On the algebraic classification of module spectra

Patchkoria, Irakli¹

¹Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany

Algebraic and Geometric Topology, Tome 12 (2012) no. 4, pp. 2329-2388

Résumé

Using methods developed by Franke in [K-theory Preprint Archives 139 (1996)], we obtain algebraic classification results for modules over certain symmetric ring spectra (S-algebras). In particular, for any symmetric ring spectrum R whose graded homotopy ring π∗R has graded global homological dimension 2 and is concentrated in degrees divisible by some natural number N ≥ 4, we prove that the homotopy category of R–modules is equivalent to the derived category of the homotopy ring π∗R. This improves the Bousfield-Wolbert algebraic classification of isomorphism classes of objects of the homotopy category of R-modules. The main examples of ring spectra to which our result applies are the p–local real connective K–theory spectrum ko(p), the Johnson–Wilson spectrum E(2), and the truncated Brown–Peterson spectrum BP〈1〉, all for an odd prime p. We also show that the equivalences for all these examples are exotic in the sense that they do not come from a zigzag of Quillen equivalences.

Détail

DOI : 10.2140/agt.2012.12.2329

Classification : 18E30, 55P42, 55P43, 18G55
Keywords: algebraic classification, model category, module spectrum, symmetric ring spectrum, stable model category

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