## Algebraic & Geometric Topology

### On the algebraic classification of module spectra

Irakli Patchkoria

#### Abstract

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.

#### Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715460

Digital Object Identifier
doi:10.2140/agt.2012.12.2329

Mathematical Reviews number (MathSciNet)
MR3020210

Zentralblatt MATH identifier
1264.18017

#### Citation

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. https://projecteuclid.org/euclid.agt/1513715460

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