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?

Question

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?

Answer 1

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.