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I have a question regarding the construction of the $p$-localization of a given spectrum $X$. I have seen in many papers people defining the dual notion of it, namely the $p$-completion of $X$, $X_p^{\wedge}:= L_{M(\mathbf{Z}/p)}X$, but never actually someone giving the precise definition/construction of the $p$-localization. So I was wondering if anyone is able to give this definition/construction of the $p$-localization associated to a spectrum $X$, or to provide a reference of it.

P.S. Regarding the reference; as it has been mentioned in the comments section, I am aware of the standard reference (Bousfield's paper "The Localization of Spectra with Respect to Homology") but I cannot understand where he defines the $p$-local spectrum construction of a spectrum $X$. Thus for a reference am looking for something more handy.

Ben Steffan
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user430191
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  • A. Bousfield, The localization of spectra with respect to homology, Topology vol. 18, 1979 – xsnl Jul 25 '18 at 18:20
  • I have checked that paper (which is the starting point of this idea); so is $X_{(p)}$ just defined the Bousfield localization with respect a prime number $p$? – user430191 Jul 25 '18 at 18:27
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    If you want some explicit construction, then you can take your favourite spectrum $M$ and take homotopy colimit of diagram looking like $M \to M \to M \dots$, where arrows are multiplication by all $p$-free natural numbers in some order. Informally, localization with respect to some good class of morphisms (e. g. giving isomorphism on some homology theory) is obtained by taking homotopy colimit by all endomorphisms in that class. (It's not true as stated, but IMO gives good sense of how it looks). – xsnl Jul 25 '18 at 18:29

1 Answers1

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This is quite explicit in Bousfield's paper, namely as Proposition 2.4, and I'd argue that it is about as handy as it gets. All important basic facts about $p$-localization are essentially summed up in the statement of that proposition. Alternatively, Tyler Lawson's "An Introduction to Bousfield Localization" might be a little friendlier as a reference, at the cost of being somewhat wordier and wider in scope. It discusses $p$-localization in section 8.

Anyway, it does not take much effort to sum up the situation: $p$-localization (or localization at $(p)$) is localization at the Moore spectrum $M\mathbb{Z}_{(p)} = \mathbb{S}_{(p)}$ where $$ \mathbb{S}_{(p)} := \operatorname{hocolim}\Bigl(\mathbb{S} \xrightarrow{\cdot p_1} \mathbb{S} \xrightarrow{\cdot p_1 p_2} \mathbb{S} \xrightarrow{\cdot p_1 p_2 p_3} \cdots\Bigr) $$ for $P = \{p_i \mid i \in \mathbb{N}_{\geq 0}\}$ an enumeration of all primes except $p$. This is a smashing localization, so for any spectrum $X$ we have $$ L_{M\mathbb{Z}_{(p)}} X \simeq X \wedge \mathbb{S}_{(p)} $$ which in turn gives $$ L_{M\mathbb{Z}_{(p)}} X \simeq \operatorname{hocolim}\Bigl(X \xrightarrow{\cdot p_1} X \xrightarrow{\cdot p_1 p_2} X \xrightarrow{\cdot p_1 p_2 p_3} \cdots\Bigr) $$ since smash products commute with homotopy colimits. Noting that $$ \mathbb{Z}_{(p)} \cong \operatorname{colim}\Bigl(\mathbb{Z} \xrightarrow{\cdot p_1} \mathbb{Z} \xrightarrow{\cdot p_1 p_2} \mathbb{Z} \xrightarrow{\cdot p_1 p_2 p_3} \cdots\Bigr) $$ and consequently that $$ G_{(p)} \cong G \otimes \mathbb{Z}_{(p)} \cong \operatorname{colim}\Bigl(G \xrightarrow{\cdot p_1} G \xrightarrow{\cdot p_1 p_2} G \xrightarrow{\cdot p_1 p_2 p_3} \cdots\Bigr) $$ for any abelian group $G$, we obtain that the homotopy groups of a $p$-local spectrum are $p$-local abelian groups after noting that homotopy groups commute with filtered colimits. In fact, the converse is also true: A spectrum $X$ is $p$-local if and only if $\pi_n X$ is $p$-local for all $n$, which is extremely useful as a criterion for detecting $p$-locality.

Ben Steffan
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