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Homological Asymptotics of Modules over Local Rings

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Commutative Algebra

Part of the book series: Mathematical Sciences Research Institute Publications ((MSRI,volume 15))

Abstract

In this paper we consider the asymptotic behavior of the ranks of the free modules in resolutions of finitely generated modules, M, over a commutative noetherian ring, R. We address two basic questions:

  • What are the possible types of asymptotics for these ranks?

  • How is any particular type related to the structure of M?

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References

  1. J. Alperin, Periodicity in groups, Illinois J. Math. 21 (1977), 776–783.

    MathSciNet  MATH  Google Scholar 

  2. J. Alperin, L. Evens, Representations, resolutions, and Quillen’s dimension theorem, J. Pure Appi. Algebra 22 (1981), 1–9.

    Article  MathSciNet  MATH  Google Scholar 

  3. D. J. Anick, A counterexample to a conjecture of Serre, Ann. of Math. 115 (1982), 1–33.

    Article  MathSciNet  MATH  Google Scholar 

  4. L. L. Avramov, Small homomorphisms of local rings, J. Algebra 50 (1978), 400–453.

    Article  MathSciNet  MATH  Google Scholar 

  5. L. L. Avramov, Obstructions to the existence of multiplicative structures on minimal free resolutions, Amer. J. Math. 103 (1981), 1–31.

    Article  MathSciNet  MATH  Google Scholar 

  6. L. L. Avramov, Free Lie subalgebras of the cohomology of local rings, Trans. AMS 270 (1982), 589–608.

    Article  MathSciNet  MATH  Google Scholar 

  7. L. L. Avramov, Local algebra and rational homotopy, Astérisque 113–114 (1984), 15–43.

    MathSciNet  Google Scholar 

  8. L. L. Avramov, Modules of finite virtual projective dimension, Invent. Math, (to appear).

    Google Scholar 

  9. L. L. Avramov and H.-B. Foxby, Locally Gorenstein homomorphisms, Københavns Universitets Matematiske Instituts Preprint Series 1988, no. 16.

    Google Scholar 

  10. L. L. Avramov, V. N. Gasharov, I. V. Peeva, A periodic module of infinite virtual projective dimension, J. Pure Appi. Algebra (to appear).

    Google Scholar 

  11. L. L. Avramov, A. R. Kustin, M. Miller, Poincaré series of modules over local rings of small embedding codepth or small linking number, J. Algebra (to appear).

    Google Scholar 

  12. H. Bass, Finitistic dimension and a homological generalization of semiprimary rings, Trans. AMS 95 (1960), 466–488.

    Article  MathSciNet  MATH  Google Scholar 

  13. L. Bieberbach, “Analytische Fortsetzung,” Springer-Verlag, Berlin, 1955.

    Google Scholar 

  14. D. A. Buchsbaum, D. Eisenbud, Some structure theorems for finite free resolutions, Advances in Math. 12 (1974), 84–139.

    Article  MathSciNet  MATH  Google Scholar 

  15. D. A. Buchsbaum, D. Eisenbud, Algebra structures for finite free resolu-tions, and some structure theorems for ideals of codimension 3, Amer. J. Math. 99 (1977), 447–485.

    Article  MathSciNet  MATH  Google Scholar 

  16. L. Burch, On ideals of finite homological dimension in local rings, Proc. Camb. Phil. Soc. 64 (1968), 941–952.

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Choi, Betti numbers and the integral closure of ideals, preprint (1988).

    Google Scholar 

  18. D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. AMS 260 (1980), 35–64.

    Article  MathSciNet  MATH  Google Scholar 

  19. Y. Felix, S. Halperin, C. Jacobsson, C. Löfwall J.C. Thomas, The radical of the homotopy Lie algebra, Amer. J. Math. 110 (1988), 301–322.

    Article  MATH  Google Scholar 

  20. H.-B. Foxby, On the μ i in a minimal injective resolution, II, Math. Scand. 41 (1977), 19–44.

    MathSciNet  MATH  Google Scholar 

  21. F. Ghione, T. H. Gulliksen, Some reduction formulas for the Poincaré series of modules, Atti Acad. Naz. Lincei Rend. CI. Sci. Fis. Natur. (8) 58 (1975), 82–91.

    MathSciNet  MATH  Google Scholar 

  22. V. N. Gasharov, I. V. Peeva, Boundedness versus periodicity over commutative local rings, Trans. AMS (to appear).

    Google Scholar 

  23. E. H. Gover, M. Ramras, Increasing sequences of Betti numbers, Pacific J. Math. 87 (1980), 65–68.

    MathSciNet  MATH  Google Scholar 

  24. T. H. Gulliksen, Massey operations and the Poincaré series of certain local rings, J. Algebra 22 (1972), 223–232.

    Article  MathSciNet  MATH  Google Scholar 

  25. T. H. Gulliksen, A change of rings theorem with applications to Poincaré series and intersection multiplicity, Math. Scand. 34 (1974), 167–183.

    MathSciNet  MATH  Google Scholar 

  26. T. H. Gulliksen, On the deviations of a local rings, Math. Scand. 47 (1980), 5–20.

    MathSciNet  MATH  Google Scholar 

  27. J. Herzog, M. Steurich, Two applications of change of rings theorems for Poincaré series, Proc. AMS 73 (1979), 163–168.

    MathSciNet  MATH  Google Scholar 

  28. C. Jacobsson, On local flat homomorphisms and the Yoneda Ext-algebra of the fibre, Astérisque 113–114 (1984), 227–233.

    MathSciNet  Google Scholar 

  29. C. Jacobsson, A. R. Kustin, M. Miller, The Poincaré series of a codimension four Gorenstein ring is rational, J. Pure Appi. Algebra 38 (1985), 255–275.

    Article  MathSciNet  MATH  Google Scholar 

  30. J. P. Jans, Some generalizations of finite projective dimension, Illinois J. Math. 5 (1961), 334–343.

    MathSciNet  MATH  Google Scholar 

  31. A. R. Kustin, Gorenstein algebras of codimension four and characteristic two, Communications in Algebra 15 (1987), 2417–2429.

    Article  MathSciNet  MATH  Google Scholar 

  32. A. R. Kustin, M. Miller, Algebra structures on minimal resolutions of Gorenstein rings of codimension four, Math. Z. 173 (1980), 171–184.

    Article  MathSciNet  MATH  Google Scholar 

  33. A. R. Kustin, Miller, Multiplicative structure on resolutions of algebras defined by Herzog ideals, J. London Math. Soc. (2) 28 (1983), 247–260.

    Article  MathSciNet  MATH  Google Scholar 

  34. A. R. Kustin, M. Miller, Classification of the Tor-algebras of codimension four Gorenstein local rings, Math. Z. 190 (1985), 341–355.

    Article  MathSciNet  MATH  Google Scholar 

  35. J. Lescot, Asymptotic properties of Betti numbers of modules over certain rings, J. Pure Appi. Algebra 38 (1985), 287–298.

    MathSciNet  MATH  Google Scholar 

  36. J. Lescot, Series de Poincaré et modules inertes, preprint (1986).

    Google Scholar 

  37. S. MacLane, “Homology,” Springer-Verlag, Berlin, 1963.

    Google Scholar 

  38. M. Ramras, Sequences of Betti numbers, J. Algebra 66 (1980), 193–204.

    Article  MathSciNet  MATH  Google Scholar 

  39. G. Scheja, Ùber die Bettizahlen lokaler Ringe, Math. Ann. 155 (1964), 155–172.

    Article  MathSciNet  MATH  Google Scholar 

  40. J. Shamash, The Poincaré series of a local ring, J. Algebra 12 (1969), 453–470.

    Article  MathSciNet  MATH  Google Scholar 

  41. J. Tate, Homology of Noetherian rings and of local rings, Illinois J. Math. 1 (1957), 14–27.

    MathSciNet  MATH  Google Scholar 

  42. J. Weyman, On the structure of free resolutions of length 3, preprint (1985).

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Mathematics, ul. “Akad. G. Bončev” bl. 8, 1113, Sofia, Bulgaria

    Luchezar L. Avramov

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  1. Luchezar L. Avramov
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Univeristy of Michigan, 48109-1003, Ann Arbor, MI, USA

    M. Hochster

  2. Department of Mathematics, Purdue University, 47907, West Lafayette, IN, USA

    C. Huneke

  3. Mathematical Sciences Research Institute, 94720, 1000 Centennial Drive, Berkeley, CA, USA

    C. Huneke

  4. Department of Mathematics, Northwestern University, 60201, Evanston, IL, USA

    J. D. Sally

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© 1989 Springer-Verlag New York Inc.

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Avramov, L.L. (1989). Homological Asymptotics of Modules over Local Rings. In: Hochster, M., Huneke, C., Sally, J.D. (eds) Commutative Algebra. Mathematical Sciences Research Institute Publications, vol 15. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3660-3_3

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  • DOI: https://doi.org/10.1007/978-1-4612-3660-3_3

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-8196-2

  • Online ISBN: 978-1-4612-3660-3

  • eBook Packages: Springer Book Archive

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