Homological dimensions of stable homotopy modules and their geometric characterizations

Lin, T. Y.

doi:10.1090/S0002-9947-1972-0380789-8

Homological dimensions of stable homotopy modules and their geometric characterizations
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Trans. Amer. Math. Soc. 172 (1972), 473-490 Request permission

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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