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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
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.

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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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