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Let $Sp$ be the category of spectra. Let $L : Sp \to Sp_L$ be the localization functor onto a reflective subcategory.

Question 1: Is it ever the case that $L(S^0)$ is not bounded below?

Question 2: What if we assume that $L$ is smashing?

Notes:

  • Basic examples like $p$-localization do not decrease connectivity.

  • Even $L_{K(n)}S^0$ is bounded below, despite $K(n)$ being periodic. I think the connectivity is related to the dimension of the Morava stabilizer group, but I don’t understand this very well.

  • I believe every finite localization has $L S^0$ bounded below, by brute force enumeration. Is there a more conceptual reason?

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    $\begingroup$ I'm pretty sure $L_{K(1)}(\mathbb{S})$ cannot be bounded below. If it were, $L_{K(1)}(\mathbb{S}/p)$ would be bounded below as well, but this one arises from $\mathbb{S}/p$ by inverting a $v_1$ self-map, hence is periodic and not bounded below. $\endgroup$ Apr 15 at 6:25
  • $\begingroup$ You probably don't want to allow multiple primes. If $L_n$ is bounded below but not uniformly bounded below, then you can combine infinitely many primes with $n_p$ going to infinity. That's smashing, right? $\endgroup$ Apr 15 at 16:22
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    $\begingroup$ The localisations of the sphere with respect to $E(n)$ or $K(n)$ are not bounded below. The analogous telescopic localisations are not bounded below either. Achim's approach is one way to see this, but there are also arguments based on the ordinary homology of the localised spheres. $\endgroup$ Apr 15 at 16:38

1 Answer 1

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EDIT: I'll leave the old answer up, see below. In the meantime, Maxime Ramzi and I have thought this through and came up with a fun general argument.

Claim. Let $L: \mathrm{Sp}\to \mathrm{Sp}$ be a Bousfield localisation (not necessarily smashing!) which takes bounded below spectra to bounded below spectra. Then it is given (for bounded below $X$) by one of the following:

  1. $LX = X[P^{-1}]$ for $P$ a set of primes.
  2. $LX = \prod_{p\notin P} X^\wedge_p$ for $P$ a set of primes.

To see this, first observe that $L$ takes $\mathbb{Z}$-modules to $\mathbb{Z}$-modules. This is because $\mathrm{Mod}_{\mathrm{Sp}}(\mathbb{Z})$ admits a description as Lawvere theory, namely as finite-product preserving functors $\mathrm{Fun}^{\Pi}(\mathrm{Latt}^{\mathrm{op}},\mathrm{Sp})$, which is functorial in exact functors.

Let $C$ denote the functor given by the cofiber $X\to LX \to CX$. Then the above shows that for an abelian group $A$, $LA$ and $CA$ are both generalized Eilenberg-MacLane. So they split as $\bigoplus \pi_nLA[n]$ and $\bigoplus \pi_nCA[n]$. So $\pi_n CA$ is $L$-acyclic and $\pi_nLA$ $L$-local. By the long exact sequence, for $n\notin\{0,1\}$, we have $\pi_nCA = \pi_nLA$, so they must be $0$. It follows directly that $LA$ and $CA$ are $0$-connective and $1$-truncated.

By Postnikov induction, this implies that for any $a$-connective and $b$-truncated $X$, $LX$ is $a$-connective and $b+1$-truncated.

We also have that there exists $n$ such that $LX$ is $-n$-connective for all $X$, where $n$ does not depend on $X$. Otherwise we would find a sequence of connective $X_n$ such that $LX_n$ is not $-n$-connective, and then $L(\bigoplus X_n)$ could not be bounded below at all, since it admits all $LX_n$ as retracts.

The sequence $\tau_{\geq n} X \to X \to \tau_{\leq n} X$ now combines with the two statements above to prove that $LX$ is connective for connective $X$. That is: A bounded below localisation automatically preserves connectivity.

Next, call an abelian group $L$-acyclic if $LA=0$, and $L$-local if $LA=A$. If all homotopy groups of $X$ are $L$-acyclic, $X$ is $L$-acyclic, by Postnikov induction, and analogously for $L$-locality. We prove the converse. Assume $X$ is $L$-acyclic, then in the sequence $$ L\tau_{\geq n} X \to 0 \to L\tau_{\leq n-1}X $$ the left term is $n$-connective and the right term is $n$-truncated, so both have to be $0$. This shows that all truncations of bounded below $L$-acyclic spectra are $L$-acyclic. The same argument using the functor $C$ proves the locality statement.

A localisation $L$ preserving bounded-below spectra is thus classified by the $L$-acyclic abelian groups. These form a localising subcategory of $\mathcal{D}(\mathbb{Z})$, which are classified by their support and generated by $\mathbb{F}_p$ for a set of primes $p\in P$, and possibly $\mathbb{Q}$. All possible combinations are given by the two alternatives in the claim.

Old answer: As discussed in the comments, the opposite of your premise seems to be true: Basically all interesting localisations seem to take non-bounded below values. To substantiate this, let's classify the smashing localisations which take the sphere to a bounded-below object and see that they are exactly given by the ordinary localisations and $0$.

We have $L\mathbb{S} \otimes L\mathbb{S} \simeq L\mathbb{S}$. So the base-change $LK = L\mathbb{S}\otimes K$ for $K$ any ordinary field (we'll use $\mathbb{Q}$ and $\mathbb{F}_p$) satisfies $LK \otimes_K LK \simeq LK$. As derived modules over a field always split as a sum of shifts of $K$, we either have that $LK$ is zero, or that base-change along $K\to LK$ is conservative. Combining with the fact $LK\otimes_K LK$, we learn that for any field, the unit map $K\to LK$ is an equivalence, or $LK\simeq 0$.

Now consider $\mathbb{Z}\to L\mathbb{Z}$. As we saw above, for any $p$ we either have $L\mathbb{F}_p=0$ or that $\mathbb{F}_p\simeq L\mathbb{F}_p$ is an equivalence. In the latter case, we learn that $L\mathbb{Z}\to L\mathbb{F}_p$ is surjective on $\pi_0$, so multiplication by $p$ on $\pi_{-1} L\mathbb{Z}$ is injective. If $L\mathbb{F}_p=0$, it is even bijective. Since $L\mathbb{Q}$ is either $0$ or $\mathbb{Q}$, and we may write it as colimit of $L\mathbb{Z}$ along multiplication with all integers, we learn that $\pi_{-1}L\mathbb{Z}=0$. So $L\mathbb{Z}$ is connective. The fact that $L\mathbb{F}_p=0$ or $\mathbb{F}_p$ also tells us that $\pi_0L\mathbb{Z}$ is $p$-torsion free. So if $L\mathbb{Z}\neq 0$, $L\mathbb{Q}$ has to be $\mathbb{Q}$ and cannot be $0$.

Let $P$ be the set of primes for which $L\mathbb{F}_p=0$, the map $\mathbb{Z}\to L\mathbb{Z}$ factors through a map $\mathbb{Z}[P^{-1}]\to L\mathbb{Z}$. Unless $L\mathbb{Z}=0$, this map is now an equivalence after base-change to any $\mathbb{F}_p$ or $\mathbb{Q}$, so by Hurewicz, it is an equivalence.

So far we have not used the assumption that $L\mathbb{S}$ is bounded below at all, we have just proved that all smashing localisations over $\mathbb{Z}$ are either $0$ or ordinary localisations. This comes in now, when relating $\mathbb{Z}$ and $\mathbb{S}$:

Recall that for bounded-below spectra, being $\mathbb{F}_p$-acyclic is the same as being trivial mod $p$ (by Hurewicz), so for any prime $p$ with $L\mathbb{F}_p=0$, $L\mathbb{S}/p=0$ as well. The map $\mathbb{S}\to L\mathbb{S}$ thus factors through $\mathbb{S}[P^{-1}]\to L\mathbb{S}$. This is now a map of bounded below spectra which is an equivalence after tensoring with $\mathbb{Z}$, if $L\mathbb{Z}\neq 0$. So $L\mathbb{S}\simeq \mathbb{S}[P^{-1}]$. If $L\mathbb{Z}=0$, we instead directly observe that $L\mathbb{S}\otimes \mathbb{Z} \simeq 0$, so $L\mathbb{S}=0$ by Hurewicz.

To summarize: Smashing localisations with $L\mathbb{S}$ bounded below are either ordinary localisations or trivial.

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  • $\begingroup$ Thanks. It’s embarrassing — even after you explaining all of this to me, I still can’t figure out which of the summands described in Thm 7.6 and 7.7 here are unbounded below. Probably one of the ones with an ``$\infty$”… $\endgroup$
    – Tim Campion
    Apr 16 at 0:38
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    $\begingroup$ This kept me up at night, but I finally have a full classification of bounded-below spectra preserving localisations. The argument is kind of fun, so I edited the more general statement in! $\endgroup$ Apr 16 at 20:56

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