Infinite Free Resolutions

  • Luchezar L. Avramov
Part of the Progress in Mathematics book series


This text is based on the notes for a series of five lectures to the Barcelona Summer School in Commutative Algebra at the Centre de Recerca Matemàtica, Institut d’Estudis Catalans, July 15–26, 1996.


Exact Sequence Spectral Sequence Local Ring Complete Intersection Betti Number 
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.


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  1. [1]
    S. S. Abhyankar, Local rings of high embedding dimension, Amer. J. Math. 89 (1967), 1073–1077.zbMATHCrossRefGoogle Scholar
  2. [2]
    J. Alperin, L. Evens, Representations, resolutions, and Quillen’s dimension theorem, J. Pure Appl. Algebra 22 (1981), 1–9.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    M. André, Méthode simpliciale en algèbre homologique et algèbre commutative, Lecture Notes Math. 32, Springer, Berlin, 1967.Google Scholar
  4. [4]
    M. André, L’algèbre de Lie d’un anneau local, Symp. Math. 4, (INDAM, Rome, 1968/69), Academic Press, London, 1970; pp. 337–375.Google Scholar
  5. [5]
    M. André, Hopf algebras with divided powers, J. Algebra 18 (1971), 19–50.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    M. André, Homologie des algèbres commutatives, Grundlehren Math. Wiss. 204, Springer, Berlin, 1974.Google Scholar
  7. [7]
    M. André, La (2p+ 1)ème déviation d’un anneau local, Enseignement Math. (2) 23 (1977), 239–248.Google Scholar
  8. [8]
    M. André, Algèbre homologique des anneaux locaux à corps résiduels de caractéristique deux, Sém. d’Algèbre P. Dubreil, Paris, 1979 (M.-P. Malliavin, ed.), Lecture Notes Math. 740, Springer, Berlin, 1979; pp. 237–242.Google Scholar
  9. [9]
    M. André, Le caractère additif des déviations des anneaux locaux, Comment. Math. Hely. 57 (1982), 648–675.CrossRefGoogle Scholar
  10. [10]
    D. Anick, Constructions d’espaces de lacets et d’anneaux locaux à séries de Poincaré-Betti non rationnelles, C. R. Acad. Sci. Paris Sér. A 290 (1980), 729–732.Google Scholar
  11. [11]
    D. Anick, Counterexample to a conjecture of Serre, Ann. of Math. (2) 115 (1982), 1–33; Comment, ibid. 116 (1982), 661.Google Scholar
  12. [12]
    D. Anick, Recent progress in Hilbert and Poincaré series, Algebraic topology. Rational homotopy, Louvain-la-Neuve, 1986 (Y. Félix, ed.), Lecture Notes Math. 1318, Springer, Berlin, 1988; pp. 1–25.Google Scholar
  13. [13]
    D. Anick, T. H. Gulliksen, Rational dependence among Hilbert and Poincaré series, J. Pure Appl. Algebra 38 (1985), 135–158.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    A. G. Aramova, J. Herzog, Koszul cycles and Eliahou-Kervaire type resolutions, J. Algebra 181 (1996), 347–370.MathSciNetzbMATHGoogle Scholar
  15. [15]
    E. F. Assmus, Jr., On the homology of local rings Illinois J. Math. 3 (1959), 187199.Google Scholar
  16. [16]
    M. Auslander, Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631–647.zbMATHGoogle Scholar
  17. [17]
    M. Auslander, D. A. Buchsbaum, Codimension and multiplicity, Ann. of Math. (2) 68 (1958), 625–657; Corrections, ibid, 70 (1959), 395–397.Google Scholar
  18. [18]
    L. L. Avramov, On the Hopf algebra of a local ring Math USSR-Izv. 8 (1974), 259–284; [translated from:] Izv. Akad. Nauk. SSSR, Ser. Mat. 38 (1974), 253–277 [Russian].Google Scholar
  19. [19]
    L. L. Avramov, Flat morphisms of complete intersections Soviet Math. Dokl. 16 (1975), 1413–1417; [translated from:] Dokl. Akad. Nauk. SSSR, 225 (1975), 11–14 [Russian].Google Scholar
  20. [20]
    L. L. Avramov, Homology of local flat extensions and complete intersection defects, Math. Ann. 228 (1977), 27–37.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    L. L. Avramov, Small homomorphisms of local rings, J. Algebra 50 (1978), 400–453.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    L. L. Avramov, Obstructions to the existence of multiplicative structures on minimal free resolutions, Amer. J. Math. 103 (1981), 1–31.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    L. L. Avramov, Local algebra and rational homotopy, Homotopie algébrique et algèbre locale; Luminy, 1982 (J.-M. Lemaire, J.-C. Thomas, eds.) Astérisque 113114, Soc. Math. France, Paris, 1984; pp. 15–43.Google Scholar
  24. [24]
    L. L. Avramov, Golod homomorphisms, Algebra, algebraic topology, and their interactions; Stockholm, 1983 (J.-E. Roos, ed.), Lecture Notes Math. 1183, Springer, Berlin, 1986; pp. 56–78.Google Scholar
  25. [25]
    L. L. Avramov, Modules of finite virtual projective dimension, Invent. Math. 96 (1989), 71–101.zbMATHGoogle Scholar
  26. [26]
    L. L. Avramov, Homological asymptotics of modules over local rings, Commutative algebra; Berkeley, 1987 (M. Hochster, C. Huneke, J. Sally, eds.), MSRI Publ. 15, Springer, New York 1989; pp. 33–62.Google Scholar
  27. [27]
    L. L. Avramov, Problems on infinite free resolutions, Free resolutions in commutative algebra and algebraic geometry; Sundance, 1990 (D. Eisenbud, C. Huneke, eds.), Res. Notes Math. 2, Jones and Bartlett, Boston 1992, pp. 3–23.Google Scholar
  28. [28]
    L. L. Avramov, Local rings over which all modules have rational Poincaré series, J. Pure Appl. Algebra 91 (1994), 29–48.MathSciNetCrossRefGoogle Scholar
  29. [29]
    L. L. Avramov, Modules with extremal resolutions, Math. Res. Lett. 3 (1996), 319–328.MathSciNetGoogle Scholar
  30. [30]
    L. L. Avramov, Locally complete intersection homomorphisms, and a conjecture of Quillen on the vanishing of cotangent homology, Preprint, 1997.Google Scholar
  31. [31]
    L. L. Avramov, R.-O. Buchweitz, Modules of complexity two over complete intersections, Preprint, 1997.Google Scholar
  32. [32]
    L. L. Avramov, V. N. Gasharov, I. V. Peeva, Complete intersection dimension Publ. Math. I.H.E.S. (to appear).Google Scholar
  33. [33]
    L. L. Avramov, S. Halperin, Through the looking glass: A dictionary between rational homotopy theory and local algebra, Algebra, algebraic topology, and their interactions; Stockholm, 1983 (J.-E. Roos, ed.), Lecture Notes Math. 1183, Springer, Berlin, 1986; pp. 1–27.Google Scholar
  34. [34]
    L. L. Avramov, S. Halperin, On the non-vanishing of cotangent cohomology Comment. Math. Helv. 62 (1987), 169–184.MathSciNetzbMATHCrossRefGoogle Scholar
  35. [35]
    L. L. Avramov, J. Herzog, Jacobian criteria for complete intersections. The graded case, Invent. Math. 117 (1994), 75–88.MathSciNetzbMATHGoogle Scholar
  36. [36]
    L. L. Avramov, A. R. Kustin, and M. Miller, Poincaré series of modules over local rings of small embedding codepth or small linking number, J. Algebra 118 (1988), 162–204.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    L. L. Avramov, L.-C. Sun, Cohomology operators defined by a deformation, Preprint, 1996.Google Scholar
  38. [38]
    I. K. Babenko, On the analytic properties of Poincaré series of loop spaces Math. Notes 29 (1980), 359–366; [translated from:] Mat. Zametki 27 (1980), 751–765 [Russian].Google Scholar
  39. [39]
    I. K. Babenko, Problems of growth and rationality in algebra and topology Russian Math. Surv. 29 (1980), no. 2, 95–142; [translated from:] Uspekhi Mat. Nauk 41 (1986), no. 2, 95–142 [Russian].Google Scholar
  40. [40]
    D. Bayer, M. E. Stillman, Macaulay A computer algebra system for computing in Algebraic Geometry and Computer Algebra, 1990; available via anonymous ftp from Scholar
  41. [41]
    A. Blanco, J. Majadas, A. G. Rodicio, On the acyclicity of the Tate complex J. Pure Appl. Algebra (to appear).Google Scholar
  42. [42]
    D. Benson, Representations and cohomology. I; II, Cambridge Stud. Adv. Math. 31; 32, Cambridge Univ. Press, Cambridge, 1991.Google Scholar
  43. [43]
    A. Bigatti, Upper bounds for the Betti numbers of a given Hilbert function, Comm. Algebra 21 (1993), 2317–2334.CrossRefGoogle Scholar
  44. [44]
    R. Bogvad, Gorenstein rings with transcendental Poincaré series, Math. Scand. 53 (1983), 5–15.MathSciNetGoogle Scholar
  45. [45]
    N. Bourbaki, Algèbre, X. Algèbre homologique, Masson, Paris, 1980.Google Scholar
  46. [46]
    W. Bruns, J. Herzog, Cohen-Macaulay rings, Cambridge Stud. Adv. Math. 39, Cambridge Univ. Press, Cambridge, 1993.Google Scholar
  47. [47]
    D. A. Buchsbaum, D. Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Amer. J. Math. 99 (1977), 447–485.MathSciNetzbMATHCrossRefGoogle Scholar
  48. [48]
    R.-O. Buchweitz, G.-M. Greuel, F.-O. Schreyer, Cohen-Macaulay modules on hypersurface singularities. II, Invent. Math. 88 (1987), 165–182.MathSciNetzbMATHGoogle Scholar
  49. [49]
    L. Carroll, Through the looking glass and what Alice found there, Macmillan, London, 1871.Google Scholar
  50. [50]
    H. Cartan, Algèbres d’Eilenberg-MacLane, Exposés 2 à 11, Sém. H. Cartan, Éc. Normale Sup. (1954-1955), Secrétariat Math., Paris, 1956; [reprinted in:] OEvres, vol. III, Springer, Berlin, 1979; pp. 1309–1394.Google Scholar
  51. [51]
    H. Cartan, S. Eilenberg, Homological Algebra, Princeton Univ. Press, Princeton, NJ, 1956.Google Scholar
  52. [52]
    S. Choi, Betti numbers and the integral closure of ideals, Math. Scand. 66 (1990), 173–184.MathSciNetGoogle Scholar
  53. [53]
    S. Choi, Exponential growth of Betti numbers, J. Algebra 152 (1992), 20–29.MathSciNetCrossRefGoogle Scholar
  54. [54]
    J. A. Eagon, M. Fraser, A note on the Koszul complex, Proc. Amer. Math. Soc. 19 (1968), 251–252.MathSciNetzbMATHGoogle Scholar
  55. [55]
    S. Eilenberg, Homological dimension and syzygies, Ann. of Math. (2) 64 (1956), 328–336.MathSciNetCrossRefGoogle Scholar
  56. [56]
    S. Eilenberg, S. MacLane, On the groups H(H, n). I, Ann. of Math. (2) 58 (1953), 55–106.MathSciNetzbMATHCrossRefGoogle Scholar
  57. [57]
    D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), 35–64.MathSciNetCrossRefGoogle Scholar
  58. [58]
    D. Eisenbud, Commutative algebra, with a view towards algebraic geometry, Graduate Texts Math. 150, Springer, Berlin, 1995.Google Scholar
  59. [59]
    D. Eisenbud, S. Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), 89–133.MathSciNetzbMATHCrossRefGoogle Scholar
  60. [60]
    S. Eliahou, M. Kervaire, Minimal resolutions of some monomial ideals, J. Algebra 129 (1990), 1–25.MathSciNetzbMATHCrossRefGoogle Scholar
  61. [61]
    E. G. Evans, P. Griffith, Syzygies, London Math. Soc. Lecture Notes Ser. 106, Cambridge Univ. Press, Cambridge, 1985.Google Scholar
  62. [62]
    L. Evens, The cohomology of groups, Oxford Math. Monographs, Clarendon Press, Oxford, 1991.Google Scholar
  63. [63]
    C. T. Fan, Growth of Betti numbers over noetherian local rings, Math. Scand. 75 (1994), 161–168.MathSciNetGoogle Scholar
  64. [64]
    Y. Félix, S. Halperin Rational L.-S. category and its applications, Trans. Amer. Math. Soc. 273 (1982), 1–37.zbMATHCrossRefGoogle Scholar
  65. [65]
    Y. Félix, J.-C. Thomas, The radius of convergence of the Poincaré series of loop spaces, Invent. Math. 68 (1982), 257–274.zbMATHGoogle Scholar
  66. [66]
    Y. Félix, S. The homotopy Lie algebra for finite complexes, Publ. Math. I.H.E.S. 56 (1982), 179–202.zbMATHGoogle Scholar
  67. [67]
    Y. Félix, S. Halperin, C. Jacobsson, C. Löfwall, J.-C. Thomas, The radical of the homotopy Lie algebra, Amer. J. Math., 110 (1988), 301–322.zbMATHCrossRefGoogle Scholar
  68. [68]
    D. Ferrand, Suite régulière et intersection complète, C. R. Acad. Sci. Paris Sér. A 264 (1967), 427–428.zbMATHGoogle Scholar
  69. [69]
    R. Fröberg, T. H. Gulliksen, C. Löfwall, Flat families of local, artinian algebras, with an infinite number of Poincaré series, Algebra, algebraic topology, and their interactions; Stockholm, 1983 (J.-E. Roos, ed.), Lecture Notes Math. 1183, Springer, Berlin, 1986; pp. 56–78.Google Scholar
  70. [70]
    V. N. Gasharov, I. V. Peeva, Boundedness versus periodicity over commutative local rings, Trans. Amer. Math. Soc, 320 (1990), 569–580.MathSciNetzbMATHCrossRefGoogle Scholar
  71. [71]
    I. M. Gelfand, A. A. Kirillov, Sur les corps liés aux algèbres enveloppantes des algèbres de Lie, Publ. Math. I.H.E.S., 31 (1966), 509–523.Google Scholar
  72. [72]
    F. Ghione, T. H. Gulliksen, Some reduction formulas for Poincaré series of modules, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Nat. (8) 58 (1975), 82–91.MathSciNetzbMATHGoogle Scholar
  73. [73]
    P. G. Goerss, On the André-Quillen cohomology of commutative F2-algebras, Astérisque 186, Soc. Math. France, Paris, 1990.Google Scholar
  74. [74]
    E. S. Golod, On the homologies of certain local rings Soviet Math. Dokl. 3 (1962), 745–748; [translated from:] Dokl. Akad. Nauk. SSSR, 144 (1962), 479–482 [Russian].Google Scholar
  75. [75]
    E. H. Cover, M. Ramras, Increasing sequences of Betti numbers, Pacific J. Math. 87 (1980), 65–68.Google Scholar
  76. [76]
    V. K. A. M. Gugenheim, J. P. May, On the theory and applications of differential torsion products, Mem. Amer. Math. Soc, 142, Amer. Math. Soc., Providence, RI, 1974.Google Scholar
  77. [77]
    T. H. Gulliksen, A proof of the existence of minimal algebra resolutions, Acta Math. 120 (1968), 53–58.MathSciNetzbMATHCrossRefGoogle Scholar
  78. [78]
    T. H. Gulliksen, A homological characterization of local complete intersections, Compositio Math. 23 (1971), 251–255.MathSciNetzbMATHGoogle Scholar
  79. [79]
    T. H. Gulliksen, Massey operations and the Poincaré series of certain local rings, J. Algebra 22 (1972), 223–232.MathSciNetzbMATHCrossRefGoogle Scholar
  80. [80]
    T. H Gulliksen, A change of ring theorem with applications to Poincaré series and intersection multiplicity, Math. Scand. 34 (1974), 167–183.MathSciNetzbMATHGoogle Scholar
  81. [81]
    T. H. Gulliksen, On the Hilbert series of the homology of differential graded algebras, Math. Scand. 46 (1980), 15–22.MathSciNetzbMATHGoogle Scholar
  82. [82]
    T. H. Gulliksen, On the deviations of a local ring, Math. Scand. 47 (1980), 5–20.MathSciNetzbMATHGoogle Scholar
  83. [83]
    T. H. Gulliksen, G. Levin, Homology of local rings, Queen’s Papers Pure Appl. Math. 20, Queen’s Univ., Kingston, ON, 1969Google Scholar
  84. [84]
    S. Halperin, On the non-vanishing of the deviations of a local ring, Comment. Math. Helv. 62 (1987), 646–653.CrossRefGoogle Scholar
  85. [85]
    R. Heitmann, A counterexample to the rigidity conjecture for rings, Bull. Amer. Math. Soc. (New Ser.) 29 (1993), 94–97.CrossRefGoogle Scholar
  86. [86]
    J. Herzog, Komplexe, Auflösungen, und Dualität in der lokalen Algebra, Habilitationsschrift, Regensburg, 1973.Google Scholar
  87. [87]
    J. Herzog, B. Ulrich, J. Backelin, Linear maximal Cohen-Macaulay modules over strict complete intersections, J. Pure Appl. Algebra 71 (1991), 187–201.MathSciNetzbMATHGoogle Scholar
  88. [88]
    D. Hilbert, Über die Theorie der algebraischen Formen, Math. Ann. 36 (1890) 473-534; [reprinted in:]Gesammelte Abhandlungen, Band II: Algebra, Invariantentheorie, Geometrie, Springer, Berlin, 1970; pp. 199–257.Google Scholar
  89. [89]
    M. Hochster, Topics in the homological study of modules over commutative rings, CBMS Regional Conf. Ser. in Math. 24, Amer. Math. Soc., Providence, RI, 1975.Google Scholar
  90. [90]
    H. Hulett, Maximum Betti numbers of homogeneous ideals with a given Hilbert function, Comm Algebra 21 (1993), 2335–2350.MathSciNetCrossRefGoogle Scholar
  91. [91]
    C. Huneke, R. Wiegand, Tensor products of modules, rigidity and local cohomology Math. Scand. (to appear).Google Scholar
  92. [92]
    S. Iyengar, Free resolutions and change of rings, J. Algebra, 190 (1997), 195–213.MathSciNetCrossRefGoogle Scholar
  93. [93]
    C. Jacobsson, Finitely presented graded Lie algebras and homomorphisms of local rings, J. Pure Appl. Algebra 38 (1985), 243–253.MathSciNetCrossRefGoogle Scholar
  94. [94]
    C. Jacobsson, A. R. Kustin, and M. Miller, The Poincaré series of a codimension four Gorenstein ideal is rational, J. Pure Appl. Algebra 38 (1985), 255–275.MathSciNetzbMATHCrossRefGoogle Scholar
  95. [95]
    D. A. Jorgensen, Complexity and Tor on a complete intersection J. Algebra (to appear).Google Scholar
  96. [96]
    R. Kiehl, E. Kunz, Vollständige Durchschnitte und p-Basen, Arch. Math. (Basel), 16 (1965), 348–362.MathSciNetzbMATHCrossRefGoogle Scholar
  97. [97]
    A. R. Kustin, Gorenstein algebras of codimension four and characteristic two, Comm. Algebra 15 (1987), 2417–2429.CrossRefGoogle Scholar
  98. [98]
    A. R. Kustin, The minimal resolution of a codimension four almost complete intersection is a DG algebra, J. Algebra 168 (1994), 371–399.MathSciNetCrossRefGoogle Scholar
  99. [99]
    A. R. Kustin, The deviation two Gorenstein rings of Huneke and Ulrich, Commutative algebra, Trieste, 1994 (A. Simis, N. V. Trung, G. Valla, eds.), World Scientific, Singapore, 1994; pp. 140–163.Google Scholar
  100. [100]
    A. R. Kustin, Huneke-Ulrich almost complete intersections of Cohen-Macaulay type two, J. Algebra 174 (1995), 373–429.MathSciNetCrossRefGoogle Scholar
  101. [101]
    A. R. Kustin, M. Miller, Algebra structures on minimal resolutions of Gorenstein rings of embedding codimension four, Math. Z. 173 (1980), 171–184.MathSciNetzbMATHCrossRefGoogle Scholar
  102. [102]
    A. R. Kustin, S. Palmer Slattery, The Poincaré series of every finitely generated module over a codimension four almost complete intersection is a rational function, J. Pure Appl. Algebra 95 (1994), 271–295.MathSciNetzbMATHCrossRefGoogle Scholar
  103. [103]
    C. Lech, Inequalities related to certain couples of local rings, Acta Math. 112 (1964), 69–89.MathSciNetCrossRefGoogle Scholar
  104. [104]
    T. Larfeldt, C. Lech, Analytic ramification and flat couples of local rings, Acta Math. 146 (1981), 201–208.MathSciNetzbMATHCrossRefGoogle Scholar
  105. [105]
    J.-M. Lemaire, Algèbres connexes et homologie des espaces de lacets, Lecture Notes Math. 422, Springer, Berlin, 1974.Google Scholar
  106. [106]
    J. Lescot, Asymptotic properties of Betti numbers of modules over certain rings, J. Pure Appl. Algebra 38 (1985), 287–298.MathSciNetCrossRefGoogle Scholar
  107. [107]
    J. Lescot, Séries de Poincaré et modules inertes, J. Algebra 132 (1990), 22–49.MathSciNetCrossRefGoogle Scholar
  108. [108]
    G. Levin, Homology of local rings, Ph. D. Thesis, Univ. of Chicago, Chicago, IL, 1965.Google Scholar
  109. [109]
    G. Levin, Local rings and Golod homomorphisms, J. Algebra 37 (1975), 266–289.MathSciNetCrossRefGoogle Scholar
  110. [110]
    G. Levin, Lectures on Golod homomorphisms, Matematiska Istitutionen, Stockholms Universitet, Preprint 15, 1975.Google Scholar
  111. [111]
    G. Levin, Modules and Golod homomorphisms, J. Pure Appl. Algebra 38 (1985), 299–304.MathSciNetzbMATHCrossRefGoogle Scholar
  112. [112]
    S. Lichtenbaum, On the vanishing of Tor in regular local rings, Illinois J. Math. 10 (1966), 220–226.zbMATHGoogle Scholar
  113. [113]
    C. Löfwall, On the subalgebra generated by the one-dimensional elements in the Yoneda Ext-algebra Algebra, algebraic topology, and their interactions; Stockholm, 1983 (J.-E. Roos, ed.), Lecture Notes Math. 1183 Springer, Berlin, 1986; pp. 291–338.Google Scholar
  114. [114]
    C. Löfwall, J.-E. Roos, Cohomologie des algèbres de Lie graduées et séries de Poincaré-Betti non-rationnelles, C. R. Acad. Sci. Paris Sér. A 290 (1980), 733–736.zbMATHGoogle Scholar
  115. [115]
    S. MacLane, Homology, Grundlehren Math. Wiss. 114 Springer, Berlin, 1967.Google Scholar
  116. [116]
    A. Martsinkovsky, A remarkable property of the (co)syzygy modules of the residue field of a non-regular local ring, J. Pure Appl. Algebra 111 (1996), 9–13.Google Scholar
  117. [117]
    H. Matsumura, Commutative ring theory, Cambridge Stud. Adv. Math. 8, Cambridge Univ. Press, Cambridge, 1986.Google Scholar
  118. [118]
    J. P. May Matric Massey products, J. Algebra 12 (1969), 533–568.MathSciNetzbMATHCrossRefGoogle Scholar
  119. [119]
    V. Mehta, Endomorphisms of complexes and modules over Golod rings, Ph. D. Thesis, Univ. of California, Berkeley, CA, 1976.Google Scholar
  120. [120]
    C. Miller, Complexity of tensor products of modules and a theorem of HunekeWiegand Proc. Amer. Math. Soc. (to appear).Google Scholar
  121. [121]
    J. W. Milnor, J. C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264.MathSciNetzbMATHCrossRefGoogle Scholar
  122. [122]
    J. C. Moore Algèbre homologique et homologie des espaces classifiants, Exposé 7, Sém. H. Cartan, Ec. Normale Sup. ( 1959–1960 ), Sectétariat Math., Paris, 1957.Google Scholar
  123. [123]
    M. P. Murthy, Modules over regular local rings, Illinois J. Math. 7 (1963), 558–565.zbMATHGoogle Scholar
  124. [124]
    M. Nagata, Local rings, Wiley, New York, 1962.zbMATHGoogle Scholar
  125. [125]
    D. G. Northcott, Finite free resolutions, Tracts in Pure Math., 71, Cambridge Univ. Press, Cambridge, 1976.Google Scholar
  126. [126]
    S. Okiyama, A local ring is CM if and only if its residue field has a CM syzygy, Tokyo J. Math. 14 (1991), 489–500.Google Scholar
  127. [127]
    S. Palmer Slattery, Algebra structures on resolutions of rings defined by grade four almost complete intersections, J. Algebra 168 (1994), 371–399.MathSciNetCrossRefGoogle Scholar
  128. [128]
    K. Pardue, Deformation classes of graded modules and maximal Betti numbers, Illinois J. Math. 40 (1996), 564–585.Google Scholar
  129. [129]
    I. Peeva, 0-Borel fixed ideals, J. Algebra 184 (1996), 945–984.MathSciNetCrossRefGoogle Scholar
  130. [130]
    I. Peeva, Exponential growth of Betti numbers J. Pure. Appl. Algebra (to appear).Google Scholar
  131. [131]
    D. Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295.MathSciNetCrossRefGoogle Scholar
  132. [132]
    D. Quillen, The spectrum of an equivariant cohomology ring I; II, Ann. of Math. (2) 94 (1971), 549–572; 573–602.Google Scholar
  133. [133]
    D. Quillen, On the (co-)homology of commutative rings, Applications of categorical algebra; New York, 1968 (A. Heller, ed.), Proc. Symp. Pure Math. 17, Amer. Math. Soc., Providence, RI, 1970; pp. 65–87.CrossRefGoogle Scholar
  134. [134]
    M Ramras, Sequences of Betti numbers J. Algebra 66 (1980), 193–204.Google Scholar
  135. [135]
    M Ramras, Bounds on Betti numbers Can. J. Math. 34 (1982), 589–592.Google Scholar
  136. [136]
    P. Roberts, Homological invariants of modules over commutative rings, Sém. Math. Sup., 72, Presses Univ. Montréal, Montréal, 1980.Google Scholar
  137. [137]
    J.-E. Roos, Relations between the Poincaré-Betti series of loop spaces and local rings, Sém. d’Algèbre P. Dubreil; Paris, 1977–78 (M.-P. Malliavin, ed.), Lecture Notes Math. 740, Springer, Berlin, 1979; pp. 285–322.Google Scholar
  138. [138]
    J.-E. Roos, Homology of loop spaces and of local rings, Proc. 18th Scand. Congr. Math. Arhus, 1980 (E. Balslev, ed.), Progress Math. 11, Birkhäuser, Basel, 1982; pp. 441–468.Google Scholar
  139. [139]
    C. Schoeller, Homologie des anneaux locaux noethériens, C. R. Acad. Sci. Paris Sér. A 265 (1967), 768–771.zbMATHGoogle Scholar
  140. [140]
    J.-P. Serre, Sur la dimension homologique des anneaux et des modules noethériens, Proc. Int. Symp., Tokyo-Nikko (1956), pp. 175–189.Google Scholar
  141. [141]
    J.-P. Serre, Algèbre locale. Multiplicités, Lecture Notes Math. 11 Springer, Berlin, 1965.Google Scholar
  142. [142]
    J. Shamash, The Poincaré series of a local ring, J. Algebra 12 (1969), 453–470.MathSciNetzbMATHCrossRefGoogle Scholar
  143. [143]
    G. Scheja, Über die Bettizahlen lokaler Ringe, Math. Ann. 155 (1964), 155–172.MathSciNetCrossRefGoogle Scholar
  144. [144]
    G. Sjödin, A set of generators for Ext R (k, k), Math. Scand. 38 (1976), 1–12.Google Scholar
  145. [145]
    G. Sjödin, Hopf algebras and derivations, J. Algebra 64 (1980), 218–229.MathSciNetCrossRefGoogle Scholar
  146. [146]
    H. Srinivasan, The non-existence of a minimal algebra resolutions despite the vanishing of Avramov obstructions, J. Algebra 146 (1992), 251–266.MathSciNetCrossRefGoogle Scholar
  147. [147]
    H. Srinivasan, A grade five Gorenstein algebra with no minimal algebra resolutions, J. Algebra 179 (1996), 362–379.MathSciNetCrossRefGoogle Scholar
  148. [148]
    D. Sullivan, Infinitesimal computations in topology, Publ. Math. I.H.E.S. 47 (1978), 269–331.Google Scholar
  149. [149]
    L.-C. Sun, Growth of Betti numbers of modules over rings of small embedding codimension or small linkage number, J. Pure Appl. Algebra, 96 (1994), 57–71.MathSciNetCrossRefGoogle Scholar
  150. [150]
    L.-C. Sun, Growth of Betti numbers of modules over generalized Golod rings, Preprint, 1996.Google Scholar
  151. [151]
    J. Tate, Homology of noetherian rings and local rings, Illinois J. Math. 1 (1957), 14–25.Google Scholar
  152. [152]
    H. Uehara, W. S. Massey, The Jacobi identity for Whitehead products, Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton Univ. Press, Princeton, NJ, 1957; pp. 361–377.Google Scholar
  153. [153]
    V. A. Ufnarovskij, Combinatorial and asymptotic methods in algebra Encyclopaedia of Math. Sci. 57, Springer, Berlin, 1995; pp. 1–196; [translated from:] Current problems in mathematics. Fundamental directions, 57, Akad. Nauk SSSR, VINITI, Moscow, 1990. pp. 5–177 [Russian].Google Scholar
  154. [154]
    W. V. Vasconcelos, Ideals generated by R-sequences, J. Algebra 6 (1970), 309–316.Google Scholar
  155. [155]
    W. V. Vasconcelos, On the homology of I/I 2, Comm Algebra 6 (1978), 1801–1809.MathSciNetzbMATHCrossRefGoogle Scholar
  156. [156]
    W. V. Vasconcelos, The complete intersection locus of certain ideals, J. Pure Appl. Algebra, 38 (1986), 367–378.Google Scholar
  157. [157]
    J. Watanabe, A note on Gorenstein rings of embedding codimension three, Nagoya Math. J. 50 (1973), 227–232.zbMATHGoogle Scholar
  158. [158]
    H. Wiebe, Ober homologische Invarianten lokaler Ringe, Math. Ann. 179 (1969), 257–274.MathSciNetzbMATHCrossRefGoogle Scholar
  159. [159]
    K. Wolffhardt, Die Betti-Reihe and die Abweichungen eines lokalen Rings, Math. Z. 114 (1970), 66–78.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1998

Authors and Affiliations

  • Luchezar L. Avramov
    • 1
  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA

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