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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asked Sep 4, 2012 at 20:03
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2$\begingroup$ 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. $\endgroup$– Fernando MuroCommented Sep 4, 2012 at 22:14
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$\begingroup$ 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. $\endgroup$– Jonathan BeardsleyCommented Sep 4, 2012 at 23:31
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1$\begingroup$ 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. $\endgroup$– Fernando MuroCommented Sep 5, 2012 at 8:46
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2$\begingroup$ 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. $\endgroup$– Greg StevensonCommented Sep 5, 2012 at 10:09
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$\begingroup$ Thanks for all the help @Greg and @Fernando. Checking out that Neeman paper now! $\endgroup$– Jonathan BeardsleyCommented Sep 5, 2012 at 18:22
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