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On the algebraic classification of module spectra

Irakli Patchkoria

Algebraic & Geometric Topology 12 (2012) 2329–2388

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 πR 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 πR. 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 BP1, 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.

algebraic classification, model category, module spectrum, symmetric ring spectrum, stable model category
Mathematical Subject Classification 2010
Primary: 18E30, 55P42, 55P43
Secondary: 18G55
Received: 4 November 2011
Revised: 19 July 2012
Accepted: 19 July 2012
Published: 9 January 2013
Irakli Patchkoria
Mathematisches Institut
Universität Bonn
Endenicher Allee 60
53115 Bonn