Bousfield localization isolates the information in a spectrum detected by a chosen homology theory.
Fix a spectrum \(E\) throughout this section. As discussed in Definition 4.4.21, it defines the \(E\)-homology groups of an anima: \[ X \mapsto E_*(X) := \pi _*(E \otimes \S [X]). \] For example, taking \(E = HA\) the Eilenberg–MacLane spectrum of an abelian group \(A\) results in singular homology with coefficients in \(A\), while taking \(E = \S \) the sphere spectrum results in the stable homotopy groups.
The homology groups \(E_*(X)\) will generally only see some aspects of the given anima \(X\), and may completely ignore others. Consider for example the case of the projective space \(X = \R \!\mathrm {P}^2\). If we considered the Eilenberg–MacLane spectrum \(E = H\Q \) of the rational numbers, its homology would give no information beyond that of a point: the reduced rational homology of \(\R \!\mathrm {P}^2\) is trivial. But taking \(E = H\Z /2\) does capture interesting information about \(\R \!\mathrm {P}^2\). The point of Bousfield localizations is to focus precisely on those parts of an anima (or spectrum) which are ‘seen by \(E\)’.
Definition 7.1.1. A morphism \(f\colon X \to Y\) is called an \(E\)-equivalence if the induced map \(E \otimes f\colon E \otimes X \to E \otimes Y\) is an isomorphism. We say that a spectrum \(X\) is \(E\)-acyclic if the map \(X \to 0\) is an \(E\)-equivalence, i.e. if \(E \otimes X\) is the zero spectrum.
Recall that a morphism of spectra is an isomorphism if and only if it induces isomorphisms on all homotopy groups. So if we write \(E_*(X) := \pi _*(E \otimes X)\) for the \(E\)-homology of \(X\), generalizing the definition for animae, we see that the \(E\)-equivalences are precisely those maps that induce isomorphisms on \(E\)-homology. Similarly, the \(E\)-acyclic spectra are those with trivial \(E\)-homology.
Example 7.1.2. For \(E = H\Q \), the \(E\)-equivalences are known as the rational equivalences: those that induce isomorphisms on rational homology.
We will now introduce a class of spectra that ‘cannot distinguish between \(E\)-equivalent spectra’:
Definition 7.1.3. A spectrum \(L\) is called \(E\)-local if for every \(E\)-equivalence \(f\colon X \to Y\), the induced map \(f^*\colon \hom (Y,L) \to \hom (X,L)\) on mapping spectra is an isomorphism. We denote by \[ \Sp _E \subseteq \Sp \] the full subcategory spanned by the \(E\)-local spectra.
Equivalently, it suffices to require that \(f\) induce an isomorphism \(\Hom _{\Sp }(Y,L) \to \Hom _{\Sp }(X,L)\) on hom animae for every \(E\)-equivalence \(f\colon X \to Y\). Indeed, every shift of an \(E\)-equivalence is again an \(E\)-equivalence, and applying this condition to all shifts of \(f\) detects all homotopy groups of the map of mapping spectra.
Exercise 7.1.4. Show the following basic properties of \(E\)-local spectra:
- (1)
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A spectrum \(L\) is \(E\)-local if and only if \(\hom (X,L) = 0\) for every \(E\)-acyclic spectrum \(X\);
- (2)
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Given an exact sequence \(X \to Y \to Z\), if two of these three spectra are \(E\)-local then so is the third;
- (3)
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In particular, if \(L\) is \(E\)-local then so are all its shifts \(L[n]\);
- (4)
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A limit \(\lim _i L_i\) of \(E\)-local spectra is \(E\)-local;
- (5)
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A retract of an \(E\)-local spectrum is \(E\)-local.
Exercise 7.1.5. Show that for every spectrum \(Y\), the mapping spectrum \(\hom (E,Y)\) is \(E\)-local. Using this, show that for a morphism \(f\colon X \to X'\) of spectra the following conditions are equivalent:
- (1)
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The morphism \(f\colon X \to X'\) is an \(E\)-equivalence;
- (2)
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The induced map \(f^*\colon \hom (X',L) \to \hom (X,L)\) is an isomorphism for all \(E\)-local spectra \(L\);
- (3)
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The induced map \(f^*\colon \Hom _{\Sp }(X',L) \to \Hom _{\Sp }(X,L)\) is an isomorphism for all \(E\)-local spectra \(L\).
Hint: use that we have \(\pi _n \hom (X,Y) \cong \pi _0 \hom (X,Y[n]) \cong \pi _0 \Hom _{\Sp }(X,Y[n])\), and similarly for \(X'\).
Lemma 7.1.6 (‘\(E_*\)-Whitehead theorem’). Let \(f\colon L \to L'\) be a morphism between \(E\)-local spectra. Then \(f\) is an isomorphism if and only if it is an \(E\)-equivalence.
Proof. By passing to fibers, we may equivalently show that an \(E\)-local spectrum \(L\) is zero if and only if it is \(E\)-acyclic. One direction is clear. For the other direction, assume \(L\) is \(E\)-acyclic. Then by assumption we have \(\hom (L,L) = 0\), hence \(\Hom _{\Sp }(L,L) = *\), hence the identity on \(L\) is the zero map. This implies that \(L\) itself is zero, as desired. □
We will now talk about the universal way of approximating an arbitrary spectrum by an \(E\)-local one:
Definition 7.1.7. A morphism \(l\colon X \to X'\) of spectra is called an \(E\)-localization if \(X'\) is \(E\)-local and the morphism \(l\) is an \(E\)-equivalence.
Corollary 7.1.8. Let \(X'\) be an \(E\)-local spectrum. A morphism \(l\colon X \to X'\) is an \(E\)-localization if and only if for every \(E\)-local spectrum \(Y\) the induced map \[ l^*\colon \Hom _{\Sp }(X',Y) \to \Hom _{\Sp }(X,Y) \] is an isomorphism.
Proof. This is immediate from Exercise 7.1.5. □
Note that the conclusion of the corollary says that \(X'\) is a left adjoint object of \(X\) under the inclusion \(\Sp _E \hookrightarrow \Sp \), in the sense of Definition 21.1.3. In particular, the \(E\)-localization of \(X\) is unique if it exists, and is then denoted by \(L_E(X)\).
Theorem 7.1.9 (Bousfield (1979), Theorem 1.1). For every object \(X\), there exists an \(E\)-localization \(l\colon X \to L_EX\). In particular, the inclusion \(\Sp _E \hookrightarrow \Sp \) admits a left adjoint \[ L_E\colon \Sp \to \Sp _E \] called the \(E\)-localization functor.
Proof. Let \(\Sp ^{E\text {-}\mathrm {acyc}}\subseteq \Sp \) be the full subcategory of \(E\)-acyclic spectra. It is the kernel of the colimit-preserving functor \[ E\otimes -\colon \Sp \to \Sp , \] so it is presentable and its inclusion \(i\colon \Sp ^{E\text {-}\mathrm {acyc}}\hookrightarrow \Sp \) preserves small colimits by Corollary 22.2.4. This kernel is stable, since \(E\otimes -\) is exact. The adjoint functor theorem Theorem 22.2.5 therefore supplies a right adjoint \(\Gamma _E\) to \(i\).
For a spectrum \(X\), define \(L_EX\) by the exact sequence \[ i\Gamma _EX\longrightarrow X\longrightarrow L_EX, \] where the first map is the counit of the adjunction. Let \(A\) be an \(E\)-acyclic spectrum. Full faithfulness of \(i\) and the adjunction give isomorphisms \[ \Hom _{\Sp }(A,i\Gamma _EX) \iso \Hom _{\Sp ^{E\text {-}\mathrm {acyc}}}(A,\Gamma _EX) \iso \Hom _{\Sp }(A,X). \] Their composite is the map induced by the counit. Mapping out of \(A\) carries the defining exact sequence for \(L_EX\) to a fiber sequence of animae, so it follows that \(\Hom _{\Sp }(A,L_EX)=*\). The same argument applies to every shift of \(A\), which is again \(E\)-acyclic. Hence \(\hom (A,L_EX)=0\), and \(L_EX\) is \(E\)-local by Exercise 7.1.4. On the other hand, \(E\otimes i\Gamma _EX=0\), so applying \(E\otimes -\) to the exact sequence shows that \(X\to L_EX\) is an \(E\)-equivalence. Thus it is an \(E\)-localization. □
The proof packages the required size argument into the general theory of presentable categories. Bousfield’s original construction instead builds the reflector explicitly by a transfinite cell-attachment process. The general notion of a Bousfield localization is discussed in Section 21.8.
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