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Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs

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Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics

Abstract

In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertex-weighted oriented graphs. These are digraphs having no oriented cycles of length two with weights on the vertices. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of its primary components. If the primary components of a monomial ideal are normal, we present a simple procedure to compute its symbolic Rees algebra using Hilbert bases, and give necessary and sufficient conditions for the equality between its ordinary and symbolic powers. We give an effective characterization of the Cohen–Macaulay vertex-weighted oriented forests. For edge ideals of transitive weighted oriented graphs we show that Alexander duality holds. It is shown that edge ideals of weighted acyclic tournaments are Cohen–Macaulay and satisfy Alexander duality.

Dedicated to Professor Antonio Campillo on the occasion of his 65th birthday

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References

  1. Bahiano, C.: Symbolic powers of edge ideals. J. Algebra 273(2), 517–537 (2004)

    Article  MathSciNet  Google Scholar 

  2. Bang-Jensen, J., Gutin, G.: Digraphs: Theory, Algorithms and Applications. Springer Monographs in Mathematics. Springer, London (2006)

    Google Scholar 

  3. Bruns, W., Ichim, B., Römer, T., Sieg, R., Söger, C.: Normaliz. Algorithms for rational cones and affine monoids. Available at http://normaliz.uos.de

  4. Carvalho, C., Lopez Neumann, V.G., López, H.H.: Projective nested cartesian codes. Bull. Braz. Math. Soc. (N.S.) 48(2), 283–302 (2017)

    Article  MathSciNet  Google Scholar 

  5. Cooper, S., Embree, R., Hà, H.T., Hoefel, A.H.: Symbolic powers of monomial ideals. Proc. Edinb. Math. Soc. (2) 60(1), 39–55 (2017)

    Article  MathSciNet  Google Scholar 

  6. Crispin Quiñonez, V.: Integral closure and other operations on monomial ideals. J. Commut. Algebra 2(3), 359–386 (2010)

    Article  MathSciNet  Google Scholar 

  7. Dao, H., Huneke, C., Schweig, J.: Bounds on the regularity and projective dimension of ideals associated to graphs. J. Algebraic Combin. 38(1), 37–55 (2013)

    Article  MathSciNet  Google Scholar 

  8. Dao, H., De Stefani, A., Grifo, E., Huneke, C., Núñez-Betancourt, L.: Symbolic powers of ideals. Preprint, arXiv:1708.03010 (2017)

    Google Scholar 

  9. Dupont, L.A., Villarreal, R.H.: Edge ideals of clique clutters of comparability graphs and the normality of monomial ideals. Math. Scand. 106(1), 88–98 (2010)

    Article  MathSciNet  Google Scholar 

  10. Escobar, C., Villarreal, R.H., Yoshino, Y.: Torsion freeness and normality of blowup rings of monomial ideals. In: Commutative Algebra. Lecture Notes in Pure and Applied Mathematics, vol. 244, pp. 69–84. Chapman & Hall/CRC, Boca Raton (2006)

    Chapter  Google Scholar 

  11. Francisco, C., Hà, H.T., Mermin, J.: Powers of Square-Free Monomial Ideals and Combinatorics. Commutative Algebra, pp. 373–392. Springer, New York (2013)

    Google Scholar 

  12. Gimenez, P., Simis, A., Vasconcelos, W.V., Villarreal, R.H.: On complete monomial ideals. J. Commut. Algebra 8(2), 207–226 (2016)

    Article  MathSciNet  Google Scholar 

  13. Gitler, I., Villarreal, R.H.: Graphs, Rings and Polyhedra. Aportaciones Mat. Textos, vol. 35. Sociedad Matemtica Mexicana, México (2011)

    Google Scholar 

  14. Gitler, I., Valencia, C., Villarreal, R.H.: A note on Rees algebras and the MFMC property. Beiträge Algebra Geom. 48(1), 141–150 (2007)

    MathSciNet  MATH  Google Scholar 

  15. Goto, S., Nishida, K.: The Cohen–Macaulay and Gorenstein Rees algebras associated to filtrations. Mem. Am. Math. Soc. 110(526), 1–134 (1994)

    MathSciNet  MATH  Google Scholar 

  16. Grayson, D., Stillman, M.: Macaulay2. Available via anonymous FTP from http://math.uiuc.edu (1996)

  17. Hà, H.T., Morey, S.: Embedded associated primes of powers of square-free monomial ideals. J. Pure Appl. Algebra 214(4), 301–308 (2010)

    Article  MathSciNet  Google Scholar 

  18. Hà, H.T., Lin, K.-N., Morey, S., Reyes, E., Villarreal, R.H.: Edge ideals of oriented graphs, Preprint (2017)

    Google Scholar 

  19. Harary, F.: Graph Theory. Addison-Wesley, Reading (1972)

    MATH  Google Scholar 

  20. Herzog, J., Hibi, T.: Monomial Ideals. Graduate Texts in Mathematics, vol. 260. Springer, London/New York (2011)

    Chapter  Google Scholar 

  21. Herzog, J., Takayama, Y., Terai, N.: On the radical of a monomial ideal. Arch. Math. 85, 397–408 (2005)

    Article  MathSciNet  Google Scholar 

  22. Herzog, J., Hibi, T., Trung, N.V.: Symbolic powers of monomial ideals and vertex cover algebras. Adv. Math. 210, 304–322 (2007)

    Article  MathSciNet  Google Scholar 

  23. Hosten, S., Smith, G.G.: Monomial Ideals. Computations in Algebraic Geometry with Macaulay 2. Algorithms and Computational Mathematics, vol. 8, pp. 73–100. Springer, Berlin (2002)

    Chapter  Google Scholar 

  24. Martínez-Bernal, J., Morey, S., Villarreal, R.H., Vivares, C.E.: Depth and regularity of monomial ideals via polarizations and combinatorial optimization. Preprint (2017)

    Google Scholar 

  25. Martínez-Bernal, J., Pitones, Y., Villarreal, R.H.: Minimum distance functions of graded ideals and Reed-Muller-type codes. J. Pure Appl. Algebra 221, 251–275 (2017)

    Article  MathSciNet  Google Scholar 

  26. Matsumura, H.: Commutative Algebra. Benjamin-Cummings, Reading (1980)

    MATH  Google Scholar 

  27. Miller, E.: Alexander duality for monomial ideals and their resolutions. Preprint, arXiv:math/9812095 (1998)

    Google Scholar 

  28. Miller, E.: The Alexander duality functors and local duality with monomial support. J. Algebra 231(1), 180–234 (2000)

    Article  MathSciNet  Google Scholar 

  29. Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Graduate Texts in Mathematics, vol. 227. Springer, New York (2004)

    Google Scholar 

  30. Morey, S., Villarreal, R.H.: Edge ideals: algebraic and combinatorial properties. In: Francisco, C., Klingler, L.C., Sather-Wagstaff, S., Vassilev, J.C. (eds.) Progress in Commutative Algebra, Combinatorics and Homology, vol. 1, pp. 85–126. De Gruyter, Berlin (2012)

    MATH  Google Scholar 

  31. Nejad, A.N., Simis, A., Zaare-Nahandi, R.: The Aluffi algebra of the Jacobian of points in projective space: torsion-freeness. J. Algebra 467, 268–283 (2016)

    Article  MathSciNet  Google Scholar 

  32. Neves, J., Vaz Pinto, M., Villarreal, R.H.: Regularity and algebraic properties of certain lattice ideals. Bull. Braz. Math. Soc. (N.S.) 45, 777–806 (2014)

    Article  MathSciNet  Google Scholar 

  33. Paulsen, C., Sather-Wagstaff, S.: Edge ideals of weighted graphs. J. Algebra Appl. 12(5), 1250223, p. 24 (2013)

    Article  MathSciNet  Google Scholar 

  34. Pitones, Y., Reyes, E., Toledo, J.: Monomial ideals of weighted oriented graphs. Preprint, arXiv:1710.03785 (2017)

    Google Scholar 

  35. Simis, A.: Combinatoria Algebrica, XVIII Coloquio Brasileiro de Matematica, IMPA (Apendice. Palimpsesto 2: Potencias simbolicas, 2.1) (1991)

    Google Scholar 

  36. Simis, A.: Effective computation of symbolic powers by Jacobian matrices. Commun. Algebra 24, 3561–3565 (1996)

    Article  MathSciNet  Google Scholar 

  37. Simis, A., Trung, N.V.: The divisor class group of ordinary and symbolic blow-ups. Math. Z. 198, 479–491 (1988)

    Article  MathSciNet  Google Scholar 

  38. Simis, A., Vasconcelos, W.V., Villarreal, R.H.: On the ideal theory of graphs. J. Algebra 167, 389–416 (1994)

    Article  MathSciNet  Google Scholar 

  39. Van Tuyl, A.: A beginner’s guide to edge and cover ideals. In: Bigatti, A., Gimenez, P., Sáenz-de-Cabezón, E. (eds.) Monomial Ideals, Computations and Applications. Lecture Notes in Mathematics, vol. 2083, pp. 63–94. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  40. Villarreal, R.H.: Cohen–Macaulay graphs. Manuscripta Math. 66, 277–293 (1990)

    Article  MathSciNet  Google Scholar 

  41. Villarreal, R.H.: Monomial Algebras. Monographs and Textbooks in Pure and Applied Mathematics, vol. 238. Chapman and Hall/CRC Press, Boca Raton (2001)

    Google Scholar 

  42. Villarreal, R.H.: Monomial Algebras. Monographs and Research Notes in Mathematics, 2nd edn. Chapman and Hall/CRC, Boca Raton (2015)

    Google Scholar 

  43. Zariski, O., Samuel, P.: Commutative Algebra, vol. II. Springer, New York (1960)

    Book  Google Scholar 

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Acknowledgements

We would like to thank Ngô Viêt Trung and the referees for a careful reading of the paper and for the improvements suggested. The first, third and fourth authors were partially supported by the Spanish Ministerio de Economía y Competitividad grant MTM2016-78881-P. The second and fourth authors were supported by SNI. The fifth author was supported by a scholarship from CONACYT

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Authors and Affiliations

  1. Facultad de Ciencias, Instituto de Investigación en Matemáticas de la Universidad de Valladolid (IMUVA), Valladolid, Spain

    Philippe Gimenez

  2. Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del IPN, México City, México

    José Martínez-Bernal, Rafael H. Villarreal & Carlos E. Vivares

  3. Departamento de Matemática, Universidade Federal de Pernambuco, Recife, PE, Brazil

    Aron Simis

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  1. Philippe Gimenez
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  2. José Martínez-Bernal
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  3. Aron Simis
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  4. Rafael H. Villarreal
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  5. Carlos E. Vivares
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Corresponding author

Correspondence to Rafael H. Villarreal .

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Editors and Affiliations

  1. Fachbereich Mathematik, Universität Kaiserslautern, Kaiserslautern, Germany

    Gert-Martin Greuel

  2. Departamento de Álgebra, Instituto de Matemáticas (IMUS), Universidad de Sevilla, Sevilla, Spain

    Luis Narváez Macarro

  3. Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain

    Sebastià Xambó-Descamps

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Gimenez, P., Martínez-Bernal, J., Simis, A., Villarreal, R.H., Vivares, C.E. (2018). Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs. In: Greuel, GM., Narváez Macarro, L., Xambó-Descamps, S. (eds) Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics. Springer, Cham. https://doi.org/10.1007/978-3-319-96827-8_21

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