Bousfield Localization of Spectra

This post is organized as a part of the UT Austin Spring 2020 DRP program.I’d like to thank my mentor Richard Wong for useful directions throughout the semester and introducing me to much of the foundational material in homotopy theory.

This post will be organized as follows:

First, I’ll present the basic definitions of the topic(spectra, homotopy groups,CW-spectra,triangulated categories etc).

Second, I’;ll introduce finite spectra, Z_{p}-localization of spectra aka. p-local spectra via Bousfield localization, thick subcategories.

Third, I’ll discuss the the Thick Subcategory Theorem but avoid technical details.

So, let’s begin.


A spectrum X is a sequence of pointed spaces \{X_{n} \} and structure maps \sigma_{n}: \sum X_{n} \to X_{n+1}. For example, one can take a pointed space X and consider the suspensions \sum^{n} X=S^{n} \wedge X. This forms a spectrum with the structure maps the identity. This is the suspension spectrum of X.

One can consider this general spectrum \{X_{n} \} and suspend it by the integer $i$ to get a spectrum \sum^{i} X whose components are given by (\sum^{i} X)_{n}=X_{n+i}.

The homotopy groups \pi_{k}(X) of a spectrum X are defined as the colimit:

\pi_{k}(X)=colim_{n} \pi_{n+k}(X_{n}).

 

Since \sum is a functor on Top, one is motivated to define maps between spectra f:X \to Y as a a collection of maps f_{n}:X_{n} \to Y_{n} such that the obvious diagrams commute.This may not actually be the best definition of maps between spectra but it doesn’t matter too much for now.

For example, one can consider S^{0} and suspend this to get \mathbb{S}, the sphere spectrum, an example of a suspension spectrum.

Given the suspension and loop space functors \sum, \Omega, one has the loop-space adjunction for pointed topological spaces, namely,

[\sum X,Y] \simeq [X,\Omega Y]

Note that \Omega Y=[S^{1},Y](with compact-open toplogy). This allows us to define \Omega-spectrum as spectrum X where the adjoint map \tilde{\sigma_{n}}:X_{n} \to \Omega X_{n+1} is a weak homotopy equivalence.

Let’s look at an example.

Eilenberg-MacLane Spectrum

Recall that an Eilenberg-Maclane space corresponding to (G,n) where G is an abelian group is a CW-complex X such that \pi_{n}(X)=G and \pi_{k}(X)=0 for k \neq n.Call this CW-complex K(G,n)

We know the following correspondence from algebraic topology : For any CW-complex X,H^{k}(X;G) \simeq [X,K(G,n)].

Collect these spaces on the index n to get \{K(G,n) \}. We claim this forms a spectra. What are the structure maps though?

In particular, note that we we have \pi_{k}(\Omega K(G,n+1))=\pi_{k+1}(K(G,n+1)). By uniqueness of the Eilemberg-McLane space upto homotopy equivalence, we have a homotopy equivalence \tilde{\sigma}_{n}:K(G,n) \to \Omega K(G,n+1). Take the adjoint of this under the loopspace-adjunction to get the structure maps \sigma_{n}:\sum K(G,n) \to K(G,n+1).

Hence, this is also an \Omega-spectrum.

CW-spectra

Well, I suppose some analogy with the classical case is required here. Consider the Quillen-Serre model structure on pointed topological spaces(aka the usual one). Here, the cofibrations are retracts of generalized relative CW inclusions(see [1] for definitions). In particular, the CW complexes are cofibrant objects in this case. We’d like our so called CW complexed to be cofibrant objects in the stable model category on topological sequential spectra(these are the same things as the spectra defined above where \sum X =S^{1} \wedge X).

A CW spectra is a sequence \{E_{n} \} of CW-complexes where the structure maps \sigma_{n}:\sum E_{n} \to E_{n+1} map isomorphically into a subcomplex of E_{n+1}. TO prove the above fact, see Prop 0.49 of this.

The cell of a CW complex is the collection \{A_{n},\sigma_{n+1}(A_{n}),\sigma_{n+2}(A_{n}),\cdots, \} where A_{n} is a cell of E_{n} and A_{n} is not of the form \sigma_{i}(A_{k}) for some cell A_{k} \subset E_{n} where k<n. Essentially, you package a cell A_{n} of some component space in the spectrum and the cells in the higher components obtained from the structure maps such that the original A_{n} itself is not obtained in this form.

A finite spectrum is a CW-spectrum which has a finite number of cells. Another important property of Top extends to the case of spectra:CW-Approximation.

A subspectrum B of a CW spectrum X is closed if B is a union of cells and for any cell in X with some suspension in B, it is in B, that is, X/B is a CW spectrum. This will be important later in Bousfield localization.

We know that if X is a topological space, then there exists a CW-Complex \Gamma X and a homotopy equivalence \Gamma X \simeq X.

This extends as follows:

CW Approximation for Sequential Spectra:

Let X be a sequential spectrum then there exists a CW-Spectrum \Gamma X such that there is a weak homotopy equivalence between \Gamma X and X.

Proof:

By CW-approximation on Top, consider a weak homotopy equivalence f_{0}: \Gamma X_{0} \sim X_{0}. Assume that there exists a collection of weak homotopy equivalences $f_{n}: \Gamma X_{k} \sim X_{k}$ for all k \leq n which respects the structure maps, we have the following

image1

Consider the composite \sum \Gamma X_{n} \to \sum X_{n} \to X_{n+1}. By CW-Approximation on Top, the second map factorizes as \sum X_{n} \to \Gamma X_{n+1} \to X_{n+1} where \Gamma X_{n+1} \to X_{n+1} is a weak homotopy equivalence and the first one is a relative CW inclusion.

 

Triangulated Category

Consider an additive category C an automorphism \sum on C(called a shift functor). A triangle in this category is a sequence:

X \to Y \to Z \to \sum X

A morphism of triangles X \to Y \to Z \to \sum X and X' \to Y' \to Z' \to \sum X' with component-wise maps that commute in the obvious diagram.

We have a distinguished class of these triangles called exact triangles/distinguished triangles with respect to which the following axioms are satisfied:

TR1(Cofiber, triangles are replete)

The triangle X \to X \to 0 \to \sum X is an exact triangle.

For any map f: X \to Y, there is an object Z \in C and an exact triangle of the form:

image2

Third, every triangle isomorphic to an exact triangle is exact.

TR2(Shift exact triangle)

If

image2

is an exact triangle if and only if the following triangle is exact:

image3

TR3(Maps between exact triangles)

If X \to Y \to Z \to \sum X,X' \to Y' \to Z' \to \sum X' are two exact triangles and if we’re given maps between the first two components X \to X',Y \to Y' which makes the square commute, then there exists a third map $latex $Z \to Z’$ which makes the entire diagram commute.

TR4(Octahedral axiom)

This is quite technical condition so I’ll just leave this here for self-explanation:

image4

Homotopy Category of Stable Model Category

The homotopy category of a stable model category, in our case, the category of spectra. This means that loop and suspension functors \Omega, \sum are equivalences. Note that this isn’t the case in Top^{*}.  The projective and injective model structures on chain complexes on an abelina category A are also a stable model category.

Smash Product of Spectra

So let SHC be the stable homotopy category discussed above. We’d like to define a smash product on homotopy category of sequential spectra equipped with its stable model structure, Ho(S^{N}) defined above. We want a symmetric monoidal product on this category, the smash product, in this case. The situation is a little complicated in general but one could define a naive smash product of sequential sepctra X,Y as follows:

X \wedge Y=

image5

where the structure maps are:

\sum (\sum X_{n} \wedge Y_{n}) \simeq \sum X_{n} \wedge \sum Y_{n} \to X_{n+1} \wedge Y_{n+1} where the last map is the smash product of the maps \sum X_{n} \to X_{n+1} and \sum Y_{n} \to Y_{n+1} and:

\sum (X_{n} \wedge Y_{n}) \to \sum X_{n} \wedge Y_{n}

To define a symmetric monoidal product, we need a unit. This is the sphere spectrum \mathbb{S}.

Ring spectra

A ring spectrum X is a spectrum equipped with a multiplication and unit map \mu: X \wedge X \to X and \eta: S \to E such that the following diagram commutes with these monad-like axioms of associativity:

image6

 

Bousfield Localization

We wish for a way to add more weak equivalences to a model category which are reflected in the stable homotopy category. To this effect, in a sense, we add more cofibrations to the data of the model category C. We’d also want to get atleast some sort of a functor C \to L_{W}C with some results on existence/uniqueness where W is the class of maps that we wish to localize on.

Just like in the case of algebraic localization ,we expect to ‘kill off’ more than just W under the localization process. A basic example is the localization with respect to a homology theory. The upshot is that the localizing at primes will prove to be computationally better to handle as we’ll see.

We’re just going to be looking at localization of CW-spectra with respect to a homology theory here.. Let Ho^{s} be the category of CW-spectra.

Let E \in Ho(S^{N}). One can prove that E_{n}(X)=\pi_{n}(E \wedge X) is a generalized homology theory. Throughout, we’ll write this as E_{*}(X) for simplicity.

Also, fix the notation [A,B]_{*}=[\sum ^{*} A,B].

A morphism of spectra f: X \to Y is an E-equivalence if the map \tilde{f}:E \wedge X \to E \wedge Y is an isomorphism in Ho^{s}, that is the induced map, E_{*}(f) is an isomorphism

A spectrum is E-acyclic if E_{*}(X) =0$.

A spectrum X is E-local if for every E*-equivalence f: A \to B, the induced map f^{*}:[B,X]_{*} \to [A,X]_{*} is a bijection.

E_{*}-Whithead Theorem:

Let X,Y \in Ho^{s} be E-local and f: X \to Y is an $E-$equivalence then f:X \simeq Y in Ho^{s}.

Proof:

The definitions give the equivalences [Y,X]_{*} \simeq [X,X]_{*} and [X,Y]_{*} \simeq [Y,Y]_{*} which gives the stable weak equivalence


Ok, we ‘re now going to get to building this localization functor. Based on everything so far, let L_{E} be the localization functor we’ll construct;For a spectrum X, we want L_{E}(X) to be E-local and a natural transformation q \Rightarrow L_{E}.Here, |X| represents the number of cells of CW spectrum X.

Lemma 1:

Let X be a CW-spectrum and B be a proper closed subspectrum of X with E_{*}(E/B)=0. Let \kappa \geq |\pi_{*}(E)| be a infinite cardinal number. Then, there is a closed subspectrum W \subset X with at most \kappa cells such that W \not \subset B and E_{*}(W/(W \cap B))=0

Proof:

Start off with some W_{1} such that |W_{1}| \leq \kappa and W_{1} \not \subset B. Proceed inductively. Given this W_{n}, choose for each x \in E_{*}(W_{n}/(W_{n} \cap B)), a subspectrum F_{x} such that x=0 in E_{*}((W_{n} \cup F_{x})/((W_{n} \cup F_{x}) \cap B)). This is possible by the hypothesis E_{*}(E/B)=0. Anyways, define W_{n+1}=W_{n} \bigcup \cup_{x} F_{x}. We get that |W_{n}| \leq \kappa for all n,W_{n} \not \subset B.

Choose W=\bigcup_{n=1}^{\infty} W_{n} to finish the proof.

Lemma 2:

There exists an E-acyclic spectrum A^{E} such that the a spectrum Y is E-local if and only if [A,Y]_{*}=0.

Proof:

By the CW approximation theorem(proven above), let \{K_{\alpha} \} be a set of CW spectrum representatives of the weak equivalence classes of E-acyclic spectra with atmost \kappa cells. Consider A^{E}=\vee_{\alpha} K_{\alpha}. If Y is $E-local$ then [A^{E},Y]_{*}=0.

Now, if [A^{E},Y]_{*}=0 then [A',Y]_{*}=0 for any A' formed from the A under weak equivalence, shift, homotopy cofiber, wedges, summands. Let C(A^{E}) be this class of spectra. We’ve to show that all E-acyclic spectra lie in C(A^{E}).

Let X be an E-acyclic spectra, upto weak equivalence consider the representative \Gamma X. By the previous Lemma(Lemma 1), we start from B_{0}=0 and get the sequence:

B_{0}=0 \subset B_{1} \subset \cdots \subset B_{\gamma}=\Gamma X where:

B_{\lambda} is an E-acyclic closed subspectrum$ and

B_{\lambda+1}=B_{\lambda} \cup W_{\lambda} where W_{\lambda} is the closed subspectrum from Lemma 1 such that |W| \leq \kappa,W \not \subset B_{\lambda} and E_{*}(W/W \cap B_{\lambda})=0 and

For \lambda a limit ordinal,B_{\lambda}=\bigcup_{\mu<\lambda} B_{\mu}

Now, let K_{\alpha_{0}} be the CW representative of W_{\lambda}/W_{\lambda}  \cap B_{\lambda}. One can see that B_{\lambda+1}/B_{\lambda} \simeq K_{\alpha_{0}}. Assume that B_{\lambda} \in C(A^{E}).

By shifting, we get the cofiber sequence:

\Omega K_{\alpha_{0}} \to B_{\lambda} \to B_{\lambda+1} hence B_{\lambda+1} \in C(A^{E}).

For \lambda a limit ordinal, consider the triangle:

\vee_{\mu < \lambda} B_{\mu} \to \vee_{\mu<\lambda}B_{\mu} \to B_{\lambda} where the first map is 1-\bigvee_{\mu<\lambda}i_{\mu} where i_{\mu} is the inclusion B_{\mu} \to B_{\mu+1}.Transfinite induction gives that X \in C(A_{E}).

Localization theorem:

Every spectrum X \in Ho^{s} sits in a homotopy cofiber sequence of the form

A^{E}(X) \to X \to A_{E}(X)

which is natural in X such that A^{E}(X) is $E-acyclic$ and A_{E}(X) is E-local

Proof:

The proof can be found using the previous lemma in Proposition 2.5 of this followed by Lemma 2.6. It uses a slightly technical ‘small object’ argument followed  by transfinite induction.

Moore spectrum of an Abelian Group

We’re slowly approaching our goal of Z_{(p)}-localization.

Consider an abelian group A, then one has a Moore spectrum SA is characterized as having the homotopy groups:

\pi_{i}(SA)=0 for i <0,\pi_{i}(SA \wedge H\mathbb{Z})=0 for i>0 and \pi_{0}(SA)=A where H\mathbb{Z} is the Eilenberg-Moore spectra of the integers.

Moreover, this is a ring spectrum

For a spectrum E, define EG :=E \wedge SG where G is an abelian group. We have the following Universal Coefficient Theorem:

Universal Coefficient Theorem:

There exists natural exact sequences:

0 \to \pi_{n}(X) \otimes G \to (EG)_{n}(X) \to Tor(\pi_{n-1}(X),G) \to 0

Two abelian groups G_{1},G_{2} have the same acyclicity type if:

  1. G_{1} is a torsion group iff G_{2} is a torsion group
  2. For each prime p,G_{1} is uniquely p-divisible iff G_{2} is uniquely p-divisible.

So, both \pi_{*}(X) \otimes G and Tor(\pi_{*}(X),G) are determined by the acyclicity type of G which gives the the following equivalence of localizations:

Proposition:

G_{1},G_{2} have same acyclicity type iff SG_{1},SG_{2} give equivalent localization functors.

Thus to determine the localization functors is equivalent to considering the cases when G=\mathbb{Z}_{(J)} and G=\bigoplus_{p \in J} Z_{p} where J is a set of primes and \mathbb{Z}-{(J)} are the integers localized at J. The case J=\{p\} gives us the so-called p-local spectra.

From ([1], Prop 2.4,Prop2.6), one can determine the following when X \in Ho^{s}:

If G=\mathbb{Z}_{(J)} then L_{SG}(X)=SG \wedge X

If G=\bigoplus_{p \in J}Z_{p} then L_{SG}(X)=\prod\limits_{p\ in J}X \wedge \Omega S \mathbb{Z}_{p^{\infty}}. In this case, we have a split exact sequence:

0 \to Ext(\mathbb{Z}_{p^{\infty}},\pi_{*}(X)) \to \pi_{*}(L_{SG}(X)) \to Hom(\mathbb{Z}_{p^{\infty}},\pi_{*-1}(X)) \to 0. Also, if \pi_{n}(X) is finitely generated for all n, then one gets \pi_{*}(L_{SG}(X))

=\prod_{p \in J} \mathbb{Z}_{p} \otimes \pi_{*}(X)

Moreover, X is SG-local if and only if \pi_{*}(X) are uniquely p-divisible for all p \not \in J.

Thick Subcategory Theorem

Ok, so one could perform Bousfield localization with respect to more general generalized cohomology theories. For example, one could consider K(n), Morava K-theory. We’re not going to get into any of the actual details. It gives an interesting answer as to what the thick subcategories of the category of finite p-local spectra F_{p} are.

A full subcategory F \subset F_{p} is said to be thick if:

  1. For every C \in F, any spectra weakly equivalent to C are also in F
  2. Given 0 \to X \to Y \to Z \to 0, a cofibration in F_{p} and two of \{X,Y,Z \} are in F then third is also.
  3. A retract of an object in F is also in F.

One can prove that if $X \in F_{p}$ and K(n)_{*}(X)=0 then K(i)_{*}(X)=0 for i<n. We say that X is of type n if K(n)_{*}(X)\neq 0 but K(n-1)_{*}(X)=0.Let C_{\geq n} be the category of spectra in F_{p} of type \geq n. A major theorem in homotopy theory which can be proven using a weaker version of the nilpotence theorem is the Thick Subcategory Theorem:

Thick Subcategory Theorem:

The thick subcategories of F_{p} are exactly C_{\geq n} for 0 \leq n <\infty.

One can find a proof of this fact in Lurie’s notes.

 

 

 

 

 

 

 

 

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