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Let $S$ be the unit object in a monoidal stable homotopy category $SH$ (we demand that the multiplication $S\times S\to S$ is commutative and associative on the level of spectra, and not just up to weak equivalence). Let $\tau$ be a $t$-structure for $SH$ such that $S$ is $\tau$-negative and that $SH^{\tau\le 0}\times SH^{\tau\le 0}\subset SH^{\tau\le 0}$. Then for any $m\le 0$ the object $S'=S^{\tau \ge m}$ (a 'factor' of $S$) can be easily seen to be a monoidal object in $SH$. My question is: can $S'$ be lifted to a commutative ring spectrum? Which restrictions on $SH,S,t$ are needed to do this? Are there any other ways of 'rigidifying' $S'$ such that one can still consider a certain triangulated category of $S'$-modules?

Basically I am interested in 'motivic' stable homotopy categories ($SH$, $MGL$-modules, and 'big Voevodsky's motives'), yet the easiest examples of my setting are the Postnikov $t$-truncations of the sphere spectrum in the 'usual' ('topological') $SH$ (for $m=0$ this is the Eilenberg-Maclane spectrum for $\mathbb{Z}$). Also, what can be said about the complex cobordism spectrum and modules over it?

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If I understand the question correctly, it is answered quite generally in, Theorem 5.16 (where $l_i$ means the $i$th truncation you're considering). This applies not only to $t$-structures but also to more general filtration such as the motivic slice filtration. – Marc Hoyois Nov 16 '12 at 6:25
There was a typo in the title, so I fixed it – David White Nov 16 '12 at 14:49
Dear Marc, thank you very much! This seems to be a very useful reference! I will try to understand it completely (in particular, I wonder whether I can always assume $S$ to be cofibrant). – Mikhail Bondarko Nov 16 '12 at 20:10
I think you can always take cofibrant replacements within the category of $A_\infty$ or $E_\infty$ objects (cf. beginning of section 5 in the paper), so the cofibrancy assumption can be safely ignored. – Marc Hoyois Nov 18 '12 at 19:43
up vote 3 down vote accepted

In the usual stable homotopy category of spectra, the Postnikov truncations of any connective commutative ring are again commutative rings, and the entire Postnikov tower can be enriched to maps of commutative rings. It follows immediately that the same result (and in particular an affirmative answer to your question) holds in the category of modules over any connective commutative ring. This is a theorem of Kriz, written up as Theorem 8.1 in Basterra's André–Quillen cohomology of commutative S-algebras. Basically, the argument is to show you can add one higher homotopy group via an extension of commutative rings by making a computation in Andre-Quillen cohomology.

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Thank you! I will try to read this. – Mikhail Bondarko Nov 16 '12 at 20:11

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