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 ; weak dimensions are shown to be 0 or . 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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Additional Information

DOI:
http://dx.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