Advertisement

Local Cohomology Modules Supported on Monomial Ideals

  • Josep Àlvarez MontanerEmail author
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2083)

Abstract

Local cohomology was introduced by A. Grothendieck in the early 1960s and quickly became an indispensable tool in Commutative Algebra. Despite the effort of many authors in the study of these modules, their structure is still quite unknown. C

Keywords

Simplicial Complex Polynomial Ring Monomial Ideal Free Resolution Characteristic Cycle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

Many thanks go to Anna M. Bigatti, Philippe Gimenez, and Eduardo Sáenz-de-Cabezón for the invitation to participate in the MONICA conference and the great enviroment they created there. I am also indebted with Oscar Fernández Ramos who agreed to develop the Macaulay 2 routines that not only allowed us to perform many computations but also enlightened part of my research.

References

  1. 3.
    J. Àlvarez Montaner, Characteristic cycles of local cohomology modules of monomial ideals. J. Pure Appl. Algebra 150, 1–25 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 4.
    J. Àlvarez Montaner, Characteristic cycles of local cohomology modules of monomial ideals II. J. Pure Appl. Algebra 192, 1–20 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 5.
    J. Àlvarez Montaner, Some numerical invariants of local rings. Proc. Am. Math. Soc. 132, 981–986 (2004)zbMATHCrossRefGoogle Scholar
  4. 6.
    J. Àlvarez Montaner, Operations with regular holonomic D-modules with support a normal crossing. J. Symb. Comput. 40, 999–1012 (2005)zbMATHCrossRefGoogle Scholar
  5. 7.
    J. Àlvarez Montaner, R. García López, S. Zarzuela, Local cohomology, arrangements of subspaces and monomial ideals. Adv. Math. 174, 35–56 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 8.
    J. Àlvarez Montaner, S. Zarzuela, Linearization of local cohomology modules, in Commutative Algebra: Interactions with Algebraic Geometry, ed. by L.L. Avramov, M. Chardin, M. Morales, C. Polini. Contemporary Mathematics, vol. 331 (American Mathematical Society, Providence, 2003), pp. 1–11Google Scholar
  7. 10.
    J. Àlvarez Montaner, A. Vahidi, Lyubeznik numbers of monomial ideals. Trans. Am. Math. Soc. (2011) [arXiv/1107.5230] (to appear)Google Scholar
  8. 16.
    D. Bayer, I. Peeva, B. Sturmfels, Monomial resolutions. Math. Res. Lett. 5, 31–46 (1998)MathSciNetzbMATHGoogle Scholar
  9. 17.
    D. Bayer, B. Sturmfels, Cellular resolutions of monomial modules. J. Reine Angew. Math. 502, 123–140 (1998)MathSciNetzbMATHGoogle Scholar
  10. 21.
    J.E. Björk, Rings of Differential Operators (North Holland Mathematics Library, Amsterdam, 1979)zbMATHGoogle Scholar
  11. 23.
    M. Blickle, Lyubeznik’s numbers for cohomologically isolated singularities. J. Algebra 308, 118–123 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 24.
    M. Blickle, R. Bondu, Local cohomology multiplicities in terms of étale cohomology. Ann. Inst. Fourier 55, 2239–2256 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 25.
    R. Boldini, Critical cones of characteristic varieties. Trans. Am. Math. Soc. 365, 143–160 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 26.
    A. Borel, in Algebraic D-Modules. Perspectives in Mathematics (Academic, London, 1987)Google Scholar
  15. 29.
    M.P. Brodmann, R.Y. Sharp, in Local Cohomology, An Algebraic Introduction with Geometric Applications. Cambridge Studies in Advanced Mathematics, vol. 60 (Cambridge University Press, Cambridge, 1998)Google Scholar
  16. 30.
    W. Bruns, J. Herzog, Cohen-Macaulay rings, revised edn. (Cambridge University Press, Cambridge, 1998)Google Scholar
  17. 45.
    S.C. Coutinho, in A Primer of Algebraic \(\mathcal{D}\) -Modules. London Mathematical Society Student Texts (Cambridge University Press, Cambridge, 1995)Google Scholar
  18. 51.
    J.A. Eagon, V. Reiner, Resolutions of Stanley-Reisner rings and Alexander duality. J. Pure Appl. Algebra 130, 265–275 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 52.
    D. Eisenbud, in Commutative Algebra with a View Towards Algebraic Geometry. GTM, vol. 150 (Springer, Berlin, 1995)Google Scholar
  20. 53.
    D. Eisenbud, G. Fløystad, F.O. Schreyer, Sheaf cohomology and free resolutions over exterior algebras. Trans. Am. Math. Soc. 355, 4397–4426 (2003)zbMATHCrossRefGoogle Scholar
  21. 55.
    D. Eisenbud, M. Mustaţă, M. Stillman, Cohomology on toric varieties and local cohomology with monomial supports. J. Symb. Comput. 29, 583–600 (2000)zbMATHCrossRefGoogle Scholar
  22. 56.
    S. Eliahou, M. Kervaire, Minimal resolutions of some monomial ideals. J. Algebra 129, 1–25 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 60.
    G. Fatabbi, On the resolution of ideals of fat points. J. Algebra 242, 92–108 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 63.
    H.B. Foxby, Bounded complexes of flat modules. J. Pure Appl. Algebra 15, 149–172 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 65.
    C. Francisco, H.T. Hà, A. Van Tuyl, Splittings of monomial ideals. Proc. Am. Math. Soc. 137, 3271–3282 (2009)zbMATHCrossRefGoogle Scholar
  26. 71.
    O. Gabber, The integrability of the Characteristic Variety. Am. J. Math. 103, 445–468 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 72.
    A. Galligo, M. Granger, Ph. Maisonobe, D-modules et faisceaux pervers dont le support singulier est un croisement normal. Ann. Inst. Fourier 35, 1–48 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 73.
    A. Galligo, M. Granger, Ph. Maisonobe, D-modules et faisceaux pervers dont le support singulier est un croisement normal. II, in Differential Systems and Singularities (Luminy, 1983). Astérisque 130, 240–259 (1985)Google Scholar
  29. 74.
    R. Garcia, C. Sabbah, Topological computation of local cohomology multiplicities. Collect. Math. 49, 317–324 (1998)MathSciNetzbMATHGoogle Scholar
  30. 75.
    V. Gasharov, I. Peeva, V. Welker, The lcm-lattice in monomial resolutions. Math. Res. Lett. 6, 521–532 (1999)MathSciNetzbMATHGoogle Scholar
  31. 77.
    M. Goresky, R. MacPherson, in Stratified Morse Theory. Ergebnisse Series, vol. 14 (Springer, Berlin, 1988)Google Scholar
  32. 78.
    S. Goto, K. Watanabe, On Graded Rings, II (\({\mathbb{Z}}^{n}\)- graded rings). Tokyo J. Math. 1, 237–261 (1978)Google Scholar
  33. 79.
    H.G. Gräbe, The canonical module of a Stanley-Reisner ring. J. Algebra 86, 272–281 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 80.
    D. Grayson, M. Stillman, Macaulay2, A software system for research in algebraic geometry (2011). Available at: http://www.math.uiuc.edu/Macaulay2
  35. 81.
    A. Grothendieck, in Local Cohomology. Lecture Notes in Mathematics, vol. 41 (Springer, Berlin, 1967)Google Scholar
  36. 82.
    A. Grothendieck, J. Dieudonné, Éléments de géométrie algébrique IV. Étude locale des schémas et des morphismes de schémas Inst. Hautes Études Sci. Publ. Math. 28 (1966)Google Scholar
  37. 83.
    H.T. Hà, A. Van Tuyl, Splittable ideals and the resolution of monomial ideals. J. Algebra 309, 405–425 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 87.
    M. Hellus, A note on the injective dimension of local cohomology modules. Proc. Am. Math. Soc. 136, 2313–2321 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 88.
    J. Herzog, T. Hibi, Componentwise linear ideals. Nagoya Math. J. 153, 141–153 (1999)MathSciNetzbMATHGoogle Scholar
  40. 94.
    J. Herzog, S. Iyengar, Koszul modules. J. Pure Appl. Algebra 201, 154–188 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 109.
    C. Huneke, Problems on local cohomology modules, in Free Resolutions in Commutative Algebra and Algebraic Geometry (Sundance, 1990). Research Notes in Mathematics, vol. 2 (Jones and Barthlett Publishers, Boston, 1994), pp. 93–108Google Scholar
  42. 110.
    C. Huneke, R.Y. Sharp, Bass numbers of local cohomology modules. Trans. Am. Math. Soc. 339, 765–779 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 113.
    S. Iyengar, G. Leuschke, A. Leykin, C. Miller, E. Miller, A. Singh, U. Walther, in Twenty-Four Hours of Local Cohomology. Graduate Studies in Mathematics, vol. 87 (American Mathematical Society, Providence, 2007)Google Scholar
  44. 116.
    M. Jöllenbeck, V. Welker, Minimal resolutions via algebraic discrete Morse theory. Mem. Am. Math. Soc. 197 vi+74 pp. (2009)Google Scholar
  45. 122.
    F. Kirwan, in An Introduction to Intersection Homology Theory. Pitman Research Notes in Mathematics (Longman Scientific and Technical, Harlow, 1988); copublished in the United States with Wiley, New YorkGoogle Scholar
  46. 123.
    S. Khoroshkin, D-modules over arrangements of hyperplanes. Commun. Algebra 23, 3481–3504 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 124.
    S. Khoroshkin, A. Varchenko, Quiver D-modules and homology of local systems over arrangements of hyperplanes. Inter. Math. Res. Papers, 2006Google Scholar
  48. 127.
    A. Leykin, M. Stillman, H. Tsai, D-modules for Macaulay 2 (2011). Available at: http://people.math.gatech.edu/~aleykin3/Dmodules
  49. 128.
    G. Lyubeznik, On the local cohomology modules \(H_{\mathfrak{A}}^{i}(R)\) for ideals \(\mathfrak{A}\) generated by monomials in an R-sequence, in Complete Intersections, ed. by S. Greco, R. Strano. Lecture Notes in Mathematics, vol. 1092 (Springer, Berlin, 1984), pp. 214–220Google Scholar
  50. 130.
    G. Lyubeznik, A new explicit finite free resolution of ideals generated by monomials in an R-sequence. J. Pure Appl. Algebra 51, 193–195 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 131.
    G. Lyubeznik, The minimal non-Cohen-Macaulay monomial ideals. J. Pure Appl. Algebra 51, 261–266 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 132.
    G. Lyubeznik, Finiteness properties of local cohomology modules (an application of D-modules to commutative algebra). Invent. Math. 113, 41–55 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 133.
    G. Lyubeznik, F-modules: Applications to local cohomology and D-modules in characteristic \(p > 0\). J. Reine Angew. Math. 491, 65–130 (1997)MathSciNetzbMATHGoogle Scholar
  54. 134.
    G. Lyubeznik, A partial survey of local cohomology modules, in Local Cohomology Modules and Its Applications (Guanajuato, 1999). Lecture Notes in Pure and Applied Mathematics, vol. 226 (Marcel Dekker, New York, 2002), pp. 121–154Google Scholar
  55. 135.
    G. Lyubeznik, On some local cohomology modules. Adv. Math. 213, 621–643 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 137.
    B. Malgrange, L’involutivité des caractéristiques des systèmes différentiels et microdifférentiels, in Lecture Notes in Mathematics, vol. 710 (Springer, Berlin, 1979), pp. 277–289Google Scholar
  57. 139.
    Z. Mebkhout, in Le formalisme des six opérations de Grothendieck pour les \(\mathcal{D}_{X}\) -modules cohérents. Travaux en Cours, vol. 35 (Hermann, Paris, 1989)Google Scholar
  58. 140.
    E. Miller, The Alexander duality functors and local duality with monomial support. J. Algebra 231, 180–234 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 141.
    E.  Miller, B.  Sturmfels, in Combinatorial Commutative Algebra. Graduate Texts in Mathematics, vol. 227 (Springer, New York, 2005)Google Scholar
  60. 146.
    M. Mustaţă, Local Cohomology at Monomial Ideals. J. Symb. Comput. 29, 709–720 (2000)zbMATHCrossRefGoogle Scholar
  61. 147.
    M. Mustaţă, Vanishing theorems on toric varieties. Tohoku Math. J. 54, 451–470 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 152.
    T. Oaku, Algorithms for b-functions, restrictions and local cohomology groups of D-modules. Adv. Appl. Math. 19, 61–105 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 153.
    T. Oaku, N. Takayama, Algorithms for D-modules—restriction, tensor product, localization, and local cohomology groups. J. Pure Appl. Algebra 156, 267–308 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 157.
    I. Peeva, M. Velasco, Frames and degenerations of monomial resolutions. Trans. Am. Math. Soc. 363, 2029–2046 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 158.
    F. Pham, in Singularités des systèmes différentiels de Gauss-Manin. Progress in Mathematics, vol. 2 (Birkhäuser, Basel, 1979)Google Scholar
  66. 166.
    T. Römer, Generalized Alexander duality and applications. Osaka J. Math. 38, 469–485 (2001)MathSciNetzbMATHGoogle Scholar
  67. 167.
    T. Römer, On minimal graded free resolutions. Ph.D. Thesis, Essen, 2001Google Scholar
  68. 168.
    M. Sato, T. Kawai, M. Kashiwara, Microfunctions and pseudo-differential equations, in Lecture Notes in Mathematics, vol. 287 (Springer, New York, 1973), pp. 265–529Google Scholar
  69. 170.
    P. Schenzel, On Lyubeznik’s invariants and endomorphisms of local cohomology modules. J. Algebra 344, 229–245 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  70. 176.
    A.M. Simon, Some homological properties of complete modules. Math. Proc. Camb. Philos. Soc. 108, 231–246 (1990)zbMATHCrossRefGoogle Scholar
  71. 177.
    G.G. Smith, Irreducible components of characteristic varieties. J. Pure Appl. Algebra 165, 291–306 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  72. 182.
    R.P. Stanley, Combinatorics and Commutative Algebra (Birkhäuser, Basel, 1983)zbMATHGoogle Scholar
  73. 187.
    N. Terai, Local cohomology modules with respect to monomial ideals (1999). PreprintGoogle Scholar
  74. 188.
    D. Taylor, Ideals genreated by monomials in an R-sequence. Ph.D. Thesis, University of Chicago, 1961Google Scholar
  75. 191.
    M. Velasco, Minimal free resolutions that are not supported by a CW-complex. J. Algebra 319, 102–114 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  76. 194.
    U. Walther, Algorithmic computation of local cohomology modules and the cohomological dimension of algebraic varieties. J. Pure Appl. Algebra 139, 303–321 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  77. 199.
    K. Yanagawa, Alexander duality for Stanley-Reisner rings and squarefree \(\mathbb{N}\)-graded modules. J. Algebra 225, 630–645 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  78. 200.
    K. Yanagawa, Bass numbers of local cohomology modules with supports in monomial ideals. Math. Proc. Camb. Philos. Soc. 131, 45–60 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  79. 201.
    K. Yanagawa, Sheaves on finite posets and modules over normal semigroup rings. J. Pure Appl. Algebra 161, 341–366 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  80. 202.
    K. Yanagawa, Derived category of squarefree modules and local cohomology with monomial ideal support. J. Math. Soc. Jpn. 56, 289–308 (2004)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Departamento de Matemàtica Aplicada IUniversitat Politècnica de CatalunyaBarcelonaSpain

Personalised recommendations