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Is it known what conditions we require of a stable homotopy category to have $\langle S\rangle = \coprod\limits_{\mathbb{N}}\langle K(n)\rangle$, where $\langle K(n)\rangle$ is some minimal Bousfield class? That is, more specifically, in which cases can we say that Proposition 3.7.5 of Hovey, Palmieri and Strickland's "Axiomatic Stable Homotopy Theory" applies?

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I think that computing the Bousfield lattice of an arbitrary stable homotopy category is a huge problem, so you can probably find explicit examples satisfying your conditions, e.g. among derived categories of commutative noetherian rings, but general conditions... that looks like very complicated. –  Fernando Muro Sep 4 '12 at 22:14
    
Yeah. I'm especially wondering if this is the case for the derived category of a Noetherian ring. I'll have to do some more reading on it. –  Jon Beardsley Sep 4 '12 at 23:31
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Probably you know MR1174255 Neeman, Amnon The chromatic tower for D(R). With an appendix by Marcel Bökstedt. Topology 31 (1992), no. 3, 519–532. –  Fernando Muro Sep 5 '12 at 8:46
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It is true for the unbounded derived category of a noetherian ring: the minimal Bousfield classes (which are not all of $D(R)$) are $\langle k(\mathfrak{p})\rangle$ and $$0 = \langle \coprod_{\mathfrak{p}} k(\mathfrak{p}) \rangle$$ as tensoring with the residue fields detects whether an object is non-zero. I am fairly sure it is not known what conditions would suffice in general; as Fernando points out this is probably a difficult issue. –  Greg Stevenson Sep 5 '12 at 10:09
    
Thanks for all the help @Greg and @Fernando. Checking out that Neeman paper now! –  Jon Beardsley Sep 5 '12 at 18:22

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