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Infinite Free Resolutions

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Six Lectures on Commutative Algebra

Part of the book series: Progress in Mathematics ((MBC))

Abstract

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.

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References

  1. S. S. Abhyankar, Local rings of high embedding dimension, Amer. J. Math. 89 (1967), 1073–1077.

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. M. André, Méthode simpliciale en algèbre homologique et algèbre commutative, Lecture Notes Math. 32, Springer, Berlin, 1967.

    Google Scholar 

  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. M. André, Hopf algebras with divided powers, J. Algebra 18 (1971), 19–50.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. André, Homologie des algèbres commutatives, Grundlehren Math. Wiss. 204, Springer, Berlin, 1974.

    Google Scholar 

  7. M. André, La (2p+ 1)ème déviation d’un anneau local, Enseignement Math. (2) 23 (1977), 239–248.

    Google Scholar 

  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. M. André, Le caractère additif des déviations des anneaux locaux, Comment. Math. Hely. 57 (1982), 648–675.

    Article  Google Scholar 

  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. D. Anick, Counterexample to a conjecture of Serre, Ann. of Math. (2) 115 (1982), 1–33; Comment, ibid. 116 (1982), 661.

    Google Scholar 

  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. D. Anick, T. H. Gulliksen, Rational dependence among Hilbert and Poincaré series, J. Pure Appl. Algebra 38 (1985), 135–158.

    Article  MathSciNet  MATH  Google Scholar 

  14. A. G. Aramova, J. Herzog, Koszul cycles and Eliahou-Kervaire type resolutions, J. Algebra 181 (1996), 347–370.

    MathSciNet  MATH  Google Scholar 

  15. E. F. Assmus, Jr., On the homology of local rings Illinois J. Math. 3 (1959), 187199.

    Google Scholar 

  16. M. Auslander, Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631–647.

    MATH  Google Scholar 

  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. 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. 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. L. L. Avramov, Homology of local flat extensions and complete intersection defects, Math. Ann. 228 (1977), 27–37.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. 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 

  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. 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. L. L. Avramov, Modules of finite virtual projective dimension, Invent. Math. 96 (1989), 71–101.

    MATH  Google Scholar 

  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. 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. L. L. Avramov, Local rings over which all modules have rational Poincaré series, J. Pure Appl. Algebra 91 (1994), 29–48.

    Article  MathSciNet  Google Scholar 

  29. L. L. Avramov, Modules with extremal resolutions, Math. Res. Lett. 3 (1996), 319–328.

    MathSciNet  Google Scholar 

  30. L. L. Avramov, Locally complete intersection homomorphisms, and a conjecture of Quillen on the vanishing of cotangent homology, Preprint, 1997.

    Google Scholar 

  31. L. L. Avramov, R.-O. Buchweitz, Modules of complexity two over complete intersections, Preprint, 1997.

    Google Scholar 

  32. L. L. Avramov, V. N. Gasharov, I. V. Peeva, Complete intersection dimension Publ. Math. I.H.E.S. (to appear).

    Google Scholar 

  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. L. L. Avramov, S. Halperin, On the non-vanishing of cotangent cohomology Comment. Math. Helv. 62 (1987), 169–184.

    Article  MathSciNet  MATH  Google Scholar 

  35. L. L. Avramov, J. Herzog, Jacobian criteria for complete intersections. The graded case, Invent. Math. 117 (1994), 75–88.

    MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  37. L. L. Avramov, L.-C. Sun, Cohomology operators defined by a deformation, Preprint, 1996.

    Google Scholar 

  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. 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. D. Bayer, M. E. Stillman, Macaulay A computer algebra system for computing in Algebraic Geometry and Computer Algebra, 1990; available via anonymous ftp from zariski.harvard.edu.

    Google Scholar 

  41. A. Blanco, J. Majadas, A. G. Rodicio, On the acyclicity of the Tate complex J. Pure Appl. Algebra (to appear).

    Google Scholar 

  42. D. Benson, Representations and cohomology. I; II, Cambridge Stud. Adv. Math. 31; 32, Cambridge Univ. Press, Cambridge, 1991.

    Google Scholar 

  43. A. Bigatti, Upper bounds for the Betti numbers of a given Hilbert function, Comm. Algebra 21 (1993), 2317–2334.

    Article  Google Scholar 

  44. R. Bogvad, Gorenstein rings with transcendental Poincaré series, Math. Scand. 53 (1983), 5–15.

    MathSciNet  Google Scholar 

  45. N. Bourbaki, Algèbre, X. Algèbre homologique, Masson, Paris, 1980.

    Google Scholar 

  46. W. Bruns, J. Herzog, Cohen-Macaulay rings, Cambridge Stud. Adv. Math. 39, Cambridge Univ. Press, Cambridge, 1993.

    Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  48. R.-O. Buchweitz, G.-M. Greuel, F.-O. Schreyer, Cohen-Macaulay modules on hypersurface singularities. II, Invent. Math. 88 (1987), 165–182.

    MathSciNet  MATH  Google Scholar 

  49. L. Carroll, Through the looking glass and what Alice found there, Macmillan, London, 1871.

    Google Scholar 

  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. H. Cartan, S. Eilenberg, Homological Algebra, Princeton Univ. Press, Princeton, NJ, 1956.

    Google Scholar 

  52. S. Choi, Betti numbers and the integral closure of ideals, Math. Scand. 66 (1990), 173–184.

    MathSciNet  Google Scholar 

  53. S. Choi, Exponential growth of Betti numbers, J. Algebra 152 (1992), 20–29.

    Article  MathSciNet  Google Scholar 

  54. J. A. Eagon, M. Fraser, A note on the Koszul complex, Proc. Amer. Math. Soc. 19 (1968), 251–252.

    MathSciNet  MATH  Google Scholar 

  55. S. Eilenberg, Homological dimension and syzygies, Ann. of Math. (2) 64 (1956), 328–336.

    Article  MathSciNet  Google Scholar 

  56. S. Eilenberg, S. MacLane, On the groups H(H, n). I, Ann. of Math. (2) 58 (1953), 55–106.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  58. D. Eisenbud, Commutative algebra, with a view towards algebraic geometry, Graduate Texts Math. 150, Springer, Berlin, 1995.

    Google Scholar 

  59. D. Eisenbud, S. Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), 89–133.

    Article  MathSciNet  MATH  Google Scholar 

  60. S. Eliahou, M. Kervaire, Minimal resolutions of some monomial ideals, J. Algebra 129 (1990), 1–25.

    Article  MathSciNet  MATH  Google Scholar 

  61. E. G. Evans, P. Griffith, Syzygies, London Math. Soc. Lecture Notes Ser. 106, Cambridge Univ. Press, Cambridge, 1985.

    Google Scholar 

  62. L. Evens, The cohomology of groups, Oxford Math. Monographs, Clarendon Press, Oxford, 1991.

    Google Scholar 

  63. C. T. Fan, Growth of Betti numbers over noetherian local rings, Math. Scand. 75 (1994), 161–168.

    MathSciNet  Google Scholar 

  64. Y. Félix, S. Halperin Rational L.-S. category and its applications, Trans. Amer. Math. Soc. 273 (1982), 1–37.

    Article  MATH  Google Scholar 

  65. Y. Félix, J.-C. Thomas, The radius of convergence of the Poincaré series of loop spaces, Invent. Math. 68 (1982), 257–274.

    MATH  Google Scholar 

  66. Y. Félix, S. The homotopy Lie algebra for finite complexes, Publ. Math. I.H.E.S. 56 (1982), 179–202.

    MATH  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  68. D. Ferrand, Suite régulière et intersection complète, C. R. Acad. Sci. Paris Sér. A 264 (1967), 427–428.

    MATH  Google Scholar 

  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. V. N. Gasharov, I. V. Peeva, Boundedness versus periodicity over commutative local rings, Trans. Amer. Math. Soc, 320 (1990), 569–580.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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.

    MathSciNet  MATH  Google Scholar 

  73. P. G. Goerss, On the André-Quillen cohomology of commutative F2-algebras, Astérisque 186, Soc. Math. France, Paris, 1990.

    Google Scholar 

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

    Google Scholar 

  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. T. H. Gulliksen, A proof of the existence of minimal algebra resolutions, Acta Math. 120 (1968), 53–58.

    Article  MathSciNet  MATH  Google Scholar 

  78. T. H. Gulliksen, A homological characterization of local complete intersections, Compositio Math. 23 (1971), 251–255.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  81. T. H. Gulliksen, On the Hilbert series of the homology of differential graded algebras, Math. Scand. 46 (1980), 15–22.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  83. T. H. Gulliksen, G. Levin, Homology of local rings, Queen’s Papers Pure Appl. Math. 20, Queen’s Univ., Kingston, ON, 1969

    Google Scholar 

  84. S. Halperin, On the non-vanishing of the deviations of a local ring, Comment. Math. Helv. 62 (1987), 646–653.

    Article  Google Scholar 

  85. R. Heitmann, A counterexample to the rigidity conjecture for rings, Bull. Amer. Math. Soc. (New Ser.) 29 (1993), 94–97.

    Article  Google Scholar 

  86. J. Herzog, Komplexe, Auflösungen, und Dualität in der lokalen Algebra, Habilitationsschrift, Regensburg, 1973.

    Google Scholar 

  87. J. Herzog, B. Ulrich, J. Backelin, Linear maximal Cohen-Macaulay modules over strict complete intersections, J. Pure Appl. Algebra 71 (1991), 187–201.

    MathSciNet  MATH  Google Scholar 

  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. 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. H. Hulett, Maximum Betti numbers of homogeneous ideals with a given Hilbert function, Comm Algebra 21 (1993), 2335–2350.

    Article  MathSciNet  Google Scholar 

  91. C. Huneke, R. Wiegand, Tensor products of modules, rigidity and local cohomology Math. Scand. (to appear).

    Google Scholar 

  92. S. Iyengar, Free resolutions and change of rings, J. Algebra, 190 (1997), 195–213.

    Article  MathSciNet  Google Scholar 

  93. C. Jacobsson, Finitely presented graded Lie algebras and homomorphisms of local rings, J. Pure Appl. Algebra 38 (1985), 243–253.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  95. D. A. Jorgensen, Complexity and Tor on a complete intersection J. Algebra (to appear).

    Google Scholar 

  96. R. Kiehl, E. Kunz, Vollständige Durchschnitte und p-Basen, Arch. Math. (Basel), 16 (1965), 348–362.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  98. A. R. Kustin, The minimal resolution of a codimension four almost complete intersection is a DG algebra, J. Algebra 168 (1994), 371–399.

    Article  MathSciNet  Google Scholar 

  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. A. R. Kustin, Huneke-Ulrich almost complete intersections of Cohen-Macaulay type two, J. Algebra 174 (1995), 373–429.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  103. C. Lech, Inequalities related to certain couples of local rings, Acta Math. 112 (1964), 69–89.

    Article  MathSciNet  Google Scholar 

  104. T. Larfeldt, C. Lech, Analytic ramification and flat couples of local rings, Acta Math. 146 (1981), 201–208.

    Article  MathSciNet  MATH  Google Scholar 

  105. J.-M. Lemaire, Algèbres connexes et homologie des espaces de lacets, Lecture Notes Math. 422, Springer, Berlin, 1974.

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  107. J. Lescot, Séries de Poincaré et modules inertes, J. Algebra 132 (1990), 22–49.

    Article  MathSciNet  Google Scholar 

  108. G. Levin, Homology of local rings, Ph. D. Thesis, Univ. of Chicago, Chicago, IL, 1965.

    Google Scholar 

  109. G. Levin, Local rings and Golod homomorphisms, J. Algebra 37 (1975), 266–289.

    Article  MathSciNet  Google Scholar 

  110. G. Levin, Lectures on Golod homomorphisms, Matematiska Istitutionen, Stockholms Universitet, Preprint 15, 1975.

    Google Scholar 

  111. G. Levin, Modules and Golod homomorphisms, J. Pure Appl. Algebra 38 (1985), 299–304.

    Article  MathSciNet  MATH  Google Scholar 

  112. S. Lichtenbaum, On the vanishing of Tor in regular local rings, Illinois J. Math. 10 (1966), 220–226.

    MATH  Google Scholar 

  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. 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.

    MATH  Google Scholar 

  115. S. MacLane, Homology, Grundlehren Math. Wiss. 114 Springer, Berlin, 1967.

    Google Scholar 

  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. H. Matsumura, Commutative ring theory, Cambridge Stud. Adv. Math. 8, Cambridge Univ. Press, Cambridge, 1986.

    Google Scholar 

  118. J. P. May Matric Massey products, J. Algebra 12 (1969), 533–568.

    Article  MathSciNet  MATH  Google Scholar 

  119. V. Mehta, Endomorphisms of complexes and modules over Golod rings, Ph. D. Thesis, Univ. of California, Berkeley, CA, 1976.

    Google Scholar 

  120. C. Miller, Complexity of tensor products of modules and a theorem of HunekeWiegand Proc. Amer. Math. Soc. (to appear).

    Google Scholar 

  121. J. W. Milnor, J. C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264.

    Article  MathSciNet  MATH  Google Scholar 

  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. M. P. Murthy, Modules over regular local rings, Illinois J. Math. 7 (1963), 558–565.

    MATH  Google Scholar 

  124. M. Nagata, Local rings, Wiley, New York, 1962.

    MATH  Google Scholar 

  125. D. G. Northcott, Finite free resolutions, Tracts in Pure Math., 71, Cambridge Univ. Press, Cambridge, 1976.

    Google Scholar 

  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. S. Palmer Slattery, Algebra structures on resolutions of rings defined by grade four almost complete intersections, J. Algebra 168 (1994), 371–399.

    Article  MathSciNet  Google Scholar 

  128. K. Pardue, Deformation classes of graded modules and maximal Betti numbers, Illinois J. Math. 40 (1996), 564–585.

    Google Scholar 

  129. I. Peeva, 0-Borel fixed ideals, J. Algebra 184 (1996), 945–984.

    Article  MathSciNet  Google Scholar 

  130. I. Peeva, Exponential growth of Betti numbers J. Pure. Appl. Algebra (to appear).

    Google Scholar 

  131. D. Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295.

    Article  MathSciNet  Google Scholar 

  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. 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.

    Book  Google Scholar 

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

    Google Scholar 

  135. M Ramras, Bounds on Betti numbers Can. J. Math. 34 (1982), 589–592.

    Google Scholar 

  136. P. Roberts, Homological invariants of modules over commutative rings, Sém. Math. Sup., 72, Presses Univ. Montréal, Montréal, 1980.

    Google Scholar 

  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. 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. C. Schoeller, Homologie des anneaux locaux noethériens, C. R. Acad. Sci. Paris Sér. A 265 (1967), 768–771.

    MATH  Google Scholar 

  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. J.-P. Serre, Algèbre locale. Multiplicités, Lecture Notes Math. 11 Springer, Berlin, 1965.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  144. G. Sjödin, A set of generators for Ext R (k, k), Math. Scand. 38 (1976), 1–12.

    Google Scholar 

  145. G. Sjödin, Hopf algebras and derivations, J. Algebra 64 (1980), 218–229.

    Article  MathSciNet  Google Scholar 

  146. H. Srinivasan, The non-existence of a minimal algebra resolutions despite the vanishing of Avramov obstructions, J. Algebra 146 (1992), 251–266.

    Article  MathSciNet  Google Scholar 

  147. H. Srinivasan, A grade five Gorenstein algebra with no minimal algebra resolutions, J. Algebra 179 (1996), 362–379.

    Article  MathSciNet  Google Scholar 

  148. D. Sullivan, Infinitesimal computations in topology, Publ. Math. I.H.E.S. 47 (1978), 269–331.

    Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  150. L.-C. Sun, Growth of Betti numbers of modules over generalized Golod rings, Preprint, 1996.

    Google Scholar 

  151. J. Tate, Homology of noetherian rings and local rings, Illinois J. Math. 1 (1957), 14–25.

    Google Scholar 

  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. 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. W. V. Vasconcelos, Ideals generated by R-sequences, J. Algebra 6 (1970), 309–316.

    Google Scholar 

  155. W. V. Vasconcelos, On the homology of I/I 2, Comm Algebra 6 (1978), 1801–1809.

    Article  MathSciNet  MATH  Google Scholar 

  156. W. V. Vasconcelos, The complete intersection locus of certain ideals, J. Pure Appl. Algebra, 38 (1986), 367–378.

    Google Scholar 

  157. J. Watanabe, A note on Gorenstein rings of embedding codimension three, Nagoya Math. J. 50 (1973), 227–232.

    MATH  Google Scholar 

  158. H. Wiebe, Ober homologische Invarianten lokaler Ringe, Math. Ann. 179 (1969), 257–274.

    Article  MathSciNet  MATH  Google Scholar 

  159. K. Wolffhardt, Die Betti-Reihe and die Abweichungen eines lokalen Rings, Math. Z. 114 (1970), 66–78.

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, Purdue University, West Lafayette, Indiana, 47907, USA

    Luchezar L. Avramov

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  1. Dept. Algebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, E-08007, Barcelona, Spain

    J. Elias , J. M. Giral , R. M. Miró-Roig  & S. Zarzuela , ,  & 

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Avramov, L.L. (1998). Infinite Free Resolutions. In: Elias, J., Giral, J.M., Miró-Roig, R.M., Zarzuela, S. (eds) Six Lectures on Commutative Algebra. Progress in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0346-0329-4_1

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  • DOI: https://doi.org/10.1007/978-3-0346-0329-4_1

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