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The localization of a space (really: homotopy type, ∞-groupoid) or spectrum with respect to some prime numbers is a homotopical analogue of the notion of localization of a commutative ring (or rather of localization of a module), specifically the ring $\mathbb{Z}$ of integers, with respect to some prime numbers. In good cases, at least, this localization acts on the homotopy and/or homology groups as algebraic localization. (For spaces, there are multiple inequivalent notions of localization, although all agree on nilpotent spaces.)
Localization in this sense is closely related to Bousfield localization. The localization of spectra is a Bousfield localization of spectra, while one of the constructions of localization of spaces is a Bousfield localization of model categories. The present notion of localization should not be confused with the completion of a space, which is a different sort of Bousfield localization.
For all of this page,
let $T$ be a set of prime numbers,
write $\neg T$ for the set of primes not in $T$.
Write $\mathbb{Z}_T$ for the ring of integers localized by inverting all primes in $\neg T$, i.e. the subring of $\mathbb{Q}$ whose denominators are products of primes in $\neg T$.
The most important cases are:
$T = \{p\}$ for a prime $p$. In this case, $T$-localization will be localization at $p$ or $p$-localization.
$T = \neg \{p\}$, the set of all primes except $p$. In this case, $T$-localization will be localization away from $p$.
$T = \emptyset$. In this case, $T$-localization will be rationalization.
In these cases:
$\mathbb{Z}_\emptyset = \mathbb{Q}$ is the rational numbers;
$\mathbb{Z}_{\neg\{p\}} = \mathbb{Z}[\frac{1}{p}]$;
$\mathbb{Z}_{\{p\}} = \mathbb{Z}_{(p)}$ is the integers localized at the prime ideal $(p)$.
The analogous theory of the completion of a space involves the cyclic groups $\mathbb{Z}/p\mathbb{Z}$ – written $\mathbb{F}_p$ when regarded as a finite field – and/or the p-adic integers $\mathbb{Z}_p$ instead of $\mathbb{Z}_T$ (e.g. Lurie “Proper morphisms”, section 4).
For the relation of that to completion see remark below.
Hence beware the subtle but crucial difference in what a subscript means, depending on which symbol is being subscribed and whether there are parenthesis or not:
$\mathbb{Z}_{(p)}$: inverting all primes except $p$;
$\mathbb{F}_p$: quotient by $p$;
$\mathbb{Z}_p$: formal completion at $p$.
A group $G$ is said to be $T$-local if the $p^{th}$ power map $G\to G$ is a bijection for all $p\in \neg T$.
If $G$ is abelian, then this map is a group homomorphism and is generally written additively as multiplication by $p$. In this case the following are equivalent:
$G$ is $T$-local;
$G$ admits a structure of $\mathbb{Z}_T$-module (necessarily unique);
The tensor product $G\otimes \mathbb{Z}/p\mathbb{Z}$ with the cyclic group of order $p$ is equal to zero for all $p\in\neg T$.
The second characterization in prop. implies that $T$-local abelian groups are reflective in Ab: the reflection is the extension of scalars functor $(\mathbb{Z}_T \otimes -)$. In fact, $T$-local nonabelian groups are also reflective in Grp, but the construction is less pretty.
A spectrum $X$ is called $T$-local (or $\mathbb{Z}_T$-local, if there is potential for confusion) if its homotopy groups are $T$-local abelian groups, def. .
The $T$-localization of a spectrum is its reflection into $T$-local spectra.
The $T$-localization, def. , may be constructed as the Bousfield localization of spectra with respect to the Moore spectrum $S(\mathbb{Z}_T)$ (e.g. Bauer 11, Example 1.7).
It can also be constructed as a Bousfield localization of model categories where we invert the maps that induce an isomorphism on generalized homology with coefficients in $H \mathbb{Z}_T$; these are called $\mathbb{Z}_T$-homology isomorphisms. See at homology localization.
The presence of the nonabelian group $\pi_1$ makes the theory of localization of unstable spaces more subtle than that of spectra. If spaces are simply connected, of course, then this is not a problem. More generally, it suffices to consider simple spaces: those where $\pi_1$ is abelian and acts trivially on the higher homotopy groups. Even more generally, it suffices to consider nilpotent spaces, whose definition is more complicated; the reader is encouraged to think about simple or even simply connected spaces.
For a nilpotent space $Z$, the following conditions are equivalent. When they hold, we say that $Z$ is $T$-local. (See May-Ponto, Theorem 6.1.1.)
Whenever $X\to Y$ induces an isomorphism on homology with $\mathbb{Z}_T$-coefficients, the induced map $[Y,Z] \to [X,Z]$ is a bijection.
Each homotopy group $\pi_n(Z)$ is a $T$-local group.
Each homology group $H_n(Z,\mathbb{Z})$ is a $T$-local group.
For a general nilpotent space $X$, the following properties of a map $\phi:X\to Y$, with $Y$ nilpotent and $T$-local, are equivalent. Such a map exists and is unique up to homotopy, and we call it the $T$-localization of $X$ at $T$. (See May-Ponto, Theorem 6.1.2.)
The induced map $[Y,Z] \to [X,Z]$ is a bijection for all $T$-local nilpotent spaces $Z$.
$\phi$ induces an isomorphism on homology with $\mathbb{Z}_T$-coefficients.
The induced map $\pi_n(X) \to \pi_n(Y)$ is an algebraic $T$-localization for all $n$.
The induced map $H_n(X,\mathbb{Z}) \to H_n(Y,\mathbb{Z})$ is an algebraic $T$-localization for all $n$.
For non-nilpotent spaces, there are multiple inequivalent definitions of localization (see May-Ponto, Remark 19.3.11). For instance:
One can do a Bousfield localization of model categories with respect to the class of $\mathbb{Z}_T$-homology isomorphisms. In this case, the “$T$-local spaces” are those $Z$ such that $[Y,Z] \to [X,Z]$ is an isomorphism for all $\mathbb{Z}_T$-homology isomorphisms $X\to Y$ (the first characterization of $T$-local nilpotent spaces above).
One can construct a totalization of the cosimplicial object induced by the “free simplicial $\mathbb{Z}_T$-module” monad. This construction is due to Bousfield-Kan.
One can do a Bousfield localization of model categories with respect to the maps $p:S^1\to S^1$ for all $p\in \neg T$. This construction is due to Casacuberta-Peschke. In this case, the “$T$-local spaces” are those such that $\pi_1(X)$ is a $T$-local group and acts “$T$-locally” on each $\pi_n(X)$.
One can of course state the other two characterizations of “$T$-local space” from the nilpotent case even in the non-nilpotent case, as is done by Sullivan, but these characterizations are no longer equivalent to the first one, and it is not clear whether corresponding “localizations” exist.
(relation to completion)
A $\neg\{p\}$-local spectrum is also called $\mathbb{Z}/p\mathbb{Z}$-acyclic. According to the general theory of Bousfield localization of spectra, they are “dual” to the “$\mathbb{Z}/p\mathbb{Z}$-local spectra”, in the sense that $X$ is $\mathbb{Z}/p\mathbb{Z}$-local if every map $Y \to X$ out of a $\mathbb{Z}/p\mathbb{Z}$-acyclic $Y$ is null homotopic.
$\mathbb{Z}/p\mathbb{Z}$-local spectra are also known as $p$-complete spectra, and are the Bousfield localization of spectra at the Moore spectrum $S \mathbb{Z}/p\mathbb{Z}$ (e.g. Bauer 11, Example 1.7).
This may be regarded as a consequence of the mod p Whitehead theorem.
See (May-Ponto, example 19.2.3, Lurie, example 8,Lurie “Proper morphisms”, section 4).
In summary:
localization of spaces/spectra at $E$
$E$ | $E$-acyclic | $E$-local |
---|---|---|
$S \mathbb{Z}_{(p)}$ | $p$-local | |
$S \mathbb{F}_p$ | $\not\{p\}$-local | $p$-complete |
In terms of arithmetic geometry this may be understood as follows:
$\mathbb{F}_p = \mathbb{Z}/(p \mathbb{Z})$ is the ring of functions exactly on the point $(p)\in$ Spec(Z)
$\mathbb{Z}_p$ is the functions on the formal neighbourhood of $(p)$.
$\mathbb{Z}_{(p)}$ is the ring of functions defined on the open complement of the point $(p)$, hence on an open neighbourhood.
So the localization at the Moore spectrum of one of these rings localizes to the formal neighbourhood of their support. For $\mathbb{F}_p$ the support is the closed point, and so the localization there is infinitesimally bigger than that, as given by the $p$-adic completion. On the other hand the support of $\mathbb{Z}_{(p)}$ is open and hence contains its own formal neighbourhood, so localizing here just gives the plain localization.
Many of the basic constructions and theorems in chromatic homotopy theory apply to finite $p$-local spectra, such as
Dennis Sullivan, Localization, Periodicity and Galois Symmetry (The 1970 MIT notes) edited by Andrew Ranicki, K-Monographs in Mathematics, Dordrecht: Springe (pdf)
Aldridge Bousfield, Daniel Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol 304, Springer 1972
Peter May, Kate Ponto, More concise algebraic topology: Localization, completion, and model categories (pdf)
Carles Casacuberta and Georg Peschke, Localizing with respect to self maps of the circle (pdf).
Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 20 Bousfield localization (pdf)
Tilman Bauer, Bousfield localization and the Hasse square (pdf)
Jacob Lurie, section 4 of Proper Morphisms, Completions, and the Grothendieck Existence Theorem
See also
and see at
Formalization in homotopy type theory: