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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2 Answers
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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.
answered Apr 10, 2014 at 16:26
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$\begingroup$ 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!) $\endgroup$ Commented Apr 10, 2014 at 16:31
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$\begingroup$ Whoops. Yes, that's a ridiculous mistake, I did mean $u \cdot u^{-1} = 1$. $\endgroup$ Commented Apr 10, 2014 at 16:42
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$\begingroup$ @TylerLawson : I think it's not true that $\tau_{\leq 0}KU\wedge \tau_{\leq 0}KU$ is rational; but the example still works (if $\tau_{\leq 0}KU$ had a ring structure it would be a module over its $\pi_0$, i.e. an Eilenberg-MacLane spectrum; but its $k$-invariants are the same as those of $KU$, so it's not an EM spectrum). The reason I think it's not rational is because $KU\wedge \tau_{\leq 0}KU$ is rational (it's a homotopy colimit of extensions of $KU\wedge H\mathbb Z$ which is rational), and so it would imply that $ku\wedge \tau_{\leq 0}KU$ is also rational. (1/2) $\endgroup$ Commented Dec 3, 2020 at 16:03
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$\begingroup$ But that would imply that $ku\wedge H\mathbb Z$ is also rational, which it's not (2/2) $\endgroup$ Commented Dec 3, 2020 at 16:03
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In addition to Tyler's example, a very nice example has been provided at the end of Section 6 in the paper "Localization of algebras over coloured operads" by Casacuberta, Gutierrez, Moerdijk, and Vogt. The idea is again to truncate the unit. The authors prohibit this behavior by insisting that the localization is stable (commutes with suspension). Alternately, they prove in Theorem 6.5 that you can get away with unstable localizations if you assume some connectivity hypotheses on the ring you are localizing.
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.
This is the maximum generality in which I can answer the OP's question. In the examples section I treat separately the cases of localizations of spaces (also addressed in this mathoverflow question), spectra, equivariant spectra, and chain complexes. I characterize Bousfield localizations of chain complexes and devote particular attention to the case of CDGAs over fields of characteristic zero. I hope to have this work up on arXiv sometime soon, but I also welcome emails from people who want to see it now. Many details can be found in my research statement.
answered Apr 11, 2014 at 3:47