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Bousfield localization of spectra

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Contents

Idea

Bousfield localization of spectra refers generally to localizations of the stable (∞,1)-category of spectra (hence Bousfield localization of model categories of spectra) at the collection of morphisms which become equivalences under smash product with a given spectrum EE. Since any such EE represents a generalized homology theory, this may also be thought of EE-homology localization.

More specifically, if the stable (∞,1)-category of spectra is presented by a (stable) model category, then the ∞-categorical localization can be presented by the operation of Bousfield localization of model categories. The original article (Bousfield 79) essentially considered localization at the level of homotopy categories.

Specifically, for ESpecE \in Spec a spectrum, the Bousfield localization at EE of another spectrum XX is the universal map

XL EX X \longrightarrow L_E X

to the EE-local spectrum L EXL_E X, with the property that for YY any EE-acyclic spectrum in that YE0Y \wedge E \simeq 0, every morphism YL EXY \longrightarrow L_E X is null-homotopic (a zero morphism in the stable (∞,1)-category of spectra). (see for instance Lurie 10, Example 4)

For E=M pE = M \mathbb{Z}_p the Moore spectrum of the cyclic group p/p\mathbb{Z}_p \coloneqq \mathbb{Z}/p\mathbb{Z} for some prime number pp, this EE-localization is p-completion. (see for instance Lurie 10, Examples 7 and 8)

Definition

We write Ho(Spectra)Ho(Spectra) for the stable homotopy category. For X,YHo(Spectra)X,Y \in Ho(Spectra) two spectra we write [X,Y][X,Y] \in Ab for its hom-abelian groups, and [X,Y] [Σ X,Y][X,Y]_\bullet \coloneqq [\Sigma^\bullet X,Y] for the corresponding graded abelian group (def.).

Definition

Let EHo(Spectra)E \in Ho(Spectra) be a spectrum. Say that

  1. a spectrum XX is EE-acyclic if the smash product with EE is zero, EX0E \wedge X \simeq 0;

  2. a morphism f:XYf \colon X \to Y of spectra is an EE-equivalence if Ef:EXEYE \wedge f \;\colon\; E \wedge X \to E \wedge Y is an isomorphism in Ho(Spectra)Ho(Spectra), hence if E (f)E_\bullet(f) is an isomorphism in EE-generalized homology;

  3. a spectrum XX is EE-local if the following equivalent conditions hold

    1. for every EE-equivalence ff then [f,X] [f,X]_\bullet is an isomorphism;

    2. every morphism YXY \longrightarrow X out of an EE-acyclic spectrum YY is zero in Ho(Spectra)Ho(Spectra);

(Bousfield 79, §1) see also for instance (Lurie, Lecture 20, example 4)

Lemma

The two conditions in the last item of def. are indeed equivalent.

Proof

Notice that AHo(Spectra)A \in Ho(Spectra) being EE-acyclic means equivalently that the unique morphism 0A0 \longrightarrow A is an EE-equivalence.

Hence one direction of the claim is trivial. For the other direction we need to show that for [,X] [-,X]_\bullet to give an isomorphism on all EE-equivalences ff, it is sufficient that it gives an isomorphism on all EE-equivalences of the form 0A0 \to A.

Given a morphism f:ABf \colon A \to B, write BB/AB \longrightarrow B/A for its homotopy cofiber. Then since Ho(Spectra)Ho(Spectra) is a triangulated category (prop.) the defining axioms of triangulated categories (def., lemma) give that there is a commuting diagram of the form

0 A id A 0 ΣA id f id Σ 1B/A A f B B/A ΣA, \array{ 0 &\longrightarrow& A &\overset{id}{\longrightarrow}& A &\overset{}{\longrightarrow}& 0 &\overset{}{\longrightarrow}& \Sigma A \\ \downarrow && \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{f}} && \downarrow && \downarrow^{\mathrlap{id}} \\ \Sigma^{-1} B/A &\longrightarrow& A &\underset{f}{\longrightarrow}& B &\longrightarrow& B/A &\longrightarrow& \Sigma A } \,,

where both the top as well as the bottom are homotopy cofiber sequences. Hence applying [,X] [-,X]_\bullet to this diagram in Ho(Spectra)Ho(Spectra) yields a diagram of graded abelian groups of the form

0 [A,X] [A,X] 0 [A,X] +1 id [f,X] id [B/A,X] +1 [A,X] [B,X] [B/A,X] [A,X] +1, \array{ 0 &\longleftarrow& [A,X]_\bullet &\longleftarrow& [A,X]_\bullet &\longleftarrow& 0 &\longleftarrow& [A,X]_{\bullet+1} \\ \uparrow && \uparrow^{\mathrlap{id}} && \uparrow^{\mathrlap{[f,X]_\bullet}} && \uparrow && \uparrow^{\mathrlap{id}} \\ [B/A,X]_{\bullet+1} &\longleftarrow& [A,X]_\bullet &\longleftarrow& [B,X]_\bullet &\longleftarrow& [B/A,X]_\bullet &\longleftarrow& [A,X]_{\bullet+1} } \,,

where now both horizontal sequences are long exact sequences (prop.).

Hence if [B/A,X] 0[B/A,X]_\bullet \longrightarrow 0 is an isomorphism, then all four outer vertical morphisms in this diagram are isomorphisms, and then the five-lemma implies that also [f,X] [f,X]_\bullet is an isomorphism.

Hence it is now sufficient to observe that with f:ABf \colon A \to B an EE-equivalence, then its homotopy cofiber B/AB/A is EE-acyclic.

To see this, notice that by the tensor triangulated structure on Ho(Spectra)Ho(Spectra) (prop.) the smash product with EE preserves homotopy cofiber sequences, so that there is a homotopy cofiber sequence

EAEfEBE(B/A)EΣA. E \wedge A \overset{E \wedge f}{\longrightarrow} E \wedge B \longrightarrow E \wedge (B/A) \longrightarrow E \wedge \Sigma A \,.

But if the first morphism here is an isomorphism, then the axioms of a triangulated category (def.) imply that EB/A0E \wedge B / A \simeq 0. In detail: by the axioms we may form the morphism of homotopy cofiber sequences

EA Ef EB EB/A EΣA id (Ef) 1 id EA id EA 0 EΣA. \array{ E \wedge A &\overset{E \wedge f}{\longrightarrow}& E \wedge B &\longrightarrow& E \wedge B/A &\longrightarrow& E \wedge \Sigma A \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{(E\wedge f)^{-1}}} && \downarrow && \downarrow^{\mathrlap{id}} \\ E \wedge A &\underset{id}{\longrightarrow}& E \wedge A &\longrightarrow& 0 &\longrightarrow& E \wedge \Sigma A } \,.

Then since two of the three vertical morphisms on the left are isomorphisms, so is the third (lemma).

Definition

Given E,XHo(Spectra)E,X \in Ho(Spectra), then an EE-localization of XX is

  1. an EE-local spectrum L EXL_E X

  2. an EE-equivalence XL EXX \longrightarrow L_E X.

according to def. .

We discuss now that EE-Localizations always exist. The key to this is the following lemma , which asserts that a spectrum being EE-local is equivalent to it being AA-null, for some “small” spectrum AA:

Lemma

For every spectrum EE there exists a spectrum AA such that any spectrum XX is EE-local (def. ) precisely if it is AA-null, i.e.

XisE-local[A,X] *=0 X \;is\; E\text{-local} \;\;\;\; \Leftrightarrow \;\;\;\; [A,X]_\ast = 0

and such that

  1. AA is EE-acyclic (def. );

  2. there exists an infinite cardinal number κ\kappa such that AA is a κ\kappa-CW spectrum (hence a CW spectrum (def.) with at most κ\kappa many cells);

  3. the class of EE-acyclic spectra (def. ) is the class generated by AA under

    1. wedge sum

    2. the relation that if in a homotopy cofiber sequence X 1X 2X 3X_1 \to X_2 \to X_3 two of the spectra are in the class, then so is the third.

(Bousfield 79, lemma 1.13 with lemma 1.14) review includes (Bauer 11, p.2,3, VanKoughnett 13, p. 8)

Proposition

For EHo(Spectra)E \in Ho(Spectra) any spectrum, every spectrum XX sits in a homotopy cofiber sequence of the form

G E(X)Xη XL E(X), G_E(X) \longrightarrow X \overset{\eta_X}{\longrightarrow} L_E(X) \,,

and natural in XX, such that

  1. G E(X)G_E(X) is EE-acyclic,

  2. L E(X)L_E(X) is EE-local,

according to def. .

(Bousfield 79, theorem 1.1) see also for instance (Lurie, Lecture 20, example 4)

Proof

Consider the κ\kappa-CW-spectrum spectrum AA whose existence is asserted by lemma . Let

I A{ACone(A)} I_A \coloneqq \{A \to Cone(A)\}

denote the set containing as its single element the canonical morphism (of sequential spectra) from AA into the standard cone of AA, i.e. the cofiber

Cone(A)cofib(AAI +)AI Cone(A) \coloneqq cofib( A \to A \wedge I_+ ) \simeq A \wedge I

of the inclusion of AA into its standard cylinder spectrum (def.).

Since the standard cylinder spectrum on a CW-spectrum is a good cylinder object (prop.) this means (lemma) that for XX any fibrant sequential spectrum, and for AXA \longrightarrow X any morphism, then an extension along the cone inclusion

A X Cone(A) \array{ A &\longrightarrow& X \\ \downarrow & \nearrow \\ Cone(A) }

equivalently exhibits a null-homotopy of the top morphism. Hence the (ACone(A))(A \to Cone(A))-injective objects in Ho(Spectra)Ho(Spectra) are precisely those spectra XX for which [A,X] 0[A,X]_\bullet \simeq 0.

Moreover, due to the degreewise nature of the smash tensoring Cone(A)=AICone(A) = A \wedge I (def), the inclusion morphism ACone(A)A \to Cone(A) is degreewise the inclusion of a CW-complex into its standard cone, which is a relative cell complex inclusion (prop.).

By this lemma the κ\kappa-cell spectrum AA is κ\kappa-small object (def.) with respect to morphisms of spectra which are degreewise relative cell complex inclusion small object argument .

Hence the small object argument applies (prop.) and gives for every XX a factorization of the terminal morphism X*X \to \ast as an I AI_A-relative cell complex (def.) followed by an I AI_A-injective morphism (def.)

XI ACellL EXI AInj*. X \overset{I_A Cell}{\longrightarrow} L_E X \overset{I_A Inj}{\longrightarrow} \ast \,.

By the above, this means that [A,L EX]=0[A, L_E X] = 0, hence by lemma that L EXL_E X is EE-local.

It remains to see that the homotopy fiber of XL EXX \to L_E X is EE-acyclic: By the tensor triangulated structure on Ho(Spectra)Ho(Spectra) (prop.) it is sufficient to show that the homotopy cofiber is EE-acyclic (since it differs from the homotopy fiber only by suspension). By the pasting law, the homotopy cofiber of a transfinite composition is the transfinite composition of a sequence of homotopy pushouts. By lemma and applying the pasting law again, all these homotopy pushouts produce EE-acyclic objects. Hence we conclude by observing that the the transfinite composition of the morphisms between these EE-acyclic objects is EE-acyclic. Since by construction all these morphisms are relative cell complex inclusions, this follows again with the compactness of the nn-spheres (lemma).

Lemma

The morphism XL E(X)X \to L_E (X) in prop. exhibits an EE-localization of XX according to def.

Proof

It only remains to show that XL EXX \to L_E X is an EE-equivalence. By the tensor triangulated structure on Ho(Spectra)Ho(Spectra) (prop.) the smash product with EE preserves homotopy cofiber sequences, so that

EG EXEXEη XEL EXEΣG EX E \wedge G_E X \longrightarrow E \wedge X \overset{E \wedge \eta_X}{\longrightarrow} E \wedge L_E X \longrightarrow E \wedge \Sigma G_E X

is also a homotopy cofiber sequence. But now EG EX0E \wedge G_E X \simeq 0 by prop. , and so the axioms (def.) of the triangulated structure on Ho(Spectra)Ho(Spectra) (prop.) imply that EηE \wedge \eta is an isomorphism.

Remark

Hence where L EL_E is traditionally called “EE-localization”, G EG_E might be called “EE-acyclification”, though that terminology is not used commonly.

Properties

Localization at Moore spectra of abelian groups

A basic special case of EE-localization of spectra is given for E=SAE = S A the Moore spectrum of an abelian group AA (Bousfield 79, section 2). For A= (p)A = \mathbb{Z}_{(p)} this is p-localization and for A=𝔽 pA = \mathbb{F}_p this is p-completion, see examples and below for more.

Proposition

For A 1A_1 and A 2A_2 two abelian groups then the following are equivalent

  1. the Bousfield localizations at their Moore spectra are equivalent

    L SA 1L SA 2; L_{S A_1} \simeq L_{S A_2} \,;
  2. A 1A_1 and A 2A_2 have the same type of acyclicity, meaning that

    1. every prime number pp is invertible in A 1A_1 precisely if it is in A 2A_2;

    2. A 1A_1 is a torsion group precisely if A 2A_2 is.

(Bousfield 79, prop. 2.3) recalled e.g. in (VanKoughnett 13, prop. 4.2).

This means that given an abelian group AA then

  • either AA is not torsion, then

    L SAL S[I 1], L_{S A} \simeq L_{S \mathbb{Z}[I^{-1}]} \,,

    where II is the set of primes invertible in AA and [I 1]\mathbb{Z}[I^{-1}] \hookrightarrow \mathbb{Q} is the localization at these primes of the integers;

  • or AA is torsion, then

    L SAL S( q𝔽 q), L_{S A }\simeq L_{S(\oplus_q \mathbb{F}_q ) } \,,

    where the direct sum is over all cyclic groups of order qq, for qq a prime not invertible in AA.

Relation to nilpotent completion

Let EE be an E-∞ ring and let XX be any spectrum

Remark

There is a canonical map

L EXlim n(E S n+1 SX) L_E X \stackrel{}{\longrightarrow} \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X)

from the EE-Bousfield localization of spectra of XX into the totalization of the canonical cosimplicial spectrum (see at nilpotent completion).

We now consider conditions for this morphism to be an equivalence.

Definition

For RR a ring, its core cRc R is the equalizer in

cRRRR. c R \longrightarrow R \stackrel{\longrightarrow}{\longrightarrow} R \otimes R \,.
Proposition

Let EE be a connective E-∞ ring such that the core of π 0(E)\pi_0(E), def. , is either of

  • the localization of the integers at a set JJ of primes, cπ 0(E)[J 1]c \pi_0(E) \simeq \mathbb{Z}[J^{-1}];

  • n\mathbb{Z}_n for n2n \geq 2.

Then the map in remark is an equivalence

L EXlim n(E S n+1 SX). L_E X \stackrel{\simeq}{\longrightarrow} \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X) \,.

(Bousfield 79) see also for instance (Bauer 11, p.2)

For more discussion of E-infinity (derived) formal completions via totalizations of Amitsur complexes, see (Carlsson 07).

Fracture theorem

The fracture theorem says how Bousfield localization at a coproduct/wedge sum of spectra is a homotopy pullback of Bousfield localization separately. See at fracture theorem for more on this.

Examples

General

Example

For EE an E-∞ ring, every ∞-module XX over EE is EE-local, def. .

(e.g. Lurie, Lecture 20, example 5)

Example

For EE an E-∞ algebra over an E-∞ ring SS and for XX an SS-∞-module, consider the dual Cech nerve cosimplicial object

E S +1 SX:ΔSpectra. E^{\wedge_S^{\bullet+1}}\wedge_S X \;\colon\; \Delta \longrightarrow Spectra \,.

By example each term is EE-local, so that the map to the totalization

XlimE S +1 SX X \longrightarrow \underset{\leftarrow}{\lim} E^{\wedge_S^{\bullet+1}} \wedge_S X

factors through the EE-localization of XX

XL EXlimE S +1 SX. X \longrightarrow L_E X \longrightarrow \underset{\leftarrow}{\lim} E^{\wedge_S^{\bullet+1}} \wedge_S X \,.

Under suitable condition the second map here is indeed an equivalence, in which case the totalization of the dual Cech nerve exhibits the EE-localization. This happens for instance in the discussion of the Adams spectral sequence, see the examples given there.

(see also e.g. Bauer 11, p. 2)

Rationalization

Example

Bousfield localization at the Moore spectrum/Eilenberg-MacLane spectrum SHS \mathbb{Q}\simeq H\mathbb{Q} of the rational numbers is rationalization to rational homotopy theory.

The corresponding \mathbb{Q}-acyclification (remark ) is torsion approximation.

e.g. (Bauer 11, example 1.7)

pp-Localization

For pp a prime number write (p)\mathbb{Z}_{(p)} for the localization of the integers at (p)(p), for the ring of integers localized at pp, hence with all primes except pp inverted; equivalently the subring of the rational numbers with denominator not divisible by pp.

Example

The Bousfield localization at the Moore spectrum S (p)S \mathbb{Z}_{(p)} is p-localization.

(Bousfield 79), Bauer 11, example 1.7). See at localization of a space for details on this.

Proposition

pp-localization is a smashing localization:

L S (p)XS (p)X. L_{S \mathbb{Z}_{(p)}} X \simeq S \mathbb{Z}_{(p)} \wedge X \,.

(Bousfield 79, prop. 2.4) recalled e.g. as (van Koughnett 13, prop. 4.3).

pp-Completion

For pp \in \mathbb{N} a prime number, write

𝔽 p=/(p) \mathbb{F}_p = \mathbb{Z}/(p)

for the cyclic group/finite field of order pp.

Definition

Write

/p ([p 1])/ \mathbb{Z}/p^\infty \coloneqq (\mathbb{Z}[p^{-1}])/\mathbb{Z}

for the localization of the integers away from pp followed by the quotient by \mathbb{Z}.

e.g. (Bousfield 79, p. 6)

Remark

The short exact sequence of abelian groups

0[p 1]/p 0 0 \to \mathbb{Z} \longrightarrow \mathbb{Z}[p^{-1}] \longrightarrow \mathbb{Z}/p^\infty \to 0

induces the homotopy fiber sequence (in spectra) of Moore spectra

ΩS(/p )SS([p 1]). \Omega S(\mathbb{Z}/p^\infty) \longrightarrow S\mathbb{Z} \longrightarrow S(\mathbb{Z}[p^{-1}]) \,.

As in Bousfield 79, p. 6 one also writes

S 1(/p )ΩS(/p ). S^{-1} (\mathbb{Z}/p^\infty) \coloneqq \Omega S(\mathbb{Z}/p^\infty) \,.
Proposition

The localization of spectra at the Moore spectrum S𝔽 pS\mathbb{F}_p is given by the mapping spectrum out of S 1/p S^{-1} \mathbb{Z}/p^\infty:

L S𝔽 pX[ΩS/p ,X]. L_{S \mathbb{F}_p} X \simeq [\Omega S \mathbb{Z}/p^\infty, X] \,.

(Bousfield 79, prop. 2.5)

Fact: 𝔽 p\mathbb{F}_p-localizaton is p-completion, e.g. Lurie “Proper Morphisms…”, section 4.

Example

Let

EH/p E \coloneqq H \mathbb{Z}/p\mathbb{Z}

be the corresponding Moore spectrum. Then a spectrum which corresponds to a chain complex under the stable Dold-Kan corespondence is EE-local, def. , if that chain complex has chain homology groups being [p 1]\mathbb{Z}[p^{-1}]-modules.

The EE-localization of a spectrum in this case is p-completion.

(e.g. Lurie, Lecture 20, example 8)

More generally

Example

Bousfield localization at the Moore spectrum S𝔽 pS \mathbb{F}_p is p-completion to p-adic homotopy theory.

E.g. (Bauer 11, example 1.7). See at localization of a space for more on this.

Telescopic localization

Chromatic localization

References

Original articles are

  • Aldridge Bousfield, The localization of spectra with respect to homology , Topology vol 18 (1979) (pdf)

  • Douglas Ravenel, Localization with respect to certain periodic homology theories, American Journal of Mathematics, Vol. 106, No. 2, (Apr., 1984), pp. 351-414 (pdf)

Discussion in terms of Bousfield localization of model categories of spectra appears in

see also

Lecture notes include

Discussion the general context of higher algebra/stable homotopy theory includes

Discussion specifically of K(n)-local spectra includes

See also

section 2.4 of

Last revised on October 10, 2018 at 07:06:22. See the history of this page for a list of all contributions to it.