In section 4.1 of Lurie's DAG VIII, he implies the existence of an $E\_\infty$-ring spectrum $A$ such that the coconnective truncation $\tau\_{\leq 0} (A)$ does not admit the structure of an $E\_\infty$-ring spectrum. What is an explicit example? Is it possible to give an intuitive explanation for why this fails? Also, does this phenomenon occur for other similar categories, such as commutative dg algebras over $\mathbb{Q}$?



Truncations of E_infinity algebras
In general the issue is that the natural map $A \to \tau\_{\leq 0} A$ often cannot be compatible with a ring structure, because on the level of homotopy groups or homology groups it acts as a quotient map that destroys all elements in positive degree. This is often not compatible with the multiplicative structure.


A direct example is to consider the graded ring $A = \Bbb Z[u^{\pm 1}]$ with $|u|=2$, viewed as a (commutative) differential graded algebra with zero differential. The map $A \to \tau\_{\leq 0} A$ annihilates the positive powers of $u$ but leaves the negative powers alone, and this can't arise as the homology of a map of chain complexes because it doesn't respect the relation $u \cdot u^{-1} = 0$.


Your question doesn't immediately ask for the multiplicative structure on $\tau\_{\leq 0} A$ to have anything to do with the multiplicative structure on $A$, and so we might have to work a little harder to come up with something like that. I don't have one from homological algebra, because it is hard to write down a chain complex whose homology is a ring but which admits no multiplication. The periodic $K$-theory spectrum $KU$ has $\tau\_{\leq 0} KU \wedge \tau\_{\leq 0} KU$ a rational spectrum, and this obstructs the possibility of getting a unital multiplication.


