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ANALYTIC SPREAD AND INTEGRAL CLOSURE OF INTEGRALLY DECOMPOSABLE MODULES

Part of: Ring extensions and related topics Singularities Local rings and semilocal rings

Published online by Cambridge University Press:  03 November 2020

CARLES BIVIÀ-AUSINA Open the ORCID record for CARLES BIVIÀ-AUSINA [Opens in a new window]  and
JONATHAN MONTAÑO Open the ORCID record for JONATHAN MONTAÑO [Opens in a new window]
CARLES BIVIÀ-AUSINA
Affiliation:
Institut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica de València Camí de Vera, s/n, 46022 València Spain carbivia@mat.upv.es
JONATHAN MONTAÑO*
Affiliation:
Department of Mathematical Sciences New Mexico State University 1290 Frenger Mall, MSC 3MB/Science Hall 236 Las Cruces, 88003-8001 New Mexico USA
*
jmon@nmsu.edu
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Abstract

We relate the analytic spread of a module expressed as the direct sum of two submodules with the analytic spread of its components. We also study a class of submodules whose integral closure can be expressed in terms of the integral closure of its row ideals, and therefore can be obtained by means of a simple computer algebra procedure. In particular, we analyze a class of modules, not necessarily of maximal rank, whose integral closure is determined by the family of Newton polyhedra of their row ideals.

MSC classification

Primary: 13B22: Integral closure of rings and ideals ; integrally closed rings, related rings (Japanese, etc.)
Secondary: 13H15: Multiplicity theory and related topics 32S05: Local singularities

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Type
Article
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Nagoya Mathematical Journal , Volume 245 , March 2022 , pp. 166 - 191
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Copyright
© The Authors, 2020. The publishing rights in this article are licenced to Foundation Nagoya Mathematical Journal under an exclusive license

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Footnotes

Carles Bivià-Ausina was partially supported by MICINN grant PGC2018-094889-B-I00

References

Bivià-Ausina, C., The analytic spread of monomial ideals , Comm. Algebra 31 (2003), 3487–3496.CrossRefGoogle Scholar
Bivià-Ausina, C., The integral closure of modules, Buchsbaum-Rim multiplicities and Newton polyhedra , J. Lond. Math. Soc. 69 (2004), no. 2, 407–427.CrossRefGoogle Scholar
Bivià-Ausina, C., Joint reductions of monomial ideals and multiplicity of complex analytic maps , Math. Res. Lett. 15 (2008), no. 2, 389–407.10.4310/MRL.2008.v15.n2.a15CrossRefGoogle Scholar
Bivià-Ausina, C., Fukui, T., and Saia, M. J., Newton graded algebras and the codimension of non-degenerate ideals , Math. Proc. Camb. Philos. Soc. 133 (2002), 55–75.CrossRefGoogle Scholar
Branco Correia, A. L. and Zarzuela, S., On equimultiple modules , Comm. Algebra 37 (2009), no. 6, 1949–1976.CrossRefGoogle Scholar
Branco Correia, A. L. and Zarzuela, S., On the depth of the Rees algebra of an ideal module , J. Algebra 323 (2010), 1503–1529.CrossRefGoogle Scholar
Brennan, J., Ulrich, B., and Vasconcelos, W. V., The Buchsbaum-Rim polynomial of a module , J. Algebra 241 (2001), no. 1, 379–392.CrossRefGoogle Scholar
Bruns, W. and Herzog, J., Cohen-Macaulay Rings, Cambridge Stud. Adv. Math. 39, Cambridge University Press, Cambridge, MA, 1993.Google Scholar
Bruns, W. and Koch, R., Computing the integral closure of an affine semigroup , Univ. Iagel. Acta Math. 39 (2001), 59–70.Google Scholar
Bruns, W. and Vetter, U., Determinantal Rings, Lecture Notes in Math. 1327, Springer-Verlag, Berlin, Germany, 1988.CrossRefGoogle Scholar
Buchsbaum, D. A. and Rim, D. S., A generalized Koszul complex II, depth and multiplicity , Trans. Am. Math. Soc. 111 (1964), 197–224.CrossRefGoogle Scholar
Chan, C.-Y. J. and Liu, J.-C., A note on reductions of monomial ideals in $k{\left[x,y\right]}_{\left(x,y\right)}$ commutative algebra and its connections to geometry , Contemp. Math. 555 (2011), 13–34.CrossRefGoogle Scholar
Crispin Quiñonez, V., Minimal Reductions of Monomial Ideals, Research Reports in Mathematics 10, Department of Mathematics, Stockholm University, 2004, available at http://www.math.su.se/reports/2004/10/.Google Scholar
De Jong, T., An algorithm for computing the integral closure , J. Symb. Comput. 26 (1998), 273–277.CrossRefGoogle Scholar
Decker, W., Greuel, G.-M., Pfister, G., and Schönemann, H., Singular 4-1-2—A computer algebra system for polynomial computations, 87 (2019). http://www.singular.uni-kl.de.Google Scholar
Eisenbud, D., Commutative algebra with a view toward algebraic geometry , Graduate Texts in Mathematics 150, Springer-Verlag, New York, NY, 2004.Google Scholar
Eisenbud, D., Huneke, C., and Ulrich, B., What is the Rees algebra of a module , Proc. Am. Math. Soc. 131 (2002), 701–708.CrossRefGoogle Scholar
Gaffney, T., Integral closure of modules and Whitney equisingularity , Invent. Math. 107 (1992), 301–322.CrossRefGoogle Scholar
Gaffney, T., Multiplicities and equisingularity of ICIS germs , Invent. Math. 123 (1996), 209–220.CrossRefGoogle Scholar
Gaffney, T., Trotman, D., and Wilson, L., Equisingularity of sections, $\left({t}^r\right)$ condition, and the integral closure of modules , J. Algebraic Geom. 18 (2009), no. 4, 651–689.CrossRefGoogle Scholar
Grayson, D. and Stillman, M., Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2.Google Scholar
Hartshorne, R., Algebraic Geometry, Graduate Texts in Math. 52, Springer-Verlag, New York, NY-Heidelberg, Germany, 1977.CrossRefGoogle Scholar
Hayasaka, F., Asymptotic vanishing of homogeneous components of multi-graded modules and its applications , J. Algebra 513 (2018), 1–26.CrossRefGoogle Scholar
Hayasaka, F., Constructing indecomposable integrally closed modules over a two-dimensional regular local ring , J. Algebra 556 (2020), 879–907.CrossRefGoogle Scholar
Henry, J.-P. and Merle, M., “Conormal space and Jacobian modules. A short dictionary ” in Singularities (Lille, 1991), London Math. Soc. Lecture Note Ser. 201, Cambridge University Press, Cambridge, MA, 1994, 147-174.CrossRefGoogle Scholar
Herrmann, M., Ikeda, S., and Orbanz, U., Equimultiplicity and Blowing Up: An Algebraic Study. B. Moonen, Springer-Verlag, Berlin, Germany, 1988.CrossRefGoogle Scholar
Huneke, C. and Swanson, I., Integral Closure of Ideals, Rings, and Modules, London Math. Soc. Lecture Note Series 336, Cambridge University Press, Cambridge, MA, 2006.Google Scholar
Hyry, E., The diagonal subring and the Cohen-Macaulay property of a multigraded ring , Trans. Am. Math. Soc. 351 (1999), 2213–2232 CrossRefGoogle Scholar
Hyry, E., Cohen-Macaulay multi-Rees algebras , Compos. Math. 130 (2002), 319-343.CrossRefGoogle Scholar
Katz, D., Reduction criteria for modules , Comm. Algebra 23 (1995), no. 12, 4543–4548.CrossRefGoogle Scholar
Kirby, D., On the Buchsbaum-Rim multiplicity associated with a matrix , J. Lond. Math. Soc. 32 (1985), no. 2, 57–61.CrossRefGoogle Scholar
Kirby, D. and Rees, D., Multiplicities in graded rings. I. The general theory commutative algebra: syzygies, multiplicities, and birational algebra , Contemp. Math. 159 (1994), 209–267.CrossRefGoogle Scholar
Kirby, D. and Rees, D., Multiplicities in graded rings II: Integral equivalence and the Buchsbaum-Rim multiplicity , Math. Proc. Camb. Philos. Soc. 119 (1996), no. 3, 425–445.CrossRefGoogle Scholar
Kodiyalam, V., Integrally closed modules over two-dimensional regular local rings , Trans. Am. Math. Soc. 347 (1995), no. 9, 3551–3573.CrossRefGoogle Scholar
Lejeune, M. and Teissier, B., Clôture intégrale des idéaux et equisingularité, with an appendix by J. J. Risler. Centre de Mathématiques, École Polytechnique (1974) , Ann. Fac. Sci. Toulouse Math. 17 (2008), no. 6, 781–859.Google Scholar
Lipman, J., “Equimultiplicity, reduction and blowing up ” in Commutative Algebra, Analytical Methods, Edited by Draper, R. N., Lecture Notes in Pure and Appl Math, Marcel Dekker, Inc, New York, NY-Basel, 1982, 111–147.Google Scholar
Matsumura, H., Commutative Ring Theory, Cambridge Studies in Advanced Math. 8, Cambridge University Press, Cambridge, MA, 1986.Google Scholar
Rees, D., a-Transforms of local rings and a theorem on multiplicities of ideals , Math. Proc. Camb. Philos. Soc. 57 (1961), 8–17.CrossRefGoogle Scholar
Rees, D., Reduction of modules , Math. Proc. Camb. Philos. Soc. 101 (1987) 431–449.CrossRefGoogle Scholar
Saia, M. J., The integral closure of ideals and the Newton filtration , J. Algebraic Geom. 5 (1996), 1–11.Google Scholar
Simis, A., Ulrich, B., and Vasconcelos, W. V., Rees algebras of modules , Proc. Lond. Math. Soc. 87 (2003), 610–646.CrossRefGoogle Scholar
Teissier, B., Cycles évanescents, sections planes et conditions of Whitney , Singularités à Cargèse, Astérisque (1973), no. 7–8, 285–362.Google Scholar
Teissier, B., Variétés polaires. I. Invariants polaires des singularités d’hypersurfaces , Invent. Math. 40 (1977), no. 3, 267–292.CrossRefGoogle Scholar
Teissier, B., “Monomial ideals, binomial ideals, polynomial ideals” in Trends in Commutative Algebra, Math. Sci. Res. Inst. Publ. 51, Cambridge University Press, Cambridge, MA, 2004, 211–246.CrossRefGoogle Scholar
Vasconcelos, W., Computational methods in commutative algebra and algebraic geometry , Algorithms and Computation in Mathematics 2, Springer-Verlag, Berlin, Germany, 1998.Google Scholar
Vasconcelos, W., Integral Closure. Rees Algebras, Multiplicities, Algorithms, Springer Monogr Math. Springer-Verlag, Berlin, Germany, 2005.Google Scholar