In algebra, one commonly studies rings and modules through their localizations, in which one inverts certain non-zero elements in the ring. The terminology is inspired by the geometric perspective of this operation: it corresponds to studying the ‘local’ behavior of a variety/scheme around a point.
In the case of abelian groups, this operation corresponds to inverting collections of prime numbers. Given an abelian group \(A\) and a natural number \(n\), there is a multiplication-by-\(n\) map \(n\colon A \to A\), sending \(a\) to the \(n\)-fold sum \(na\). Abelian groups for which this map is an isomorphism for certain \(n\) are usually easier to study than general abelian groups:
- If multiplication by \(n\) is an isomorphism for all \(n > 0\), then \(A\) acquires a unique structure of a rational vector space, hence is simply a direct sum of copies of \(\Q \).
- If for some prime \(p\), multiplication by \(q\) is an isomorphism for all primes \(q \neq p\), then \(A\) is called \(p\)-local. In particular, \(A\) does not admit \(q\)-torsion for any \(q \neq p\).
More generally, for a set \(P\) of primes we write \(A[P^{-1}]\) for the universal abelian group under \(A\) on which multiplication by every \(p \in P\) is invertible. When \(P\) consists of all primes, this is the rationalization \(A_{\Q }\) of \(A\). When \(P\) consists of all primes except a single prime \(p\), it is the \(p\)-localization \(A_{(p)}\) of \(A\). Some questions about \(A\) can be reduced to questions about \(A_{\Q }\) or \(A_{(p)}\).
The goal of this section is to introduce an analogous theory of localizations for spectra, and to exhibit it as a special case of the general formalism of Bousfield localizations introduced in the previous section.
Notation 7.2.1. Let \(C\) be a semiadditive \(\infty \)-category. We saw in Proposition 5.3.25 that the hom anima \(\Hom _C(X,Y)\) admits a canonical structure of a commutative monoid for all \(X,Y \in C\). In particular, there is for any \(n \in \N \) a multiplication-by-\(n\) map \[ n \cdot -\colon \Hom _C(X,Y) \to \Hom _C(X,Y), \] given explicitly by sending a map \(f\colon X \to Y\) to the composite \[ X \xrightarrow {\Delta } \bigoplus _n X \xrightarrow {\bigoplus _n f} \bigoplus _n Y \xrightarrow {\nabla } Y. \] For \(f = \id _X \colon X \to X\), we refer to this map as multiplication by \(n\) on \(X\), and denote it by \[ n\colon X \to X. \]
Note that when \(C = \Ab \) is the category of abelian groups, this precisely recovers the ordinary multiplication-by-\(n\) map.
We record the following compatibility because we will apply it both to spectra and to their homotopy groups.
Exercise 7.2.2. Show that any semiadditive functor \(F\colon C \to D\) between semiadditive \(\infty \)-categories preserves the multiplication-by-\(n\) maps.
We now specialize to spectra. Let \(P \subseteq \N \) be a set of primes. We say a spectrum \(X\) is \(P\)-divisible if multiplication by \(p\) on \(X\) is an isomorphism for every \(p\in P\). We write \[ \Sp [P^{-1}] \subseteq \Sp \] for the full subcategory of \(P\)-divisible spectra.
Definition 7.2.3. Consider a spectrum \(X\).
- For \(n > 0\), we define \(X[\frac {1}{n}]\) as the colimit \[ X[\frac {1}{n}] \; := \; \colim (X \xrightarrow {n} X \xrightarrow {n} X \xrightarrow {n} \dots ) \qin \Sp . \]
- If \(P\) consists of finitely many primes, we set \(N := \prod _{p \in P} p\) and set \[ X[P^{-1}] \; := \; X[\frac {1}{N}] \qin \Sp . \]
- If \(P\) consists of infinitely many primes, say \(p_1, p_2, p_3, \dots \), we define \(X[P^{-1}]\) as the colimit \[ X[P^{-1}] \; := \; \colim (X \xrightarrow {\cdot p_1} X \xrightarrow {\cdot p_1p_2} X \xrightarrow {\cdot p_1p_2p_3} \dots ) \qin \Sp . \]
When \(P\) consists of all primes, we refer to \(X[P^{-1}]\) as the rationalization of \(X\) and denote it by \(X_{\Q }\).
When \(P\) consists of all primes except a single prime \(p\), we refer to \(X[P^{-1}]\) as the \(p\)-localization of \(X\), and denote it by \(X_{(p)}\).
Note that \(X[P^{-1}]\) always comes with a canonical map \(X \to X[P^{-1}]\), by including the first entry of the colimit diagram. Moreover, for every spectrum \(X\), the functor \(X \otimes -\colon \Sp \to \Sp \) preserves colimits, and so there is an isomorphism \[ X[P^{-1}] \quad \cong \quad X \otimes \S [P^{-1}]. \]
Lemma 7.2.4. Let \(X\) be a spectrum. Then there is an isomorphism \[ \pi _*(X[P^{-1}]) \cong \pi _*(X)[P^{-1}]. \]
Proof. In each case of the definition, \(X[P^{-1}]\) is a filtered colimit whose transition maps are multiplication by products of primes in \(P\). By Exercise 7.2.2, the functors \(\pi _n\colon \Sp \to \Ab \) carry these to the corresponding multiplication maps of abelian groups. Since they preserve filtered colimits by Lemma 4.4.28, and since the resulting colimit of abelian groups has the universal property defining localization, we obtain \[ \pi _*(X[P^{-1}]) \cong \pi _*(X)[P^{-1}]. \qedhere \] □
Proof. We need to show that for every prime \(p\in P\), the map \(\cdot p\colon X[P^{-1}] \to X[P^{-1}]\) is an isomorphism of spectra. We may check this on homotopy groups, where it becomes the statement that the map \[ p\colon \pi _*(X)[P^{-1}] \to \pi _*(X)[P^{-1}] \] is an isomorphism. This holds by elementary properties of localization of abelian groups. □
Recall that an abelian group \(A\) is called \(P\)-power torsion if every element of \(A\) is annihilated by a positive integer all of whose prime factors lie in \(P\). It is not difficult to see that \(A\) is \(P\)-power torsion if and only if \(A[P^{-1}] = 0\). Together with Lemma 7.2.4, this implies:
Corollary 7.2.6. A spectrum \(X\) is \(\S [P^{-1}]\)-acyclic if and only if the homotopy groups \(\pi _*(X)\) are \(P\)-power torsion. □
We next show that the assignment \(X \mapsto X[P^{-1}]\) is a reflection onto the subcategory of \(P\)-divisible spectra.
Proposition 7.2.7. Let \(X\) be a spectrum and let \(Y\) be a \(P\)-divisible spectrum. Then precomposition with the canonical map \(X \to X[P^{-1}]\) induces an isomorphism of mapping spectra \[ \hom (X[P^{-1}],Y) \iso \hom (X,Y). \] In particular, the inclusion \(\Sp [P^{-1}] \hookrightarrow \Sp \) admits a left adjoint given by \[ (-)[P^{-1}]\colon \Sp \to \Sp [P^{-1}]. \]
Proof. The mapping spectrum \(\hom (X[P^{-1}],Y)\) is the limit obtained by applying \(\hom (-,Y)\) to the defining diagram for \(X[P^{-1}]\). The transition maps in this limit are given by precomposition with the corresponding multiplication maps on \(X\). By bilinearity of composition in spectra, these maps agree with postcomposition by the corresponding multiplication maps on \(Y\).
All transition maps are induced by multiplication by products of primes in \(P\) on \(Y\), hence are isomorphisms since \(Y\) is \(P\)-divisible. The limit is therefore isomorphic to the first term \(\hom (X,Y)\).
Passing to underlying mapping animae and using Corollary 7.2.5 gives the claim. □
The previous proposition establishes the functor \((-)[P^{-1}]\colon \Sp \to \Sp \) as the reflector onto the \(P\)-divisible spectra. We will now identify it with the \(E\)-localization functor for \(E = \S [P^{-1}]\).
Proof. First assume \(X\) is \(P\)-divisible. To see that \(X\) is \(\S [P^{-1}]\)-local, consider an \(\S [P^{-1}]\)-acyclic spectrum \(Y\). We must show that \(\hom (Y,X) = 0\). This follows from the following sequence of isomorphisms: \[ \hom (Y,X) \cong \hom (Y[P^{-1}],X) \cong \hom (0,X) \cong 0; \] Here the first isomorphism follows from Proposition 7.2.7, and the second from the observation that \(Y[P^{-1}] \cong Y\otimes \S [P^{-1}]\).
Conversely, assume that \(X\) is \(\S [P^{-1}]\)-local. To show that \(p\colon X \to X\) is an isomorphism, it suffices by Lemma 7.1.6 to show that the map \(p\colon X[P^{-1}] = X \otimes \S [P^{-1}] \to X \otimes \S [P^{-1}] = X[P^{-1}]\) is an isomorphism, which is the content of Corollary 7.2.5. □
Corollary 7.2.9. For every spectrum \(X\), the map \(X \to X[P^{-1}]\) is an \(\S [P^{-1}]\)-localization of \(X\).
Proof. By Corollary 7.1.8, a map \(X \to Y\) is an \(\S [P^{-1}]\)-localization if and only if it exhibits \(Y\) as a reflection into the subcategory of \(\S [P^{-1}]\)-local spectra. By Lemma 7.2.8, this subcategory agrees with the \(P\)-divisible spectra, and so the claim is an instance of Proposition 7.2.7. □
For \(E = \S [P^{-1}]\), the previous corollary implies that the \(E\)-localization of \(X\) is given by the map \(X \cong \S \otimes X \to L_E(\S ) \otimes X\). Localizations of this form have a special name:
Definition 7.2.10. The localization functor \(L_E\colon \Sp \to \Sp _E\) is called a smashing localization if the spectrum \(L_E\S \otimes X\) is \(E\)-local for every spectrum \(X\).
Exercise 7.2.11. Assume \(L_E\) is a smashing localization. Construct an isomorphism \(L_E(X) \iso L_E(\S ) \otimes X\) for every spectrum \(X\).
We next record the expected fact that reducing a spectrum modulo a power of \(p\) leaves only \(p\)-primary information, in the sense that all other primes act invertibly. The analogous claim for abelian groups is clear, since any prime different from \(p\) is invertible modulo \(p^n\). The following proposition extends this claim to spectra by expressing each homotopy group of \(X/p^n\) as an extension of two groups killed by \(p^n\), allowing us to reduce to the abelian setting.
Proposition 7.2.12. Let \(X\) be a spectrum, let \(p\) be a prime number and let \(n \geq 1\). Then the spectrum \(X/p^n := \cofib (X \xrightarrow {\cdot p^n} X)\) is \(p\)-local.
Proof. Let \(q\) be a prime number different from \(p\). We must show that the map \(q\colon X/p^n \to X/p^n\) is an isomorphism, which we may check on homotopy groups.
The long exact sequence on homotopy groups associated to the exact sequence \(X \xrightarrow {\cdot p^n} X \to X/p^n\) contains, for every \(m \in \Z \), a natural short exact sequence \[ 0 \to \pi _m(X)/p^n \to \pi _m(X/p^n) \to \pi _{m-1}(X)[p^n] \to 0. \] Here \(\pi _m(X)/p^n\) denotes the cokernel of multiplication by \(p^n\), and \(\pi _{m-1}(X)[p^n]\) denotes the kernel of multiplication by \(p^n\). By Exercise 7.2.2, homotopy groups carry multiplication by \(q\) on spectra to multiplication by \(q\) on abelian groups, so this natural short exact sequence is compatible with multiplication by \(q\).
Now multiplication by \(q\) is an isomorphism on every abelian group killed by \(p^n\): choose integers \(a,b\) with \(aq+bp^n=1\), and then multiplication by \(a\) is inverse to multiplication by \(q\). It follows that multiplication by \(q\) is an isomorphism on the two outer terms of the short exact sequence above. By the short five lemma, it is therefore also an isomorphism on \(\pi _m(X/p^n)\).
Since this holds for all \(m\), multiplication by \(q\) is an isomorphism of spectra. As this is true for every prime \(q\neq p\), the spectrum \(X/p^n\) is \(p\)-local. □
Remark 7.2.13. The previous argument shows more generally that \(X/N\) is \(P\)-divisible whenever \(\gcd (q,N) = 1\) for every \(q \in P\).
Theorem 7.2.14 (Serre). The rational sphere \(\S _{\Q }\) is isomorphic to the Eilenberg–MacLane spectrum \(H\Q \) of the rational numbers.
Proof. Consider the map \(\S \to H\Q \) corresponding to \(1 \in \Q \cong \pi _0(H\Q )\). All primes act invertibly on the homotopy groups of \(H\Q \), so \(H\Q \) is \(\S _{\Q }\)-local by Lemma 7.2.8, and this map therefore uniquely refines to a map \(\S _{\Q } \to H\Q \). We have to show that this map induces isomorphisms on homotopy groups. By Lemma 7.2.4, the induced map on \(\pi _0\) is the isomorphism \(\Q \cong \pi _0(\S _{\Q }) \to \pi _0(H\Q ) \cong \Q \) which sends \(1\) to \(1\). It thus remains to show that \(\pi _n(\S _{\Q }) = 0\) for all \(n \neq 0\). For \(n > 0\), the stable homotopy group \(\pi _n(\S ) = \pi _n^{\st }\) is finite by a theorem of Serre (1953), while for \(n < 0\) it vanishes since \(\S \) is connective. In either case it is killed by rationalization. □
Definition 7.2.15. We write \(\Sp _{\Q } \subseteq \Sp \) for the full subcategory of rational spectra, i.e. the \(\S _{\Q }\)-local spectra, or equivalently the spectra on which all primes act invertibly. By Serre’s theorem, these are also precisely the \(H\Q \)-local spectra.
Proposition 7.2.16. For every rational spectrum \(X\), there is a noncanonical isomorphism \[ \bigoplus _{n \in \Z } H(\pi _n X)[n] \iso X. \]
Proof. For every integer \(n \in \Z \), we pick a basis of \(\pi _n(X)\) as a rational vector space. An element of \(\pi _n(X)\) corresponds to a map of spectra \(\S [n] \to X\). Since \(X\) is rational, this uniquely extends to a map \(H\Q [n] = \S _{\Q }[n] \to X\). By taking a direct sum over all basis vectors, this induces a map \(H(\pi _nX)[n] \cong \bigoplus H\Q [n] \to X\) which by construction induces the identity on \(\pi _n(X)\). We now take the direct sum over all \(n \in \Z \). Direct sums of spectra are filtered colimits of finite biproducts, so Lemma 4.4.28 shows that the resulting map \(\bigoplus _{n \in \Z } H(\pi _n X)[n] \to X\) induces an isomorphism on every homotopy group. It is therefore an isomorphism. □
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