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
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