Bousfield Localization (Lecture 20)

April 27, 2010
Let C be a full subcategory of the category Sp of spectra, which is closed under shifts and homotopy
colimits and satisfies the following technical condition:(∗) There exists a small subcategory C0 ⊆ C which generates C under homotopy colimits.
In this case, the inclusion C ⊆ Sp preserves homotopy colimits; using a version of the adjoint functor
theorem one deduces that this inclusion admits a right adjoint G (at the level of homotopy categories). We
can think of G as a functor from Sp to itself, which takes values in C.
Remark 1. Roughly speaking, if X is a spectrum then we want to define G(X) to be the homotopy colimitof all objects Y ∈ C with a map to X. Condition (∗) is used to make this homotopy colimit sensible (thatis, to replace it by a homotopy colimit indexed by a small category).
For every spectrum X, we have a counit map v : G(X) → X. We let L(X) denote the cofiber of v, sothat we have a cofiber sequence
G(X) → X → L(X).
By construction, for every object Y ∈ C, the map of function spectra G(X)Y → X Y is a homotopy equivalence; it follows that L(X)Y ' 0.
Definition 2. A spectrum X is C-local if every map Y → X is nullhomotopic when Y ∈ C. We denote thecategory of C-local spectra by C⊥ .
Remark 3. The full subcategory C⊥ ⊆ Sp is stable under shifts and homotopy limits.
The above analysis shows that for every X, the spectrum L(X) is C-local. Moreover, for every C-local
spectrum Z, we have Z G(X) ' 0, so that the map Z L(X) → Z X is a homotopy equivalence. It follows thatL can be viewed as a left adjoint to the inclusion C⊥ ⊆ Sp.
Example 4 (Bousfield). Fix a spectrum E. We say that another spectrum X is E-acyclic if the smashproduct X ⊗ E is zero. The collection CE of E-acyclic spectra is clearly stable under shifts and homotopy
colimits, and one can show that it satisfies (∗). We say that a spectrum X is E-local if every map Y → X
is nullhomotopic whenever Y is E-acyclic. The above analysis shows that every spectrum X sits in anessentially unique cofiber sequence
GE (X) → X → LE (X)
where G(X) is E-acyclic and LE (X) is E-local. The functor LE is called Bousfield localization with respect
to E. The map X → LE (X) is characterized up to equivalence by two properties:(a) The spectrum LE (X) is E-local.
(b) The map X → LE (X) is an E-equivalence: that is, it induces an isomorphism on E-homology groupsE∗ (X) ' E∗ LE (X).

1

Example 5. Let E be a ring spectrum. If X is an E-module spectrum, then X is E-local. Indeed, supposethat Y is E-acyclic and we are given a map f : Y → X. Then f can be written as the composition
fY → X → E ⊗ X → X.
The composition of the first pair of morphisms factors as a compositionid ⊗f

Y → E ⊗ Y → E ⊗ X,
and is therefore nullhomotopic since E ⊗ Y ' 0.
Remark 6. Let E be an A∞ -ring spectrum and X an arbitrary spectrum, and let X • be the cosimplicialspectrum given by X n = E ⊗n+1 ⊗ X. Each X n is E-local, so the totalization lim X • of X • is E-local. It←−follows that the canonical map X → lim X • factors through a map α : LE X → lim X • . In many cases, one←−←−
can show that α is a homotopy equivalence: that is, the cosimplicial object X • is a means of computing theE-localization of X.
Example 7. Let E be the Eilenberg-MacLane spectrum H Q. Then a spectrum X is E-acyclic if and only
if the homotopy groups π∗ X consist entirely of torsion. A spectrum X is E-local if and only if the homotopy
groups π∗ X are rational vector spaces.
Example 8. The theory of Bousfield localization works in a very general context. For example, rather thanworking with spectra, we can work with chain complexes of abelian groups. Fix a prime number p. We
say that a projective chain complex A• is Z/pZ-acyclic if A• ⊗ Z/pZ is nullhomotopic: equivalently, A• isZ/pZ-acyclic if each homology group Hn (A• ) is a Z[ p1 ]-module. We say that A• is Z/pZ-local if every map
from a projective Z/pZ-acyclic chain complex into A• is nullhomotopic.
b• to be the homotopy limit
For any projective chain complex A• , we define its completion Alim A• ⊗ Z/pn Z.
←−nb• is Z/pZ-local. On the other hand,
As a homotopy limit of Z/pZ-local chain complexes, we conclude that Abb• cana simple calculation shows that the map A• → A• induces a quasi-isomorphism modulo p, so that A
be identified with the Z/pZ-localization of A• .
In general, it is good to think of Bousfield localization as involving a mix of Examples 7 and 8. In
algebro-geometric terms, it can behave sometimes like restriction to an open subscheme (as in Example 7)
and sometimes like completion along a closed subscheme (Example 8). Our next goal is to describe Bousfield
localizations of the first type more precisely.
Lemma 9. Let C, C⊥ , G, and L be as above. The following conditions are equivalent:(1) The subcategory C⊥ ⊆ Sp is stable under homotopy colimits.
(2) The functor L preserves homotopy colimits.
(3) The functor G preserves homotopy colimits.
(4) The functor L has the form L(X) = K ⊗ X for some spectrum K.
Proof. We first prove (1) ⇒ (2). Assume C⊥ is stable under homotopy colimits. For any diagram of spectra{Xα }, we have canonical maps
γβlim Xα → lim L(Xα ) → L lim Xα .
−→−→−→
The fiber of γ belongs to C (since C is stable under homotopy colimits), and lim L(Xα ) ∈ C⊥ by (1). It−→follows that β is an equivalence.
2

To prove that (2) ⇒ (1), we note that if {Xα } is a diagram in C⊥ , then L(lim Xα ) ' lim L(Xα ) ' lim Xα−→−→−→
so that lim Xα ∈ C⊥ .
−→The equivalence of (2) and (3) follows from the cofiber sequence of functors
G → id → L.
Finally, the equivalence of (2) and (4) follows from from the following observation: every functor F : Sp → Sp
which preserves homotopy colimits has the form F (X) ' K ⊗ X, for some spectrum K.
We say that a Bousfield localization L is smashing if it satisfies the equivalent conditions of Lemma 9.
Remark 10. In the situation of Lemma 9, the spectrum K can be recovered as the image L(S) of the sphere
spectrum S under the localization functor L.
Remark 11. Let C ⊆ Sp be a subcategory satisfying the conditions of Lemma 9. Then a spectrum X
belongs to C if and only if L(X) = L(S) ⊗ X ' 0. In other words, C can be identified with the collection ofL(S)-acyclic spectra, so that L = LE for E = L(S).
Example 12. Let C ⊆ Sp be a subcategory which is stable under shifts and homotopy colimits, which isgenerated under homotopy colimits by a subcategory C0 ⊆ C consisting of finite spectra. Then it is easy tosee that C satisfies condition (1) of Lemma 9, so that C determines a smashing localization functor.

3