Bousfield localisation of bounded below spectra

The goal of this short post is to extract the following result from Bousfield’s paper [Bou79]. This result is well known, but not stated explicitly as such in Bousfield’s paper. However, the condition on $E$ is sometimes overlooked, but is crucial. (In fact, this subtlety prompted the writing of this note — see the warning below.)

Theorem 1. Let $X$ be a bounded below spectrum, and let $E$ be a bounded below homotopy associative ring spectrum (or more generally, a bounded below spectrum for which $\pi_0 E$ admits a ring structure and $\pi_n E$ admits a module structure over $\pi_0 E$ for every $n$). Then we have an equivalence \[ L_E X \simeq L_{\mathrm{H}(\pi_0 E)}X. \]

Warning 2. If $E$ is not a ring (in the above weak sense), then this result can completely fail. This was pointed out to me by Maite Carli. For example, take $E = \mathbf{F}_2 \oplus \Sigma \mathbf{F}_3$.

Recall the following terminology from [Bou79, Section 2].

Definition 3. Let $G$ and $H$ be two abelian groups. We say that $G$ and $H$ have the same type of acyclicity if

  1. the group $G$ is a torsion group (i.e., consists only of elements of finite order) if and only if $H$ is a torsion group,
  2. for every prime $p$, the group $G$ is uniquely $p$-divisible (in other words, $G$ is a $\mathbf{Z}[\tfrac{1}{p}]$-module) if and only if $H$ is uniquely $p$-divisible.

The hard computation that Bousfield does is the following.

Theorem 4. Let $X$ and $E$ be bounded below spectra. If $G$ is an abelian group with the same type of acyclicity as $\bigoplus_n \pi_n E$, then \[ L_E X \simeq L_{\mathbf{S}G} X \simeq L_{\mathrm{H}G} X, \] where $\mathbf{S}G$ denotes the $G$-Moore spectrum.

Proof. Bousfield states the first equivalence in [Bou79, Theorem 3.1], and the second equivalence is stated in its proof. (Note that he uses the term ‘connective’ to mean ‘bounded below’.) $\hspace{1em}\blacksquare$

The main result follows from the following observation.

Lemma 5. Let $A$ be a ring, and let $M$ be an $A$-module. Then $A$ has the same type of acyclicity as $A \oplus M$.

Proof. Clearly if $A\oplus M$ is torsion (respectively uniquely $p$-divisible), then the same is true for $A$. Conversely, if $A$ is torsion, then so is any $A$-module: if $1\in A$ is torsion, this means there is an $n \in \mathbf{Z}$ such that $n\cdot 1 = 0$ in $A$, which implies that $nM = 0$ for any $A$-module $M$. Similarly, if $A$ is uniquely $p$-divisible, then $A$ is a $\mathbf{Z}[\tfrac{1}{p}]$-module, and therefore so is any $A$-module. $\hspace{1em}\blacksquare$

Corollary 6. Let $E$ be a spectrum such that $\pi_0 E$ admits the structure of a ring and for which $\pi_n E$ admits the structure of a $\pi_0 E$-module for every $n$. Then $\bigoplus_n \pi_n E$ has the same type of acyclicity as $\pi_0 E$.

Theorem 1 now follows immediately from Theorem 4 and Corollary 6.

Remark 7. The flexibility afforded by Theorem 4 is more than just reducing to $\pi_0 E$-localisation. Indeed, we can pick an abelian group $G$ with the same type of acyclicity as $\pi_0 E$ and localise with respect to $\mathrm{H}G$. This choice can be made particularly simply: let $J$ denote the set of primes $p$ for which $\pi_0 E$ is not uniquely $p$-divisible. If $\pi_0E$ is torsion, then we can pick $G = \bigoplus_{p\in J} \mathbf{Z}/p$; if $\pi_0E$ is not torsion, then $G = \mathbf{Z}_{(J)}$ does the job. Bousfield [Bou79, Section 2] discusses how to compute localisations of this type. In fact, we can apply the same procedure to simplify $\bigoplus_n \pi_n E$ without any assumptions on $E$.

Remark 8. As Bousfield remarks after the proof of [Bou79, Theorem 3.1], one can generalise these results to the case where $X$ that is $\mathrm{H}\mathbf{Z}$-local. Any bounded below spectrum is $\mathrm{H}\mathbf{Z}$-local. Indeed, it is the limit of its Postnikov tower. Since $X$ is bounded below, the bottom stage is an Eilenberg-MacLane spectrum, and all other associated graded spectra are also Eilenberg-MacLane. Local spectra are closed under limits, so it suffices to show that Eilenberg-MacLane spectra are local. But Eilenberg-MacLane spectra are modules over $\mathrm{H}\mathbf{Z}$, and are therefore in particular $\mathrm{H}\mathbf{Z}$-local.

References

[Bou79] A.K. Bousfield. The localization of spectra with respect to homology. 1979. doi.