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Homological dimensions of stable homotopy modules and their geometric characterizations


Author: T. Y. Lin
Journal: Trans. Amer. Math. Soc. 172 (1972), 473-490
MSC: Primary 55E45
DOI: https://doi.org/10.1090/S0002-9947-1972-0380789-8
MathSciNet review: 0380789
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Abstract: Projective dimensions of modules over the stable homotopy ring are shown to be either 0, 1 or $ \infty $; weak dimensions are shown to be 0 or $ \infty $. Also geometric charactetizations are obtained for projective dimensions 0, 1 and weak dimension 0. The geometric characterizations are interesting; for projective modules they are about the cohomology of geometric realization; while for flat modules they are about homology. This shows that the algebraic duality between ``projective'' and ``flat'' is strongly connected with the topological duality between ``cohomology'' and ``homology". Finally, all the homological numerical invariants of the stable homotopy ring--the so-called finitistic dimensions--are completely computed except the one on injective dimension.


References [Enhancements On Off] (What's this?)

  • [1] H. Bass, Big projective modules are free. Illinois J. Math. 7 (1963), 24--31. MR 26 #1341. MR 0143789 (26:1341)
  • [2] -, Finitistic dimension and homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466-488. MR 28 #1212. MR 0157984 (28:1212)
  • [3] J. Boardman, Stable homotopy theory, University of Warwick, 1965. (mimeograph notes).
  • [4] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N. J., 1956. MR 17, 1040. MR 0077480 (17:1040e)
  • [5] J. M. Cohen, The decomposition of stable homotopy, Ann. of Math. (2) 87 (1968), 305-320. MR 37 #6932. MR 0231377 (37:6932)
  • [6] P. Hilton, Homotopy theory and duality, Gordon and Breach, New York, 1965. MR 33 #6624. MR 0198466 (33:6624)
  • [7] T. Y. Lin, Homological algebra of stable homotopy ring $ {\pi _ \ast }$ of spheres, Pacific J. Math. 38 (1971), 117-143. MR 0307233 (46:6353)
  • [8] -, Cohomological characterization of stable spheres (mimeograph notes).
  • [9] -, Inadequacy of ordinary homology theory, Proc. Amer. Math. Soc. 31 (1972), 617-619. MR 0307241 (46:6361)
  • [10] -, Stable homotopy rings, localization and $ Z$-graded modules, Purdue University, West Lafayette, Ind. (mimeograph notes).
  • [11] J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211-264. MR 30 #4259. MR 0174052 (30:4259)
  • [12] F. P. Peterson, Functional cohomology operations, Trans. Amer. Math. Soc. 86 (1957), 197-211. MR 21 #4417. MR 0105679 (21:4417)
  • [13] G. W. Whitehead, Generalized homology theories, Trans. Amer. Math. Soc. 102 (1962), 227-283. MR 25 #573. MR 0137117 (25:573)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0380789-8
Keywords: Stable homotopy ring, stable homotopy module, finitistic dimensions, projective dimension, weak dimension, injective dimension, higher order homology operation, cohomology operation, Eilenberg-Mac Lane spectrum, Postnikov system, projective module, flat module, injective module, Puppe sequence, mapping cone sequence, spectral sequence, Hurewicz homomorphism
Article copyright: © Copyright 1972 American Mathematical Society

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