Algebraic & Geometric Topology

On the algebraic classification of module spectra

Irakli Patchkoria

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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 N4, 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.

Article information

Algebr. Geom. Topol., Volume 12, Number 4 (2012), 2329-2388.

Received: 4 November 2011
Revised: 19 July 2012
Accepted: 19 July 2012
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18E30: Derived categories, triangulated categories 55P42: Stable homotopy theory, spectra 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.)
Secondary: 18G55: Homotopical algebra

algebraic classification model category module spectrum symmetric ring spectrum stable model category


Patchkoria, Irakli. On the algebraic classification of module spectra. Algebr. Geom. Topol. 12 (2012), no. 4, 2329--2388. doi:10.2140/agt.2012.12.2329.

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