Skip to main content
SpringerLink
Log in

Homological dimension theory of small categories

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

REFERENCES

  1. M. André, “Limites et fibrés,” C.R. Acad. Sci. Paris, Sér. A, 260, 756-759 (1965).

    Google Scholar 

  2. M. André, “Méthode simpliciale en algébre homologique et algébre commutative,” Lect. Notes Math., 32 (1967).

  3. S. Araki and Z. Yosimura, “A spectral sequence associated with a cohomology theory of infinite CW-complexes,” Osaka J. Math., 9, No. 3, 351-365 (1972).

    Google Scholar 

  4. K. Baclawski, “Galois connections and the Leray spectral sequence,” Adv. Math., 25, No. 3, 191-215 (1977).

    Google Scholar 

  5. S. Balcerzyk, “The cohomological dimension of the ordered set of real numbers equals three,” Fund. Math., 111, No. 1, 37-44 (1981).

    Google Scholar 

  6. M. G. Barrat, “A note on the cohomology of semigroups,” J. London Math. Soc., 36, 496-498 (1961).

    Google Scholar 

  7. H. Bass, “Descending chains and the Krull ordinal of commutative Noetherian rings,” J. Pure Appl. Algebra, 5, No. 4, 347-360 (1971).

    Google Scholar 

  8. H.-J. Baues, “On the cohomology of categories, universal Toda brackets and homotopy pairs,” Preprint, MPI No. 94-141, 1-30 (1994).

  9. H.-J. Baues, Homotopy Types, in: Handbook of Algebraic Topology. Elsevier, Amsterdam, pp. 1-72 (1995).

    Google Scholar 

  10. H.-J. Baues, “Non-Abelian extensions and homotopies,” K-theory, 10, 107-133 (1996).

    Google Scholar 

  11. H.-J. Baues, “On the group of homotopy equivalences of a manifold,” Trans. Amer. Math. Soc., 348, No. 12, 4737-4773 (1996).

    Google Scholar 

  12. H.-J. Baues and D. Conduché, “On the 2-type of an iterated loop space,” Forum Math., 9, 721-738 (1997).

    Google Scholar 

  13. H.-J. Baues and W. Dreckmann, “The cohomology of homotopy categories and the general linear group,” K-theory, 3, 307-338 (1989).

    Google Scholar 

  14. H.-J. Baues and M. Hartl, “The homotopy category of Moore spaces and the cohomology of the category of Abelian groups,” Preprint, MPI, 1-28 (1995).

  15. H.-J. Baues and A. Tonks, “On sum-normalized cohomology of categories, twisted homotopy pairs and universal Toda brackets,” Quart. J. Math. Oxford (2), 47, 405-433 (1996).

    Google Scholar 

  16. H.-J. Baues and A. Tonks, “On the twisted cobar construction,” Math. Proc. Camb. Phil. Soc., 121, 229-245 (1997).

    Google Scholar 

  17. H.-J. Baues and G. Wirsching, “Cohomology of small categories,” J. Pure Appl. Algebra, 38, No. 2/3, 187-211 (1985).

    Google Scholar 

  18. S. Betley, “Homology of Gl(R) with coe.cients in a functor of finite degree,” J. Algebra, 150, 73-86 (1992).

    Google Scholar 

  19. A. K. Bousfield and D. M. Kan, “Homotopy limits, completions and localizations,” Lect. Notes Math.,304 (1972).

  20. H. Brune, “Einige globale Dimensionen bei geordneten Mengen und die Exaktheit der Finktoren \(\underrightarrow {lim}\) und \(\underleftarrow {lim}\),” Dissertation, Gesamthochschule Paderborn (1977).

  21. H. Brune, “Flache darstellungen von geordnete mengen,” Manuscripta Math., 28, 141-154 (1978).

    Google Scholar 

  22. H. Brune, “Some left pure semisimple ringoids which are not right pure semisimple,” Comm. Algebra, 7, No. 17, 1795-1803 (1979).

    Google Scholar 

  23. H. Brune, “On the global dimension of the functor category,” J. Pure Appl. Algebra, 28, No. 1 31-39 (1983).

    Google Scholar 

  24. N. H. Chen, “Global dimension of skew group rings,” Dissertation, Rutgers University, New Brunswick, NJ (1978).

    Google Scholar 

  25. C. C. Cheng, “Finite partially ordered sets of cohomological dimension one,” J. Algebra, 40, No. 2, 340-347 (1976).

    Google Scholar 

  26. C. C. Cheng, “Deltas of Hochschild dimension one,” Proc. Amer. Math. Soc., 67, No. 2, 221-223 (1977).

    Google Scholar 

  27. C. C. Cheng, “Exact limits,” J. Pure Appl. Algebra, 28, No. 1, 65-70 (1983).

    Google Scholar 

  28. C. C. Cheng, “Separable semigroup algebra,” J. Pure Appl. Algebra, 33, 151-158 (1984).

    Google Scholar 

  29. C. C. Cheng, “DCC categories of cohomological dimension one,” J. Pure Appl. Algebra, 80, No. 2, 107-116 (1992).

    Google Scholar 

  30. C. C. Cheng and B. Mitchell, “Flatness and projectivity of modules that come from C-sets,” in: Algebra, Topology and Category Theory, Academic Press, New York (1976), pp. 33-44.

    Google Scholar 

  31. C. C. Cheng and B. Mitchell, “DCC posets of cohomological dimension one,” J. Pure Appl. Algebra, 13, No. 2, 125-137 (1978).

    Google Scholar 

  32. C. C. Cheng and B. Mitchell, “Posets of cohomological dimension one with finitely many tails,” J. Algebra, 77, 382-391 (1982).

    Google Scholar 

  33. C. C. Cheng and J. Shapiro, “Cohomological dimension of an Abelian monoid,” Proc. Amer. Math. Soc., 88, No. 4, 547-551 (1980).

    Google Scholar 

  34. C. C. Cheng and J. Shapiro, “A converse of the Hilbert syzygy theorem,” Proc. Amer. Math. Soc., 89, No. 1, 11-15 (1983).

    Google Scholar 

  35. C. C. Cheng and R. W. Wong, “Hereditary monoid rings,” Amer. J. Math., 104, No. 5, 935-942 (1982).

    Google Scholar 

  36. C. C. Cheng and R. W. Wong, “Hereditary deltas,” J. Algebra, 167, No. 1, 1-8(1994).

    Google Scholar 

  37. C. C. Cheng, Y.-C. Wu, and B. Mitchell, “Categories of fractions preserve dimension one,” Comm. Algebra, 8, 927-939 (1980).

    Google Scholar 

  38. C. Cibils, “Cohomology of incidence algebras and simplicial complexes,” J. Pure Appl. Algebra, 56, No. 3, 221-232 (1989).

    Google Scholar 

  39. P. M. Cohn, Free Rings and Their Relations, Academic Press, New York (1971).

    Google Scholar 

  40. A. Connes, “Cohomologie cyclicue at foncteurs Extn,” C.R. Acad. Sci. Paris. Ser. I Math., 296, 953-958 (1983).

    Google Scholar 

  41. A. Connes and H. Moscovici, “Cyclic cohomology and Hopf algebras,” Preprint, (http:// xxx.itep.ru) (1999).

  42. T. Datuashvili, “On cohomology of small categories,” in: Proc. Mat. Institut Tbilisi [in Russian], 62, pp. 28-37 (1979).

    Google Scholar 

  43. R. Deheuvels, “Homologie des ensembles ordonnés et espaces topologiques,” Bull. Soc. Math. France, 90, 261-321 (1962).

    Google Scholar 

  44. A. Dow, P. Simon, and J. E. Vaughan, “Strong homology and the proper forcing axiom,” Proc. Amer. Math. Soc., 106, No. 3. 821-828 (1989).

    Google Scholar 

  45. W. G. Dwyer and D. M. Kan, “Function complexes for diagrams of simplicial sets,” Indag. Math., 45, 139-147 (1983).

    Google Scholar 

  46. W. G. Dwyer and D. M. Kan, “Centric maps and realization of diagrams in the homotopy category,” Proc. Amer. Math. Soc., 114, No. 2, 575-584 (1992).

    Google Scholar 

  47. D. A. Edwards and H. M. Hastings, “Čhech and Steenrod homotopy theories with applications to geometric topology,” in: Lecture Notes in Math., 542, Springer, Berlin etc. (1976).

    Google Scholar 

  48. D. A. Edwards and H. M. Hastings, “On topological method in homological algebra,” Proc. Amer. Math. Soc., 59, No. 2, 389-393 (1976).

    Google Scholar 

  49. P. C. Eklof, “Set theoretic methods in homological algebra and Abelian groups,” in: Sem. de Math. Superieures, 69. Les Presses de l'Universite de Montreal, Montreal (1980).

    Google Scholar 

  50. Z. Fiedorowicz, “Classifying spaces of topological monoids and categories,” Amer. J. Math., 106, No. 2, 301-350 (1984).

    Google Scholar 

  51. F. J. Ford, “A Mayer-Vietoris sequence for modules over a small category,” Comm. Algebra, 23, No. 11, 3977-3982 (1995).

    Google Scholar 

  52. P. Gabriel and M. Zisman, “Calculus of fractions and homotopy theory,” Springer-Verlag, Berlin-Heidelberg-New York (1967).

    Google Scholar 

  53. D. George, “Homology with models,” Fund. Math., 122, No. 1, 33-45 (1984).

    Google Scholar 

  54. M. Gerstenhaber and S. D. Schack, “Simplicial cohomology is Hochschild cohomology,” J. Pure Appl. Algebra, 30, 143-156 (1983).

    Google Scholar 

  55. R. Goblot, “Sur les dérivés de certaines limites projectives. Applications aux modules,” Bull. Sci. Math., 94, 251-255 (1970).

    Google Scholar 

  56. M. Golasínski, “n-Fold extensions and cohomologies of small categories,” in: Math. Rev. Anal. Numer. Theor. Approxim. Math., 31, No. 1, 53-59 (1989).

    Google Scholar 

  57. B. I. Gray, “Spaces of the same n-type, for all n,” Topology, 5, No. 3, 241-243 (1966).

    Google Scholar 

  58. B. Gray and C. A. McGibbon, “Universal phantom maps,” Topology, 32, No. 2, 371-394 (1993).

    Google Scholar 

  59. A. Grothendieck, “Sur quelques points d'algébre homologique,” Tohoku Math. J., 9, 119-221 (1957).

    Google Scholar 

  60. L. Gruson and C. U. Jensen, “Modules algébriquement compacts et foncteurs \(\underleftarrow {lim}^{\left( i \right)} \),” C. R. Acad. Sci. Paris, Sér. A, 276, No. 26, 1651-1653 (1973).

    Google Scholar 

  61. L. Gruson and C. U. Jensen, “Dimensions cohomologiques reliées and foncteurs \(\underleftarrow {lim}^{\left( i \right)} \),” in: Sem. d'Algebre Paul Dubreil et Marie-Paule Malliavin, Lec. Notes Math., 867, 234-294 (1980).

    Google Scholar 

  62. P. J. Hilton and U. Stammbach, A Course in Homological Algebra, Springer, Berlin (1971).

    Google Scholar 

  63. G. Ho., “On the cohomology of categories,” Rend. Mat., VI. Ser. 7, No. 2, 169-192 (1974).

    Google Scholar 

  64. G. Ho., “Cohomologies et extensions de categories,” Math. Scand., 74, No. 2, 191-207 (1994).

    Google Scholar 

  65. M. Höppner, “Deltas of dimension one,” Comm. Algebra, 14, 1287-1298 (1986).

    Google Scholar 

  66. M. Höppner and H. Lenzing, “Projective diagrams over partially ordered sets are free,” J. Pure Appl. Algebra, V. 20, 17-22 (1981).

    Google Scholar 

  67. A. A. Husainov, “Cohomology of small categories,” Quest. Answ. General Topology, 8, Special Issue, 179-184 (1990).

    Google Scholar 

  68. A. A. Husainov, “On the Hochschild-Mitchell dimension of subsets of the reals,” Contemp. Math., 184 207-213 (1995).

    Google Scholar 

  69. A. A. Husainov and A. Pancar, “Methods of calculating cohomological and Hochschild-Mitchell dimensions of finite partially ordered sets,” Theory Appl. Categories (to appear).

  70. K. Igusa and D. Zacharia, “On the cohomology of incidence algebras of partially ordered sets,” Comm. Algebra, 18, No. 3, 873-887 (1990).

    Google Scholar 

  71. J. Isbell, “A note on exact colimits,” Can. Math. Bull., 11, 569-572 (1968).

    Google Scholar 

  72. J. Isbell, “Exact colimits I,” Ann. Math., 100, No. 2, 633-637 (1974).

    Google Scholar 

  73. J. Isbell and B. Mitchell, “Exact colimits,” Bull. Amer. Math. Soc., 79, 994-996 (1973).

    Google Scholar 

  74. J. Isbell and B. Mitchell, “Exact colimits and fixed points,” Trans. Amer. Math. Soc., 220, 289-298 (1976).

    Google Scholar 

  75. C. U. Jensen, “On the vanishing of \(\underleftarrow {lim}^{\left( i \right)} \),” J. Algebra, 15, No. 2, 151-166 (1970).

    Google Scholar 

  76. C. U. Jensen, “Les foncteurs dérivés de \(\underleftarrow {lim}\) et leurs applications en théorie des modules,” Lect. Notes Math., 254 (1972).

  77. C. U. Jensen, “On the global dimension of the functor category (mod R, Ab),” J. Pure Appl. Algebra, 11, 45-51 (1977).

    Google Scholar 

  78. C. U. Jensen and H. Lenzing, “Model theoretic algebra: with particular emphasis in fields, rings, modules,” in: Algebra, Logic and Applications, Vol. 2, Gordon and Breach, New York (1989).

    Google Scholar 

  79. M. A. Jibladze (M. A. Dzhibladze), “Bicategorical interpretation of third group of cohomologies,” Trudy Tbilis. Mat. Inst., 97, 3-9 (1992).

    Google Scholar 

  80. M. A. Jibladze and T. I. Pirashvili, “Some linear extensions of the category of finite generated free modules,” Soobsh. Akad. Nauk GSSR, 123, No. 3, 481-484 (1986).

    Google Scholar 

  81. M. Jibladze and T. Pirashvili, “Cohomology of algebraic theories,” J. Algebra, 137, No. 2, 253-296 (1991).

    Google Scholar 

  82. A. A. Khusainov, “Cohomology of small categories with coefficients in an Abelian category with exact products,” Sib. Mat. Zh., 30, No. 4, 210-215 (1989); English transl.: Sib. Math. J., 30, No. 4, 668-672 (1989).

    Google Scholar 

  83. A. A. Khusainov, “On the Hochschild-Mitchell dimension of the real line,” Dokl. Akad. Nauk SSSR, 322, No. 2, 259-261 (1992).

    Google Scholar 

  84. A. A. Khusainov, “On extension groups in the category of Abelian diagrams,” Sib. Mat. Zh., 33, No. 1, 179-185 (1992); English transl.: Sib. Math. J., 33, No. 1, 149-154 (1992).

    Google Scholar 

  85. A. A. Khusainov, “On the Hochschild-Mitchell dimension of ordered sets,” Sib. Mat. Zh., 33, No. 6, 211-215 (1992); English transl.: Sib. Math. J., 33, No. 6, 1140-1143 (1992).

    Google Scholar 

  86. A. A. Khusainov, “The Hochschild-Mitchell dimension of the set of reals is equal to 3,” Sib. Mat. Zh., 34, No. 4, 217-227 (1993); English transl.: Sib. Math. J., 34, No. 4, 786-793 (1993).

    Google Scholar 

  87. A. A. Khusainov, “The Hochschild-Mitchell dimension of linearly ordered sets and the continuum hypothesis,” Sib. Mat. Zh., 35, No. 5, 1171-1184 (1994); English transl.: Sib. Math. J., 35, No. 5, 1040-1051 (1994).

    Google Scholar 

  88. A. A. Khusainov, “Not lim-acyclic sums of copies,” Sib. Mat. Zh., 37, No. 2, 464-474 (1996); English transl.: Sib. Math. J., 37, No. 2, 405-413 (1996).

    Google Scholar 

  89. A. A. Khusainov, “On the global dimension of the category of commutative diagrams in an Abelian category,” Sib. Mat. Zh., 37, No. 5, 1181-1194 (1996); English transl.: Sib. Math. J., 37, No. 5, 1041-1051 (1996).

    Google Scholar 

  90. A. A. Khusainov, “On the sums of copies of an acyclic spectrum,” Dokl. Ross. Akad. Nauk, 356, No. 2, 176-177 (1997).

    Google Scholar 

  91. A. A. Khusainov, “Cancellative categories of dimension 1 are partially free,” in: Urgent Problems in Contemporary Math. [in Russian], Vol. 3, Novosibirsk State University, Novosibirsk, (1997), pp. 193-201.

    Google Scholar 

  92. A. A. Khusainov, “Comparison of the dimensions of a small category,” Sib. Mat. Zh., 38, No. 6, 1413-1426 (1997); English transl.: Sib. Math. J., 38, No. 6, 1230-1240 (1997).

    Google Scholar 

  93. A. A. Khusainov, “On locally finite partially ordered sets of dimension 3,” Sib. Mat. Zh., 39, No. 1, 201-205 (1998); English transl.: Sib. Math. J., 39, No. 1. 176-180 (1998).

    Google Scholar 

  94. A. A. Khusainov, “On the global dimension of the category of Abelian group valued diagrams over a totally ordered set,” Fund. Prikl. Mat., 4, No. 2, 717-723 (1998).

    Google Scholar 

  95. A. A. Khusainov, “On the global dimension of the category of diagrams in a module category,” Sib. Mat. Zh., 40, No. 1, 214-225 (1999).

    Google Scholar 

  96. A. A. Khusainov, “On the relative extension groups in the category of commutative diagrams,” Sib. Mat. Zh. (to appear).

  97. Kourovka Notebook. Unsolved Problems of Group Theory, in: 11th ed., Siberian Branch of the Academy of Sciences of the SSSR, Novosibirsk (1990).

  98. V. I. Kuz'minov, “The limit of an exact sequence of inverse spectra,” Dokl. Akad. Nauk SSSR, 163, 565-568 (1965); English transl.: Sov. Math. Dokl., 6, 959-962 (1965).

    Google Scholar 

  99. V. I. Kuz'minov, “Derived functors of the projective limit functor,” Sib. Mat. Zh., 8, No. 2, 333-345 (1967).

    Google Scholar 

  100. V. I. Kuz'minov and I. A. Shvedov, “On the completion of Abelian groups in the P-topology,” Algebra Logika, 8, 436-457 (1970); English transl.: Algebra and Logic, 9, 262-274 (1971).

    Google Scholar 

  101. V. I. Kuz'minov, “Derived functors of inverse limits and classes of extensions,” Sib. Mat. Zh., 12, No. 2, 384-396 (1971); English transl.: Sib. Math. J., 12, 272-280 (1971).

    Google Scholar 

  102. D. M. Latch, “On derived functors of limit,” Trans. Amer. Math. Soc., 181, 155-163 (1973).

    Google Scholar 

  103. D. M. Latch, “The uniqueness of homology of the categories,” J. Pure Appl. Algebra, 9, 221-237 (1977).

    Google Scholar 

  104. D. M. Latch, “A fibred homotopy equivalence and homotopy theories for the category of small categories,” J. Pure Appl. Algebra, 15, 247-269 (1979).

    Google Scholar 

  105. D. M. Latch and B. Mitchell, “On the difference between cohomological and homological dimension,” J. Pure Appl. Algebra, 5, 333-343 (1974).

    Google Scholar 

  106. O. A. Laudal, “Sur la limite projective et la theorie de la dimension,” in: Sem. de Top. Geom. Diff., 3, 1-23 (1961).

    Google Scholar 

  107. O. A. Laudal, “Cohomologie locale. Applications,” Math. Scand., 12, No. 2, 147-162 (1963).

    Google Scholar 

  108. O. A. Laudal, “Sur les limites projectives et inductives,” Ann. Sci. Ecole Norm. Sup., 3e Ser., 82, 241-296 (1965).

    Google Scholar 

  109. O. A. Laudal, “Projective systems on trees and valuation theory,” Can. J. Math., 20, 984-1000 (1068).

    Google Scholar 

  110. O. A. Laudal, “Note on the projective limit on small categories,” Proc. Amer. Math. Soc., 33, No. 2, 307-309 (1972).

    Google Scholar 

  111. J. Leech, “Cohomology theory for monoid congruences,” Houston J. Math., 11, No. 2, 207-223 (1985).

    Google Scholar 

  112. S. MacLane, Homology, Springer, Berlin (1963).

    Google Scholar 

  113. S. MacLane, Categories for the Working Mathematician (Graduate Texts in Mathematics), Springer-Verlag, New York (1971).

    Google Scholar 

  114. S. Mardešić and V. Prasolov, “Strong homology is not additive,” Trans. Amer. Math. Soc., 307, No. 2, 725-744 (1988).

    Google Scholar 

  115. C. A. McGibbon and J. Roitberg, “Phantom maps and rational equivalences,” Amer. J. Math., 116, No. 6, 1365-1379 (1994).

    Google Scholar 

  116. C. A. McGibbon and R. Steiner, “Some questions about the first derived functor of the inverse limit,” J. Pure Appl. Algebra, 103, 325-340 (1995).

    Google Scholar 

  117. C. A. McGibbon and R. Steiner, “Phantom maps and the towers which determine them,” J. London Math. Soc. (2), 55, 601-608 (1997).

    Google Scholar 

  118. J. Milnor, “On axiomatic homology theories,” Pacific J. Math., 12, No. 1, 337-341 (1962).

    Google Scholar 

  119. M. P. Miminoshvili, “On a sequence of exact and half-exact homologies of arbitrary spaces,” Soobsh. Akad. Nauk GSSR, 113, No. 1, 41-44 (1984).

    Google Scholar 

  120. M. P. Miminoshvili, “On cohomologies of directed sets,” Soobsh. Akad. Nauk GSSR, 120, No. 3, 489-492 (1985).

    Google Scholar 

  121. B. Mitchell, Theory of Categories, Academic Press, New York (1965).

    Google Scholar 

  122. B. Mitchell, “On the dimension of objects and categories I. Monoids,” J. Algebra, 9, No. 3, 314-340 (1968).

    Google Scholar 

  123. B. Mitchell, “On the dimension of objects and categories II. Finite ordered sets,” J. Algebra, 9, No. 3, 341-368 (1968).

    Google Scholar 

  124. B. Mitchell, “On the dimension of object and categories III,” in: Lect. Notes Math., 92, 165-196 (1969).

    Google Scholar 

  125. B. Mitchell, “The dominion of Isbell,” Trans. Amer. Math. Soc., 167, 319-331 (1972).

    Google Scholar 

  126. B. Mitchell, “Rings with several objects,” Adv. Math., 8, 1-161 (1972).

    Google Scholar 

  127. B. Mitchell, “The cohomological dimension of a directed set,” Can. J. Math., 25, No. 2, 233-238 (1973).

    Google Scholar 

  128. B. Mitchell, “Some applications of module theory to functor categories,” Bull. Amer. Math. Soc., 84, 867-885 (1978).

    Google Scholar 

  129. B. Mitchell, “A remark on projectives in functor categories,” J. Algebra, 69, No. 1, 24-31 (1981).

    Google Scholar 

  130. B. Mitchell, “Low dimensional group cohomology as monoidal structures,” Amer. J. Math., 105, 1049-1066 (1983).

    Google Scholar 

  131. B. Mitchell, “Separable algebroids,” Mem. Amer. Math. Soc., 57, No. 333 (1985), 96 pp.

  132. B. V. Novikov, “On partial cohomologies of semigroups,” Semigroup Forum, 28, No. 1-3, 355-364 (1984).

    Google Scholar 

  133. B. V. Novikov, “Partial cohomology of semigroups and applications,” Izv. Vuzov, Mat. No. 11 (318), 25-32 (1988).

  134. B. V. Novikov, “Commutative cancellative semigroups of dimension 1,” Mat. Zametki, 48, No. 1, 148-149 (1990).

    Google Scholar 

  135. B. V. Novikov, “On semigroups of cohomological dimension one,” Dopov. Akad. Nauk Ukr., No. 8, 6-8 (1996).

  136. B. V. Novikov, “Semigroups of cohomological dimension one,” J. Algebra, 204, No. 2, 386-393 (1998).

    Google Scholar 

  137. B. V. Novikov, “On the weak Mitchell conjecture,” Dopov. Akad. Nauk Ukr., No. 3, 26-27 (1998).

  138. B. V. Novikov, “Semigroup cohomologies: Review,” Fund. Prikl. Mat. (to appear).

  139. U. Oberst, “Basisweiterung in der Homologie kleiner Kategorien,” Math. Z., 100, 36-58 (1967).

    Google Scholar 

  140. U. Oberst, “Exactness of inverse limits,” Bull. Amer. Math. Soc., 74, No. 6, 1129-1132 (1968).

    Google Scholar 

  141. U. Oberst, “Homology of categories and exactness of direct limits,” Math. Z., 107, 87-115 (1968).

    Google Scholar 

  142. U. Oberst and H. Rohrl, “Flat and coherent functors,” J. Algebra, 14, 91-105 (1970).

    Google Scholar 

  143. B. Oliver, “Higher limits of functors on categories of elementary Abelian p-groups,” Preprint, Ser. Mat. Inst. Aarhus Univ., No. 6. 1-13 (1992).

  144. B. Oliver, “Higher limits via Steinberg representations,” Comm. Algebra, 22, No. 4, 1381-1393 (1994).

    Google Scholar 

  145. B. Osofsky, “Global dimension of valuation rings,” Trans. Amer. Math. Soc., 127, 136-149 (1967).

    Google Scholar 

  146. B. Osofsky, The subscript of ℵn, projective dimension, and the vanishing of \(\underleftarrow {lim}^{\left( n \right)} \),” Bull. Amer. Math. Soc., 80, No. 1, 8-26 (1974).

    Google Scholar 

  147. B. Osofsky, “Projective dimension of 'nice' directed unions,” J. Pure Appl. Algebra, 13, 179-219 (1978).

    Google Scholar 

  148. B. Parshall and L. Scott, “Derived categories, quasi-hereditary algebras, and algebraic groups,” Carleton Univ. Math. Notes., 3, 1-105 (1988).

    Google Scholar 

  149. T. I. Pirashvili, “Models for a homotopy theory and cohomology of small categories,” Soobsh. Akad. Nauk GSSR, 129, No. 2, 261-264.

  150. T. I. Pirashvili, “H 3 and models for a homotopy theory,” Trudy Tbilis. Mat. Inst., 94, 73-85 (1990).

    Google Scholar 

  151. T. I. Pirashvili, “Cohomology of small categories in homotopical algebra,” in: K-Theory and Homological Algebra (Lect. Notes Math.), 1437, Springer, Berlin (1990) pp. 268-302.

    Google Scholar 

  152. T. I. Pirashvili, “Category of Eilenberg-MacLane fibrations and cohomology of Grothendieck constructions,” Comm. Algebra, 21, No. 1, 309-341 (1993).

    Google Scholar 

  153. T. Pirashvili and F. Waldhausen, “MacLane homology and topological Hochschild homology,” J. Pure Appl. Algebra, 82, 81-98 (1992).

    Google Scholar 

  154. P. Polo, “On Cohen-Macaulay posets, Koszul algebras and certain modules associated to Schubert varieties,” Bull. London Math. Soc., 27, No. 5, 425-434 (1995).

    Google Scholar 

  155. T. Porter, “Categories of promodules,” Cah. Topol. Géom. Differ., 16, 295-300 (1976).

    Google Scholar 

  156. T. Porter, “Torsion theoretic results in procategories. I. Primary decomposition of bounded promodules,” Proc. Royal Irish Acad., 76A, 145-154 (1976).

    Google Scholar 

  157. T. Porter, “Torsion theoretic results in procategories. II-III,” Proc. Royal Irish Acad., 76A, 155-172 (1976).

    Google Scholar 

  158. T. Porter, “Stability of algebraic inverse systems. I: Stability, weak stability and the weakly-stable socle,” Fundam. Math., 100, 17-33 (1978).

    Google Scholar 

  159. T. Porter, “Essential properties of pro-objects in Grothendieck categories,” Cah. Topol. Géom. Differ., 20, No. 1, 3-57 (1979).

    Google Scholar 

  160. T. Porter, “Reduced powers, ultrapowers and exactness of limits,” J. Pure Appl. Algebra, 26, 325-330 (1982).

    Google Scholar 

  161. T. Porter, “The derived functors of lim and protorsion modules,” Cah. Topol. Géom. Differ., 24, 115-131 (1983).

    Google Scholar 

  162. A. V. Prasolov, “A spectral sequence for strong homology,” Glas. Mat. III, Ser. 24, 44, No. 1, 17-24 (1989).

    Google Scholar 

  163. A. V. Prasolov, “Additivity of strong homology and the continuum hypothesis,” Dokl. Akad. Nauk SSSR, 310, 799-802 (1990); English transl.: Sov. Math. Dokl., 41, No. 1, 133-135 (1990).

    Google Scholar 

  164. D. G. Quillen, “Higher algebraic K-theory,” in: Algebraic K-Theory I (Lect. Notes Math.), 341, Evanston (1972), Springer, Berlin (1973), pp. 85-147.

    Google Scholar 

  165. D. G. Quillen, “Homotopy properties of the poset of nontrivial p-subgroups of a group,” Adv. Math., 28, No. 2, 101-128 (1978).

    Google Scholar 

  166. J.-E. Roos, “Sur les foncteurs dérivés de lim. Applications,” C.R. Acad. Sci. Paris. Sér. A, 252, No. 24, 3702-3704 (1961).

    Google Scholar 

  167. J.-E. Roos, “Bidualité et structure des foncteurs dérivés de lim dans la catégorie des modules sur un anneau réguler,” C.R. Acad. Sci. Paris. Sér. A, 1962, 254, No. 9, 1556-1558 (1962).

    Google Scholar 

  168. J.-E. Roos, “Bidualité et structure des foncteurs dérivés de lim dans la catégorie des modules sur un anneau réguler,” C.R. Acad. Sci. Paris. Sér. A, 254, No. 10, 1720-1722 (1962).

    Google Scholar 

  169. J.-E. Roos, “Sur les foncteurs dérivés des produits in.nis dans les catégories de Grothendieck. Exemples et contre-exemples,” C.R. Acad. Sci. Paris. Sér. A, 263, 895-898 (1966).

    Google Scholar 

  170. J.-E. Roos, “Locally Noetherian categories and generalized strictly linearly compact rings. Applications,” in: Lect. Notes Math.,92, Springer, Berlin (1969), pp. 197-277.

    Google Scholar 

  171. H. Schubert, Kategorien I, II, Akademie-Verlag, Berlin (1970).

    Google Scholar 

  172. S. Shelah, “Infinite Abelian groups, Whitehead problem and some construction,” Israel J. Math., 18, 243-256 (1974).

    Google Scholar 

  173. E. G. Sklyarenko, “Some applications of the functor \(\underleftarrow {lim}^{\left( i \right)} \),” Mat. Sb., Nov. Ser., 123 (165), No. 3 369-390 (1984); English transl.: Math. USSR Sb., 51, 367-387 (1985).

    Google Scholar 

  174. E. G. Sklyarenko, “Hyper(co)homology for exact left covariant functors and a homology theory for topological spaces,” Usp. Mat. Nauk, 50, No. 3. 109-146 (1995); English transl.: Russ. Math. Surv., 50, No. 3, 575-611 (1995).

    Google Scholar 

  175. E. G. Sklyarenko, “Homology and cohomology. Limit passages,” Tr. Mat. Inst. Steklova, 193, 162-168 (1992); English transl.: Proc. Steklov Inst. Math., 193, 179-185 (1993).

    Google Scholar 

  176. W. T. Spears, “Global dimension in categories of diagrams,” J. Algebra, 22, 219-222 (1972).

    Google Scholar 

  177. J. R. Stallings, “On torsion free groups with infinitely many ends,” Ann. Math.(2), 88, 312-334 (1968).

    Google Scholar 

  178. B. Stenström, “Purity in functor categories,” J. Algebra, 8, 352-361 (1968).

    Google Scholar 

  179. R. G. Swan, “Groups of cohomological dimension one,” J. Algebra, 12, 585-610 (1969).

    Google Scholar 

  180. R. M. Vogt, “Homotopy limits and colimits,” Math. Z., 134, No. 1, 11-52 (1973).

    Google Scholar 

  181. C. E. Watts, “A homology theory for small categories,” in: Proc. Conf. Categorical Algebra, La Jolla (1965), pp. 331-335.

  182. D. Wigner, “Inverse limits and the completeness of quotient groups,” Mich. Math. J., 22, No. 2, 165-170 (1975).

    Google Scholar 

  183. R. Wong, “Some projective free ringoids,” Dissertation, Rutgers University. New Brunswick, NJ (1977).

    Google Scholar 

  184. R. Wong, “Free ideal monoid rings,” J. Algebra, 53, 21-35 (1978).

    Google Scholar 

  185. Z. Z. Yeh, “Higher inverse limits and homology theories,” Thesis, Princeton (1959).

  186. N. Yoneda, “On Ext and exact sequences,” J. Fac. Sci. Tokyo, Sec. I, 8, 507-576 (1960).

    Google Scholar 

Download references

Authors
  1. A. A. Husainov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Husainov, A.A. Homological dimension theory of small categories. Journal of Mathematical Sciences 110, 2273–2321 (2002). https://doi.org/10.1023/A:1014945907335

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1014945907335

Keywords

Search

Navigation