The normalized depth function of squarefree powers

• Erey, Nursel ^[1] ; Herzog, Jürgen ^[2] ; Hibi, Takayuki ^[3] ; Madani, Sara Saeedi ^[4]
  1. [1] Gebze Technical University

     Gebze Technical University

     Turquía

     
  2. [2] University of Duisburg-Essen

     University of Duisburg-Essen

     Kreisfreie Stadt Essen, Alemania

     
  3. [3] Osaka University

     Osaka University

     Kita Ku, Japón

     
  4. [4] Amirkabir University of Technology

     Amirkabir University of Technology

     Irán

     

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• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 75, Fasc. 2, 2024, págs. 409-423
• Idioma: inglés
• DOI: 10.1007/s13348-023-00392-x
• Texto completo no disponible (Saber más ...)
• Resumen

  • The depth of squarefree powers of a squarefree monomial ideal is introduced. Let I be a squarefree monomial ideal of the polynomial ring S=K[x_1,\ldots ,x_n] . The k-th squarefree power I^{[k]} of I is the ideal of S generated by those squarefree monomials u_1\cdots u_k with each u_i\in G(I) , where G(I) is the unique minimal system of monomial generators of I. Let d_k denote the minimum degree of monomials belonging to G(I^{[k]}) . One has {\text {depth}}(S/I^{[k]}) \ge d_k -1 . Setting g_I(k) = {\text {depth}}(S/I^{[k]}) - (d_k - 1) , one calls g_I(k) the normalized depth function of I. The computational experience strongly invites us to propose the conjecture that the normalized depth function is nonincreasing. In the present paper, especially the normalized depth function of the edge ideal of a finite simple graph is deeply studied.

• Referencias bibliográficas
  • Abbott, J. , Bigatti, A. M., Robbiano, L.: CoCoA: a system for doing computations in commutative algebra. Available at http://cocoa.dima.unige.it
  • Bigdeli, M., Herzog, J., Zaare-Nahandi, R.: On the index of powers of edge ideals. Commun. Algebra 46, 1080–1095 (2018)
  • Brodmann, M.: The asymptotic nature of the analytic spread. Math. Proc. Cambridge Philos. Soc. 86, 35–39 (1979)
  • Erey, N., Faridi, S.: Betti numbers of monomial ideals via facet covers. J. Pure Appl. Algebra 220(5), 1990–2000 (2016)
  • Erey, N., Hibi, T.: Squarefree powers of edge ideals of forests. Electron. J. Combin. 28(2), P2.32 (2021)
  • Erey, N., Hibi, T., Herzog, J., Saeedi Madani, S.: Matchings and squarefree powers of edge ideals. J. Comb. Theory Ser. A 188, 105585 (2022)
  • Francisco, C.A., Hà, H.T., Van Tuyl, A.: A conjecture on critical graphs and connections to the persistence of associated primes. Discrete...
  • Hà, H.T., Hibi, T.: Max Min vertex cover and the size of Betti tables. Ann Comb. 25, 115–132 (2021)
  • Hà, H.T., Nguyen, H., Trung, N., Trung, T.: Depth functions of powers of homogeneous ideals. Proc. AMS 149, 1837–1844 (2021)
  • Herzog, J., Hibi, T.: The depth of powers of an ideal. J. Algebra 291, 534–550 (2005)
  • Herzog, J., Hibi, T.: Monomial Ideals, Graduate Text in Mathematics. Springer, Berlin (2011)
  • Herzog, J., Takayama, Y.: Resolutions by mapping cones. Homol. Homotopy Appl. 4(2), part 2, 277-294 (2002)
  • Herzog, J., Hibi, T., Zheng, X.: Dirac’s theorem on chordal graphs and Alexander duality. Eur. J. Comb. 25, 949–960 (2004)
  • Herzog, J., Hibi, T., Zheng, X.: Monomial ideals whose powers have a linear resolution. Math. Scand. 95, 23–32 (2004)
  • Herzog, J., Rahimbeigi, M., Römer, T.: Classes of cut ideals and their Betti numbers. São Paulo J. Math. Sci. 1, 1–16 (2022)
  • Kaiser, T., Stehlík, M., Šrekovski, R.: Replication in critical graphs and the persistence of monomial ideals. J. Comb. Theory Ser. A 123,...
  • Kimura, K.: Non-vanishingness of Betti numbers of edge ideals, Harmony of Gröbner bases and the modern industrial society, pp. 153–168. World...
  • Terai, N.: Alexander duality theorem and Stanley-Reisner rings, Free resolutions of coordinate rings of projective varieties and related topics...