: Introduction to Algebraic Geometry, Graduate studies in Mathematics 188. American Mathematical Society, Providence (2018)
Cutkosky, S.D.: Asymptotic multiplicities of graded families of ideals and linear series. Adv. Math. 264, 55–113 (2014)
Cutkosky, S.D.: Asymptotic Multiplicities. J. Algebra 442, 260–298 (2015)
Cutkosky, S.D.: Mixed multiplicities of divisorial filtrations. Adv. Math. 358, 106842 (2019)
Cutkosky, S.D.: The Minkowski equality of filtrations. Adv. Math. 388, 107869 (2021)
Cutkosky, S.D.: Examples of multiplicities and mixed multiplicities of filtrations. In Commutative Algebra - 150 years with Roger and Sylvia Wiegand, Contemporary Mathematics 773, pp. 19–34, American Mathematical Society (2021)
Cutkosky, S.D., Sarkar, P., Srinivasan, H.: Mixed multiplicities of filtrations. Trans. Am. Math. Soc. 372, 6183–6211 (2019)
Ein, L., Lazarsfeld, R., Smith, K.: Uniform approximation of Abhyankar valuation ideals in smooth function fields. Am. J. Math. 125, 409–440 (2003)
Goel, K.: Mixed Multiplicities of Ideals, Lecture Notes, IIT Bombay
Goel, K., Gurjar, R.V., Verma, J.K.: The Minkowskis equality and inequality for multiplicity of ideals of finite length in Noetherian local rings. Contemp. Math. 738 ,(2019)
Goto, S., Nishida, K., Watanabe, K.: Non Cohen-Macaulay symbolic blow-ups for space monomial curves and counterexamples to Cowsik’s question. Proc. Am. Math. Soc. 120, 383–392 (1994)
Grothendieck, A., Dieudonné, J.: Éléments de Géométrie Algébrique IV, Parts 2 and 3, Publ. Math. IHES 24 (1965) and 28 (1966)
Hardy, G., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1952)
Hartshorne, R.: Algebraic Geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg (1977)
Herrmann, M., Ikeda, S., Orbanz, U.: Equimultiplicity and Blowing Up: An Algebraic Study. Springer-Verlag, Berlin (1988)
Herrman, M., Schmidt, R., Vogel, W.: Theorie der Normalen Flaxhheit. B.G. Teubner, Leipzig (1977)
Huh, J.: Milnor numbers of projective hypersurfaces and the chromatic polynomials of graphs. J. Am. Math. Soc. 25, 907–927 (2012)
Katz, D.: Note on multiplicity. Proc. Am. Math. Soc. 104, 1021–1026 (1988)
Katz, D., Verma, J.: Extended Rees algebras and mixed multiplicities. Math. Z. 202, 111–128 (1989)
Kaveh, K., Khovanskii, G.: Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory. Ann. Math. 176, 925–978 (2012)
Lazarsfeld, R., Mustaţă, M.: Convex bodies associated to linear series. Ann. Sci. Ec. Norm. Super 42, 783–835 (2009)
Lipman, J.: Equimultiplicity, reduction and blowing up. In R.N. Draper (ed.) Commutative Algebra, Lecture Notes in Pure and Applied Mathematics, vol. 68, pp. 111–147, Marcel Dekker, New York (1982)
Mustaţă, M.: On multiplicities of graded sequences of ideals. J. Algebra 256, 229–249 (2002)
Okounkov, A.: Why would multiplicities be log-concave?, in The orbit method in geometry and physics. Progr. Math. 213, 329–347 (2003)
Ratliff, J.L., Jr.: Locally quasi-unmixed Noetherian rings and ideals of the principal class. Pacific J. Math. 52, 185–205 (1974)
Rees, D.: \({\cal{A}}\)-transforms of local rings and a theorem on multiplicities of ideals. Proc. Camb. Philos. Soc. 57, 8–17 (1961)
Rees, D.: Multiplicities, Hilbert, functions and degree functions. In Commutative algebra: Durham,: (Durham 1981), London Mathematical Society Lecture Note Series 72, pp. 170–178, Cambridge University Press, Cambridge 1982, (1981)
Rees, D.: Izumi’s theorem. In Commutative Algebra, pp. 407–416. Springer-Verlag, Berlin (1989)
Rees, D., Sharp, R.: On a Theorem of B Teissier on Multiplicities of Ideals in Local Rings. J. Lond. Math. Soc. 18, 449–463 (1978)
Roberts, P.C.: A prime ideal in a polynomial ring whose symbolic blow-up is not Noetherian. In Proceedings of the AMS pp. 589–592, (1985)
Serre, J.-P.: Algèbre Locale, Multiplicitiés. Lecture Notes in Mathematics, vol. 11. Springer-Verlag, Berlin (1965)
Swanson, I.: Mixed multiplicities, joint reductions and a theorem of Rees. J. Lond. Math. Soc. 48, 1–14 (1993)
Swanson, I., Huneke, C.: Integral Closure of Ideals. Rings and Modules. Cambridge University Press, Cambridge (2006)
Teissier, B.: Cycles évanescents, sections planes et conditions de Whitney, Singularités à Cargèse 1972. Astérisque. pp. 7–8, (1973)
Teissier, B.: Sur une inégalité à la Minkowski pour les multiplicités (Appendix to a paper by D. Eisenbud and H. Levine). Ann. Math. 106, 38–44 (1977)
Teissier, B.: On a Minkowski Type Inequality for Multiplicities II. In C.P. Ramanujam - a tribute, Tata Inst. Fund. Res. Studies in Math. 8, Springer, Berlin-New York (1978)
Trung, N.V., Verma, J.: Mixed multiplicities of ideals versus mixed volumes of polytopes. Trans. Am. Math. Soc. 359, 4711–4727 (2007)
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Dedicated to Jürgen Herzog on the occasion of his 80th birthday.
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The first author was partially supported by NSF Grant DMS-1700046.
The second author was partially supported by the DST, India: INSPIRE Faculty Fellowship.
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Cutkosky, S.D., Sarkar, P. Multiplicities and mixed multiplicities of arbitrary filtrations. Res Math Sci 9, 14 (2022). https://doi.org/10.1007/s40687-021-00307-x
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DOI: https://doi.org/10.1007/s40687-021-00307-x