# If a t-truncation of the unit object in a stable homotopy category is a ring object up to homotopy, can it be lifted to a ring spectrum? What about the Postnikov t-truncations of the sphere spectrum?

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 arxiv.org/pdf/1012.3301v2.pdf, 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