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localization of a space

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Localization of a space

Idea

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.

Preliminaries

Sets of primes

For all of this page,

  • let TT be a set of prime numbers,

  • write ¬T\neg T for the set of primes not in TT.

Definition

Write T\mathbb{Z}_T for the ring of integers localized by inverting all primes in ¬T\neg T, i.e. the subring of \mathbb{Q} whose denominators are products of primes in ¬T\neg T.

Example

The most important cases are:

  • T={p}T = \{p\} for a prime pp. In this case, TT-localization will be localization at pp or pp-localization.

  • T=¬{p}T = \neg \{p\}, the set of all primes except pp. In this case, TT-localization will be localization away from pp.

  • T=T = \emptyset. In this case, TT-localization will be rationalization.

In these cases:

  • =\mathbb{Z}_\emptyset = \mathbb{Q} is the rational numbers;

  • ¬{p}=[1p]\mathbb{Z}_{\neg\{p\}} = \mathbb{Z}[\frac{1}{p}];

  • {p}= (p)\mathbb{Z}_{\{p\}} = \mathbb{Z}_{(p)} is the integers localized at the prime ideal (p)(p).

Remark

The analogous theory of the completion of a space involves the cyclic groups /p\mathbb{Z}/p\mathbb{Z} – written 𝔽 p\mathbb{F}_p when regarded as a finite field – and/or the p-adic integers p\mathbb{Z}_p instead of T\mathbb{Z}_T (e.g. Lurie “Proper morphisms”, section 4).

For the relation of that to completion see remark 3 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:

  • (p)\mathbb{Z}_{(p)}: inverting all primes except pp;

  • 𝔽 p\mathbb{F}_p: quotient by pp;

  • p\mathbb{Z}_p: formal completion at pp.

Local groups

Definition

A group GG is said to be TT-local if the p thp^{th} power map GGG\to G is a bijection for all p¬Tp\in \neg T.

Proposition

If GG is abelian, then this map is a group homomorphism and is generally written additively as multiplication by pp. In this case the following are equivalent:

  • GG is TT-local;

  • GG admits a structure of T\mathbb{Z}_T-module (necessarily unique);

  • The tensor product G/pG\otimes \mathbb{Z}/p\mathbb{Z} with the cyclic group of order pp is equal to zero for all p¬Tp\in\neg T.

Remark

The second characterization in prop. 1 implies that TT-local abelian groups are reflective in Ab: the reflection is the extension of scalars functor ( T)(\mathbb{Z}_T \otimes -). In fact, TT-local nonabelian groups are also reflective in Grp, but the construction is less pretty.

Definitions

Localization of spectra

Definition

A spectrum XX is called TT-local (or T\mathbb{Z}_T-local, if there is potential for confusion) if its homotopy groups are TT-local abelian groups, def. 2.

The TT-localization of a spectrum is its reflection into TT-local spectra.

Remark

The TT-localization, def. 3, may be constructed as the Bousfield localization of spectra with respect to the Moore spectrum S( T)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 TH \mathbb{Z}_T; these are called T\mathbb{Z}_T-homology isomorphisms. See at homology localization.

Localization of nilpotent spaces

The presence of the nonabelian group π 1\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 π 1\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 ZZ, the following conditions are equivalent. When they hold, we say that ZZ is TT-local. (See May-Ponto, Theorem 6.1.1.)

  1. Whenever XYX\to Y induces an isomorphism on homology with T\mathbb{Z}_T-coefficients, the induced map [Y,Z][X,Z][Y,Z] \to [X,Z] is a bijection.

  2. Each homotopy group π n(Z)\pi_n(Z) is a TT-local group.

  3. Each homology group H n(Z,)H_n(Z,\mathbb{Z}) is a TT-local group.

For a general nilpotent space XX, the following properties of a map ϕ:XY\phi:X\to Y, with YY nilpotent and TT-local, are equivalent. Such a map exists and is unique up to homotopy, and we call it the TT-localization of XX at TT. (See May-Ponto, Theorem 6.1.2.)

  1. The induced map [Y,Z][X,Z][Y,Z] \to [X,Z] is a bijection for all TT-local nilpotent spaces ZZ.

  2. ϕ\phi induces an isomorphism on homology with T\mathbb{Z}_T-coefficients.

  3. The induced map π n(X)π n(Y)\pi_n(X) \to \pi_n(Y) is an algebraic TT-localization for all nn.

  4. The induced map H n(X,)H n(Y,)H_n(X,\mathbb{Z}) \to H_n(Y,\mathbb{Z}) is an algebraic TT-localization for all nn.

Localization of non-nilpotent spaces

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 T\mathbb{Z}_T-homology isomorphisms. In this case, the “TT-local spaces” are those ZZ such that [Y,Z][X,Z][Y,Z] \to [X,Z] is an isomorphism for all T\mathbb{Z}_T-homology isomorphisms XYX\to Y (the first characterization of TT-local nilpotent spaces above).

  • One can construct a totalization of the cosimplicial object induced by the “free simplicial T\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 1S 1p:S^1\to S^1 for all p¬Tp\in \neg T. This construction is due to Casacuberta-Peschke. In this case, the “TT-local spaces” are those such that π 1(X)\pi_1(X) is a TT-local group and acts “TT-locally” on each π n(X)\pi_n(X).

One can of course state the other two characterizations of “TT-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.

Properties

Relation to formal completion

Remark

(relation to completion)

A ¬{p}\neg\{p\}-local spectrum is also called /p\mathbb{Z}/p\mathbb{Z}-acyclic. According to the general theory of Bousfield localization of spectra, they are “dual” to the “/p\mathbb{Z}/p\mathbb{Z}-local spectra”, in the sense that XX is /p\mathbb{Z}/p\mathbb{Z}-local if every map YXY \to X out of a /p\mathbb{Z}/p\mathbb{Z}-acyclic YY is null homotopic.

/p\mathbb{Z}/p\mathbb{Z}-local spectra are also known as pp-complete spectra, and are the Bousfield localization of spectra at the Moore spectrum S/pS \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 EE

EEEE-acyclicEE-local
S (p)S \mathbb{Z}_{(p)}pp-local
S𝔽 pS \mathbb{F}_p¬{p}\not\{p\}-localpp-complete
Remark

In terms of arithmetic geometry this may be understood as follows:

  1. 𝔽 p=/(p)\mathbb{F}_p = \mathbb{Z}/(p \mathbb{Z}) is the ring of functions exactly on the point (p)(p)\in Spec(Z)

  2. p\mathbb{Z}_p is the functions on the formal neighbourhood of (p)(p).

  3. (p)\mathbb{Z}_{(p)} is the ring of functions defined on the open complement of the complement of (p)(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 𝔽 p\mathbb{F}_p the support is the closed point, and so the localization there is infinitesimally bigger than that, as given by the pp-adic completion. On the other hand the support of (p)\mathbb{Z}_{(p)} is open and hence contains its own formal neighbourhood, so localizing here just gives the plain localization.

Basic properties

In chromatic homotopy theory

Many of the basic constructions and theorems in chromatic homotopy theory apply to finite pp-local spectra, such as

Examples

References

See also

Revised on August 12, 2014 04:56:36 by Urs Schreiber (146.96.130.201)