On the algebraic classification of K-local spectra

In 1996, Jens Franke proved the equivalence of certain triangulated categories possessing an Adams spectral sequence. One particular application of this theorem is that the K(p)-local stable homotopy category at an odd prime can be described as the derived category of an abelian category. We explain this proof from a topologist’s point of view. In 1983 Bousfield published a paper about the category of E(1)-local (or, equivalently, K-local) spectra at an odd prime. There, he gave an algebraic description of isomorphism classes of E(1)-local spectra in their homotopy category via E(1)-homology and a certain “k-invariant” coming from a d2-differential in the Adams spectral sequence. However, with this setup he could only describe the morphisms up to Adams filtration. In 1996, Jens Franke constructed an abstract equivalence between certain triangulated categories possessing an Adams spectral sequence. Applying Franke’s main theorem to the special case of E(1)-local spectra, one obtains an algebraic description of the homotopy category of E(1)-local spectra also covering the morphisms. In this paper, we give a streamlined exposition of Franke’s result adapted to this special case: Theorem[Franke] There is an equivalence of categories R : D(B) −→ Ho(L1S) whereD2p−2(B) denotes the derived category of twisted cochain complexes over the abelian category B, and Ho(L1S) the homotopy category of E(1)-local spectra. This paper is organised as follows: In the first chapter, the categories playing the main role for the construction are introduced: firstly, the category of so-called twisted cochain complexes of E(1)∗E(1)comodules and secondly, a certain diagram category of spectra with a fixed diagram shape and a model structure related to the model structure of E(1)-local spectra. In the next section, a functor Q is constructed which gives an equivalence of twisted cochain complexes and the homotopy category of above diagram spectra. In the third section this equivalence Q is extended to an equivalence of the derived category of twisted cochain complexes and the homotopy category of E(1)-local spectra. Further, as section 4 1 will show, this equivalences gives an “exotic model” for E(1)-local spectra: the homotopy categories of the cochain complexes and E(1)-local spectra are equivalent as categories, yet there is no Quillen equivalence between them. We do not claim any originality, it is just the proof of Franke’s Main Uniqueness Theorem applied to Bousfield’s case with the notation adapted and some technical details filled in. My special thanks go to Stefan Schwede for his motivation and support. 1 The main ingredients 1.1 E(1)∗E(1)-comodules We begin with describing an abelian category denotedA which is equivalent to the category of E(1)∗E(1)-comodules (see [Bou85], 10.3). Bousfield describes A as follows: Let p be an odd prime and let B = B(p) denote the category of Z(p)-modules together with Adams operations ψ, k ∈ Z∗(p) satisfying the following: For each M ∈ B(p), • There is an eigenspace decomposition