On reduction numbers and Castelnuovo–Mumford regularity of blowup rings and modules

• Miranda-Neto, Cleto B. ^[1] ; Queiroz, Douglas S. ^[1]
  1. [1] Universidade Federal da Paraíba

     Universidade Federal da Paraíba

     Brasil

     
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 76, Fasc. 2, 2025, págs. 373-395
• Idioma: inglés
• DOI: 10.1007/s13348-024-00436-w
• Texto completo no disponible (Saber más ...)
• Resumen

  • We prove new results on the interplay between reduction numbers and the Castelnuovo–Mumford regularity of blowup algebras and blowup modules, the key basic tool being the operation of Ratliff–Rush closure. First, we answer in two particular cases a question of M. E. Rossi, D. T. Trung, and N. V. Trung about Rees algebras of ideals in two-dimensional Buchsbaum local rings, and we even ask whether one of such situations always holds. In another theorem we largely generalize a result of A. Mafi on ideals in two-dimensional Cohen–Macaulay local rings, by extending it to arbitrary dimension (and allowing for the setting relative to a Cohen–Macaulay module). We derive a number of applications, including a characterization of (polynomial) ideals of linear type, progress on the theory of generalized Ulrich ideals, and improvements of results by other authors.

• Referencias bibliográficas
  • Aberbach, I.M., Huneke, C., Trung, N.V.: Reduction numbers, Briançon–Skoda theorems and the depth of Rees rings. Compos. Math. 97, 403–434...
  • Bayer, D., Stillman, M.: On the complexity of computing syzygies. J. Symb. Comput. 6, 135–147 (1988)
  • Brodmann, M., Linh, C.H.: Castelnuovo–Mumford regularity, postulation numbers and relation types. J. Algebra 419, 124–140 (2014)
  • Brodmann, M., Sharp, R.: Local Cohomology: An Algebraic Introduction with Geometric Applications. Cambridge University Press, Cambridge (1998)
  • Corso, A., Polini, C., Rossi, M.E.: Depth of associated graded rings via Hilbert coefficients of ideals. J. Pure Appl. Algebra 201, 126–141...
  • Dung, L.X., Hoa, L.T.: Castelnuovo–Mumford regularity of associated graded modules and fiber cones of filtered modules. Commun. Algebra 40,...
  • Eisenbud, D., Goto, S.: Linear free resolutions and minimal multiplicity. J. Algebra 8(8), 89–133 (1984)
  • Giral, J.M., Planas-Vilanova, F.: Integral degree of a ring, reduction numbers and uniform Artin–Rees numbers. J. Algebra 319, 3398–3418 (2008)
  • Goto, S., Ozeki, K., Takahashi, R., Watanabe, K.I., Yoshida, K.I.: Ulrich ideals and modules. Math. Proc. Camb. Philos. Soc. 156, 137–166...
  • Goto, S., Ozeki, K., Takahashi, R., Watanabe, K.-I., Yoshida, K.-I.: Ulrich ideals and modules over two-dimensional rational singularities....
  • Heinzer, W., Johnston, B., Lantz, D., Shah, K.: The Ratliff–Rush ideals in a Noetherian ring: a survey. In: Methods in Module Theory, Colorado...
  • Heinzer, W., Lantz, D., Shah, K.: The Ratliff–Rush ideals in a Noetherian ring. Commun. Algebra 20, 591–622 (1992)
  • Herzog, J., Popescu, D., Trung, N.V.: Regularity of Rees algebras. J. Lond. Math. Soc. 65, 320–338 (2002)
  • Hoa, L.T.: A note on the Hilbert function in a two-dimensional local ring. Acta Math. Vietnam 21, 335–347 (1996)
  • Huckaba, S.: Reduction numbers for ideals of higher analytic spread. Math. Proc. Camb. Philos. Soc. 102, 49–57 (1987)
  • Huneke, C.: Hilbert functions and symbolic powers. Michigan Math. J. 34, 293–318 (1987)
  • Itoh, S.: Hilbert coefficients of integrally closed ideals. J. Algebra 176, 638–652 (1995)
  • Johnson, M., Ulrich, B.: Artin-Nagata properties and Cohen–Macaulay associated graded rings. Compos. Math. 108, 7–29 (1996)
  • Linh, C.H.: Upper bound for the Castelnuovo–Mumford regularity of associated graded modules. Commun. Algebra 33, 1817–1831 (2005)
  • Lipman, J.: Rational singularities with applications to algebraic surfaces and unique factorization. Inst. Hautes Études Sci. Publ. Math....
  • Mafi, A.: On the computation of the Ratliff–Rush closure, associated graded ring and invariance of a length. J. Commun. Algebra 10, 547–557...
  • Mafi, A.: Ratliff–Rush ideal and reduction numbers. Commun. Algebra 46, 1272–1276 (2018)
  • Marley, T.: The coefficients of the Hilbert polynomial and the reduction number of an ideal. J. Lond. Math. Soc. 40, 1–8 (1989)
  • Marley, T.: The reduction number of an ideal and the local cohomology of the associated graded ring. Proc. Am. Math. Soc. 117, 335–341 (1993)
  • Naghipour, R.: Ratliff-Rush closures of ideals with respect to a Noetherian module. J. Pure Appl. Algebra 195, 167–172 (2005)
  • Naghipour, R.: Ratliff-Rush closures of ideals with respect to a Noetherian module. J. Pure Appl. Algebra 195, 167–172 (2005)
  • Northcott, D.G., Rees, D.: Reductions of ideals in local rings. Math. Proc. Camb. Philos. Soc. 50, 145–158 (1954)
  • Ooishi, A.: Genera and arithmetic genera of commutative rings. Hiroshima Math. J. 17, 47–66 (1987)
  • Polini, C., Xie, Y.: j-Multiplicity and depth of associated graded modules. J. Algebra 379, 31–49 (2013)
  • Puthenpurakal, T.J.: Ratliff–Rush filtration, regularity and depth of higher associated graded modules: part I. J. Pure Appl. Algebra 208,...
  • Ratliff, L.J., Rush, D.E.: Two notes on reductions of ideals. Indiana Univ. Math. J. 27, 929–934 (1978)
  • Rossi, M.E., Trung, D.T., Trung, N.V.: Castelnuovo–Mumford regularity and Ratliff–Rush closure. J. Algebra 504, 568–586 (2018)
  • Rossi, M.E., Trung, N.V., Valla, G.: Castelnuovo–Mumford regularity and extended degree. Trans. Am. Math. Soc. 355, 1773–1786 (2003)
  • Rossi, M.E., Valla, G.: Hilbert Functions of Filtered Modules, UMI Lecture Notes 9. Springer, New York (2010)
  • Sally, J.D.: Reductions, local cohomology and Hilbert functions of local rings. In: Commutative Algebra: Durham 1981: London Mathematical...
  • Strunk, B.: Castelnuovo–Mumford regularity, postulation numbers, and reduction numbers. J. Algebra 311, 538–550 (2007)
  • Swanson, I., Huneke, C.: Integral Closure of Ideals, Rings, and Modules. Cambridge University Press, Cambridge (2006)
  • Trung, D.T.: On the computation of Castelnuovo–Mumford regularity of the Rees algebra and of the fiber ring. J. Pure Appl. Algebra 224, 402–410...
  • Trung, N.V.: The Castelnuovo regularity of the Rees algebra and the associated graded ring. Trans. Am. Math. Soc. 350, 2813–2832 (1998)
  • Trung, N.V.: Castelnuovo–Mumford regularity and related invariants. In: Commutative Algebra, Lecture Notes Series, vol. 4, pp. 157–180. Ramanujan...
  • Valla, G.: On form rings which are Cohen–Macaulay. J. Algebra 58, 247–250 (1979)
  • Vasconcelos, W.V.: Multiplicities and reduction numbers. Compos. Math. 139, 361–379 (2003)
  • Wu, Y.: Reduction numbers and Hilbert polynomials of ideals in higher-dimensional Cohen–Macaulay local rings. Math. Proc. Camb. Philos. Soc....
  • Zamani, N.: A formula for reduction number of an ideal relative to a Noetherian module. Int. Electron. J. Algebra 5, 114–120 (2009)
  • Zamani, N.: Regularity of the Rees and associated graded modules. Eur. J. Pure Appl. Math. 7, 429–436 (2014)