# Truncations of E_infinity algebras

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 $\mathbb{Q}$?

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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.

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I assume you mean $u \cdot u^{-1} = 1$ in the relation? In any case, thank you! I was indeed wondering about compatible $E_\infty$-structures, so your example is wonderful. (And I feel stupid!) –  user49450 Apr 10 at 16:31
Whoops. Yes, that's a ridiculous mistake, I did mean $u \cdot u^{-1} = 1$. –  Tyler Lawson Apr 10 at 16:42
Another excellent example was presented by Mike Hill at Oberwolfach in 2011. In this example a localization of equivariant ring spectra is given which destroys genuine $E_\infty$ structure but preserves naive $E_\infty$-structure. In my thesis I try to understand this example in maximum generality. It turns out the failure is again a failure of the localization to be stable (this time with respect to non-trivial representation spheres). The fact that the localization does preserve naive $E_\infty$ structure follows from the fact that it is stable with respect to the monoidal unit. In my thesis I describe other ways to view what is going on in this example, I introduce the notion of a monoidal Bousfield localization (these will in particular preserve the structure of algebras over $\Sigma$-cofibrant operads such as $E_\infty$), and I characterize monoidal Bousfield localizations via the following:
Theorem: Let $M$ be a cofibrantly generated, left proper, monoidal model category in which cofibrant objects are flat. Let $C$ be a class of maps such that the Bousfield localization $L_C(M)$ exists. Then $L_C(M)$ has cofibrant objects flat and satisfies the pushout product axiom if and only if $L_C$ is a monoidal Bousfield localization.