# nLab Bousfield localization of spectra

Contents

cohomology

### Theorems

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Higher algebra

higher algebra

universal algebra

# 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 $E$. Since any such $E$ represents a generalized homology theory, this may also be thought of $E$-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 $E \in Spec$ a spectrum, the Bousfield localization at $E$ of another spectrum $X$ is the universal map

$X \longrightarrow L_E X$

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

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

## Definition

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

###### Definition

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

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

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

3. a spectrum $X$ is $E$-local if the following equivalent conditions hold

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

2. every morphism $Y \longrightarrow X$ out of an $E$-acyclic spectrum $Y$ is zero in $Ho(Spectra)$;

(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 $A \in Ho(Spectra)$ being $E$-acyclic means equivalently that the unique morphism $0 \longrightarrow A$ is an $E$-equivalence.

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

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

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

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

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

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

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

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

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

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

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

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

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

###### Definition

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

1. an $E$-local spectrum $L_E X$

2. an $E$-equivalence $X \longrightarrow L_E X$.

according to def. .

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

###### Lemma

For every spectrum $E$ there exists a spectrum $A$ such that any spectrum $X$ is $E$-local (def. ) precisely if it is $A$-null, i.e.

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

and such that

1. $A$ is $E$-acyclic (def. );

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

3. the class of $E$-acyclic spectra (def. ) is the class generated by $A$ under

1. wedge sum

2. the relation that if in a homotopy cofiber sequence $X_1 \to X_2 \to X_3$ 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 $E \in Ho(Spectra)$ any spectrum, every spectrum $X$ sits in a homotopy cofiber sequence of the form

$G_E(X) \longrightarrow X \overset{\eta_X}{\longrightarrow} L_E(X) \,,$

and natural in $X$, such that

1. $G_E(X)$ is $E$-acyclic,

2. $L_E(X)$ is $E$-local,

according to def. .

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

###### Proof

Consider the $\kappa$-CW-spectrum spectrum $A$ whose existence is asserted by lemma . Let

$I_A \coloneqq \{A \to Cone(A)\}$

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

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

of the inclusion of $A$ 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 $X$ any fibrant sequential spectrum, and for $A \longrightarrow X$ any morphism, then an extension along the cone inclusion

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

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

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

By this lemma the $\kappa$-cell spectrum $A$ is $\kappa$-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 $X$ a factorization of the terminal morphism $X \to \ast$ as an $I_A$-relative cell complex (def.) followed by an $I_A$-injective morphism (def.)

$X \overset{I_A Cell}{\longrightarrow} L_E X \overset{I_A Inj}{\longrightarrow} \ast \,.$

By the above, this means that $[A, L_E X] = 0$, hence by lemma that $L_E X$ is $E$-local.

It remains to see that the homotopy fiber of $X \to L_E X$ is $E$-acyclic: By the tensor triangulated structure on $Ho(Spectra)$ (prop.) it is sufficient to show that the homotopy cofiber is $E$-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 $E$-acyclic objects. Hence we conclude by observing that the the transfinite composition of the morphisms between these $E$-acyclic objects is $E$-acyclic. Since by construction all these morphisms are relative cell complex inclusions, this follows again with the compactness of the $n$-spheres (lemma).

###### Lemma

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

###### Proof

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

$E \wedge G_E X \longrightarrow E \wedge X \overset{E \wedge \eta_X}{\longrightarrow} E \wedge L_E X \longrightarrow E \wedge \Sigma G_E X$

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

###### Remark

Hence where $L_E$ is traditionally called “$E$-localization”, $G_E$ might be called “$E$-acyclification”, though that terminology is not used commonly.

## Properties

### Localization at Moore spectra of abelian groups

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

###### Proposition

For $A_1$ and $A_2$ two abelian groups then the following are equivalent

1. the Bousfield localizations at their Moore spectra are equivalent

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

1. every prime number $p$ is invertible in $A_1$ precisely if it is in $A_2$;

2. $A_1$ is a torsion group precisely if $A_2$ is.

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

This means that given an abelian group $A$ then

• either $A$ is not torsion, then

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

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

• or $A$ is torsion, then

$L_{S A }\simeq L_{S(\oplus_q \mathbb{F}_q ) } \,,$

where the direct sum is over all cyclic groups of order $q$, for $q$ a prime not invertible in $A$.

### Relation to nilpotent completion

Let $E$ be an E-∞ ring and let $X$ be any spectrum

###### Remark

There is a canonical map

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

from the $E$-Bousfield localization of spectra of $X$ 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 $R$ a ring, its core $c R$ is the equalizer in

$c R \longrightarrow R \stackrel{\longrightarrow}{\longrightarrow} R \otimes R \,.$
###### Proposition

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

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

• $\mathbb{Z}_n$ for $n \geq 2$.

Then the map in remark is an equivalence

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

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 $E$ an E-∞ ring, every ∞-module $X$ over $E$ is $E$-local, def. .

###### Example

For $E$ an E-∞ algebra over an E-∞ ring $S$ and for $X$ an $S$-∞-module, consider the dual Cech nerve cosimplicial object

$E^{\wedge_S^{\bullet+1}}\wedge_S X \;\colon\; \Delta \longrightarrow Spectra \,.$

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

$X \longrightarrow \underset{\leftarrow}{\lim} E^{\wedge_S^{\bullet+1}} \wedge_S X$

factors through the $E$-localization of $X$

$X \longrightarrow L_E X \longrightarrow \underset{\leftarrow}{\lim} E^{\wedge_S^{\bullet+1}} \wedge_S X \,.$

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

### Rationalization

###### Example

Bousfield localization at the Moore spectrum/Eilenberg-MacLane spectrum $S \mathbb{Q}\simeq H\mathbb{Q}$ 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)

### $p$-Localization

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

###### Example

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

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

###### Proposition

$p$-localization is a smashing localization:

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

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

### $p$-Completion

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

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

for the cyclic group/finite field of order $p$.

###### Definition

Write

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

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

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

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

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

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

As in Bousfield 79, p. 6 one also writes

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

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

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

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

###### Example

Let

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

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

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

More generally

###### Example

Bousfield localization at the Moore spectrum $S \mathbb{F}_p$ 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.

## References

Original articles are

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

Review:

Discussion in the general context of higher algebra/stable homotopy theory:

Discussion specifically of K(n)-local spectra includes

• Jeroen van der Meer, Chapter 4 of: A Stack-Theoretic Perspective on $\mathrm{KU}$-Local Stable Homotopy Theory, 2019 (pdf)