Localizations of ring spectra

Proof. For every \(s \in S\) of degree \(d\) and every \(n \in \Z \), consider the shifted multiplication map \[ s[n]\colon R[d+n] \longrightarrow R[n]. \] An \(R\)-module is local with respect to this small collection of morphisms precisely when it is \(S\)-local: allowing all shifts means that locality tests multiplication by \(s\) on every homotopy group. Since \(\Mod _R\) is presentable by Proposition 22.5.5, Theorem 22.2.2(4) provides the asserted reflection \(S^{-1}(-)\).

The symmetric monoidal \(\infty \)-category \(\Mod _R\) is closed, as recalled after Corollary 19.2.17. If \(N\) is \(S\)-local and \(P\) is any \(R\)-module, then \(\iHom _R(P,N)\) is again \(S\)-local. Indeed, under the natural isomorphism \[ \iHom _R(P,N)[d] \iso \iHom _R(P,N[d]), \] multiplication by \(s\) is obtained by applying \(\iHom _R(P,-)\) to the isomorphism \(s\colon N[d] \iso N\). The internal-hom criterion Lemma 14.5.6 therefore gives the symmetric monoidal structure and the symmetric monoidality of the localization.

It remains to prove the smashing formula. The unit \(\eta _R\colon R \to S^{-1}R\) is an \(S\)-local equivalence, and the proof of the internal-hom criterion shows that \(S\)-local equivalences remain so after tensoring with any \(R\)-module. Consequently, \[ \id _M\otimes \eta _R\colon M\longrightarrow M\otimes _R S^{-1}R \] is an \(S\)-local equivalence. Its target is \(S\)-local: multiplication by \(s\) on \(M\otimes _R S^{-1}R\) is obtained by tensoring \(\id _M\) with the isomorphism \(s\colon (S^{-1}R)[\abs {s}]\iso S^{-1}R\). A local equivalence from \(M\) to a local object exhibits that object as the reflection of \(M\), which proves the claim. □

Proof. By Proposition 8.4.4, the reflection \[ S^{-1}(-)\colon \Mod _R \longrightarrow \Mod _R^{\Loc (S)} \] is a symmetric monoidal Bousfield localization. The Part II result Corollary 19.2.17 therefore equips \(S^{-1}R\) with a commutative \(R\)-algebra structure and characterizes it as initial among commutative \(R\)-algebras whose underlying modules are \(S\)-local. Under the equivalence \[ \CAlg (\Mod _R)\simeq \CAlg (\Sp )_{R/}, \] this gives the asserted morphism \(\eta \colon R\to R[S^{-1}]\) of commutative ring spectra.

Let \(f\colon R\to A\) be a morphism of commutative ring spectra. Its underlying \(R\)-module is \(S\)-local if and only if \(\pi _*(f)\) sends every \(s\in S\) to a unit. One direction is immediate. Conversely, suppose that \(A\) is \(S\)-local and let \(s'=\pi _*(f)(s)\) have degree \(d\). Since multiplication by \(s'\) is bijective, there is an element \(u\in \pi _{-d}(A)\) such that \(s'u=1\). Graded commutativity gives \(us'=(-1)^d\), so \((-1)^du\) is a left inverse of \(s'\), while \(u\) is a right inverse. Since left and right inverses coincide, \(s'\) is a unit.

The initial property of \(S^{-1}R\) among commutative \(R\)-algebras with local underlying module now gives the claimed factorization property. Conversely, every morphism that factors through \(R[S^{-1}]\) inverts \(S\), since the image of \(S\) in \(\pi _*(R[S^{-1}])\) consists of units and ring morphisms preserve units. □

Proof. The transition maps are shifts of the module map of Observation 8.4.2, so the telescope is a colimit in \(\Mod _R\). Since homotopy groups preserve filtered colimits by Lemma 4.4.28, we have \[ \pi _k(M[s^{-1}]) \cong \colim \bigl (\pi _k(M)\xrightarrow {\ s\cdot \ }\pi _{k+d}(M) \xrightarrow {\ s\cdot \ }\pi _{k+2d}(M)\longrightarrow \cdots \bigr ), \] which is the degree-\(k\) part of \(\pi _*(M)[s^{-1}]\). In particular, multiplication by \(s\) on \(M[s^{-1}]\) is an isomorphism, so the telescope is \(S\)-local.

Let \(N\) be any \(S\)-local \(R\)-module. Mapping out of the telescope gives \[ \Hom _{\Mod _R}(M[s^{-1}],N) \simeq \lim _j\Hom _{\Mod _R}(M[-jd],N). \] Every transition map in this limit is an equivalence, because under the shift adjunction it is induced by the isomorphism \(s\colon N[d]\iso N\). Evaluation at the zeroth term therefore gives an equivalence \[ \Hom _{\Mod _R}(M[s^{-1}],N) \iso \Hom _{\Mod _R}(M,N), \] induced by precomposition with the canonical map \(M\to M[s^{-1}]\). This is the universal property of the \(S\)-localization. □

Proof. Throughout we use that for a homogeneous element \(s \in \pi _d(R)\) the operation \(x \mapsto s \cdot x\) is a natural transformation \(\pi _k(-) \to \pi _{k+d}(-)\) of functors \(\LMod _R \to \Ab \), and that \(\pi _*\) carries exact sequences of left \(R\)-modules to long exact sequences of graded \(\pi _*(R)\)-modules.

Both subcategories evidently contain \(0\) and are closed under shifts. For closure under cofibers, consider an exact sequence \(M' \to M \to M''\) in \(\LMod _R\).

If \(M'\) and \(M\) are \(S\)-local, then in the long exact sequence \[ \pi _*(M') \to \pi _*(M) \to \pi _*(M'') \to \pi _{*-1}(M') \to \pi _{*-1}(M) \] multiplication by \(s\) acts as an isomorphism on four of the five terms, so the five lemma shows that it is an isomorphism on \(\pi _*(M'')\) as well; hence \(M''\) is \(S\)-local. The same argument applied to the rotated sequence shows closure under fibers, so \(\LMod _R^{\Loc (S)}\) is stable.

If \(M'\) and \(M\) are \(S\)-nilpotent, let \(x \in \pi _n(M'')\). Its image \(\partial (x) \in \pi _{n-1}(M')\) is annihilated by some \(s \in S\), so \(\partial (s\cdot x) = s \cdot \partial (x) = 0\) and therefore \(s\cdot x\) lifts to an element \(y \in \pi _{n+\abs {s}}(M)\). Choosing \(t \in S\) with \(t\cdot y = 0\), we obtain \((ts)\cdot x = t\cdot (s \cdot x) = 0\), and \(ts \in S\) since \(S\) is multiplicative. Hence \(M''\) is \(S\)-nilpotent.

Similarly, suppose that \(M'\) and \(M''\) are \(S\)-nilpotent and let \(x\in \pi _n(M)\). Choose \(s\in S\) which annihilates the image of \(x\) in \(\pi _n(M'')\). Then \(s\cdot x\) is the image of an element \(y\in \pi _{n+\abs {s}}(M')\). Choosing \(t\in S\) with \(t\cdot y=0\) gives \((ts)\cdot x=0\). Thus \(M\) is \(S\)-nilpotent as well, so \(\LMod _R^{S\dnil }\) is stable and closed under extensions.

Being stable, both subcategories are in particular closed under finite coproducts: a biproduct \(M_1 \oplus M_2\) sits in an exact sequence \(M_1 \to M_1 \oplus M_2 \to M_2\), and both subcategories are closed under extensions. Consequently both are closed under all finite colimits.

Next, both are closed under filtered colimits, since \(\pi _*\) preserves these by Lemma 4.4.28. Indeed, a filtered colimit of isomorphisms is an isomorphism, which handles the \(S\)-local case; and in the \(S\)-nilpotent case every element of \(\pi _n(\colim _i M_i) \cong \colim _i \pi _n(M_i)\) is the image of some \(x \in \pi _n(M_i)\), and any \(s \in S\) annihilating \(x\) also annihilates its image. Since an arbitrary coproduct is the filtered colimit of its finite subcoproducts, both subcategories are closed under small coproducts. As an \(\infty \)-category admitting pushouts and small coproducts admits all small colimits, this completes the proof. □

Proof. Assume first that (1) is satisfied. Let \(x \in \pi _n(R/Rs)\); we wish to show that it is annihilated by some element of \(S\). Using the exact sequence \[ \pi _n(R/Rs) \xrightarrow {\phi } \pi _{n-d-1}(R) \xrightarrow {\cdot s} \pi _{n-1}(R), \] we deduce that \(\phi (x) \in \pi _{n-d-1}(R)\) is annihilated by right multiplication by \(s\). Using condition (b) of the left Ore condition, we deduce that there exists an element \(t \in S\) of degree \(d'\) such that \(0 = t\phi (x) = \phi (tx) \in \pi _{n+d'-d-1}(R)\). Using the exactness of the sequence \[ \pi _{n+d'}(R) \xrightarrow {\psi } \pi _{n+d'}(R/Rs) \xrightarrow {\phi } \pi _{n+d'-d-1}(R), \] we conclude that \(tx = \psi (y)\) for some \(y \in \pi _{n+d'}(R)\). Using condition (a) of the left Ore condition, we can find an element \(u \in S\) of degree \(d''\) and an element \(z \in \pi _{n+d'+d''-d}(R)\) such that \(uy=zs\). It follows that the image of \(uy\) in \(\pi _{n+d'+d''}(R/Rs)\) vanishes, so that \((ut)x = u(tx) = u\psi (y) = \psi (uy)=0\). This completes the proof that (1) implies (2).

Now suppose that (2) is satisfied. We will show that \(S\) satisfies conditions (a) and (b) of the left Ore condition. For (a), suppose that \(x \in \pi _*(R)\) is a homogeneous element of degree \(n\) and \(s \in S\) has degree \(d\). We wish to show that there exist \(y \in \pi _*(R)\) and \(t \in S\) such that \(tx=ys\). By \(S\)-nilpotence, the image of \(x\) in \(\pi _n(R/Rs)\) is annihilated by multiplication by some homogeneous element \(t \in S\) of degree \(d'\). It follows that \(tx\) belongs to the kernel of the map \(\pi _{n+d'}(R) \to \pi _{n+d'}(R/Rs)\), and therefore to the image of the map \(\cdot s\colon \pi _{n+d'-d}(R) \to \pi _{n+d'}(R)\) given by right multiplication by \(s\). Thus \(tx=ys\) for some \(y\in \pi _{n+d'-d}(R)\), which proves (a).

We now verify (b). It suffices to show that if \(x \in \pi _n(R)\) is annihilated by right multiplication by some element \(s \in S\) of degree \(d\), then \(tx=0\) for some \(t \in S\). The exactness of the sequence \[ \pi _{n+d+1}(R/Rs) \xrightarrow {\phi } \pi _n(R) \xrightarrow {\cdot s} \pi _{n+d}(R) \] shows that \(x = \phi (y)\) for some \(y \in \pi _{n+d+1}(R/Rs)\). Since \(R/Rs\) is \(S\)-nilpotent, there exists an element \(t \in S\) such that \(ty=0\), from which it follows immediately that \(tx = t\phi (y) = \phi (ty) = 0\). □

Proof. Let \(s \in S\) be an element of degree \(d\). Applying the mapping spectrum functor \(\hom _{\LMod _R}(-,N)\) to the exact sequence \[ R[d] \xrightarrow {\cdot s} R \to R/Rs \] and using the identifications \(\hom _{\LMod _R}(R,N) \simeq N\) and \(\hom _{\LMod _R}(R[d],N)\simeq N[-d]\) gives an exact sequence of spectra \[ \hom _{\LMod _R}(R/Rs,N)\longrightarrow N \xrightarrow {\ s\cdot -\ }N[-d]. \] The second map is left multiplication by \(s\): precomposition with right multiplication \(\cdot s\colon R[d]\to R\) sends a map \(f\colon R \to N\) to the map determined by \(f(s) = s \cdot f(1)\).

Condition (1) says precisely that the second map is an isomorphism for every \(s\in S\), which by exactness is equivalent to the vanishing of \(\hom _{\LMod _R}(R/Rs,N)\). Since a spectrum vanishes if and only if all the animae \(\Omega ^{\infty }(-[-n])\) do, this is in turn equivalent to \(\Hom _{\LMod _R}((R/Rs)[n],N)\) being contractible for all \(n\). This proves \((1) \Leftrightarrow (2)\).

To prove the implication \((3) \Rightarrow (2)\), we observe that since \(S\) satisfies the left Ore condition, the module \(R/Rs\) is \(S\)-nilpotent for every \(s \in S\), as established in Lemma 8.4.14. As \(\LMod _R^{S\dnil }\) is closed under shifts (Observation 8.4.10), condition (3) applied to \(M = (R/Rs)[n]\) immediately yields condition (2).

We now prove the implication \((2) \Rightarrow (3)\). Let \(M\) be an \(S\)-nilpotent left \(R\)-module, and write it as the colimit of a sequence \[ 0 = M_0 \to M_1 \to M_2 \to \cdots \] whose successive fibers are coproducts of modules \((R/Rs_\alpha )[n_\alpha ]\), as provided by Lemma 8.4.15. Since hom animae turn colimits in the first variable into limits, we have \[ \Hom _{\LMod _R}(M,N) \iso \lim _{i} \Hom _{\LMod _R}(M_i,N). \] It will therefore suffice to show that each hom anima \(\Hom _{\LMod _R}(M_i,N)\) is contractible. We proceed by induction on \(i\). The case \(i=0\) is clear, since \(M_0=0\). For the inductive step, assume that \(\Hom _{\LMod _R}(M_i,N)\) is contractible. Let \(K_i\) denote the fiber of the map \(M_i \to M_{i+1}\). By construction, we have an exact sequence of modules \(K_i \to M_i \to M_{i+1}\). Applying the functor \(\Hom _{\LMod _R}(-,N)\) yields a fiber sequence of animae \[ \Hom _{\LMod _R}(M_{i+1},N) \to \Hom _{\LMod _R}(M_i,N) \to \Hom _{\LMod _R}(K_i,N). \] By the inductive hypothesis, the middle term is contractible. It therefore suffices to show that the right-hand term, \(\Hom _{\LMod _R}(K_i,N)\), is also contractible. By construction, \(K_i\) is a coproduct of modules of the form \((R/Rs_\alpha )[n_\alpha ]\). Since hom animae send coproducts in the first variable to products, we have \[ \Hom _{\LMod _R}(K_i,N) \iso \prod _\alpha \Hom _{\LMod _R}((R/Rs_\alpha )[n_\alpha ],N). \] By condition (2), each factor in this product is contractible. It follows that the entire product is contractible, which completes the inductive step and the proof. □

Proof sketch. By Lemma 8.4.15, the \(\infty \)-category \(\LMod _R^{S\dnil }\) is the localizing subcategory generated by the set of objects \(\{ (R/Rs)[n] \mid s \in S, n \in \Z \}\). It is therefore presentable by Proposition 22.2.7(2). Its inclusion into \(\LMod _R\) preserves colimits by Observation 8.4.10, so Theorem 22.2.5 gives a right adjoint \(G\). For any module \(M\), the counit of this adjunction gives a map \(G(M) \to M\). We define \(M' := G(M)\) and we let \(M''\) be the cofiber of this map. It remains to show that \(M''\) is \(S\)-local, for which we use criterion (3) from Proposition 8.4.16. Let \(N\) be an \(S\)-nilpotent left \(R\)-module. Applying \(\hom _{\LMod _R}(N,-)\) provides an exact sequence of spectra \[ \hom _{\LMod _R}(N,M') \to \hom _{\LMod _R}(N,M) \to \hom _{\LMod _R}(N,M''). \] Both functors in the adjunction are exact, so the adjunction is enriched in spectra. The first map is therefore an isomorphism, and it follows that \(\hom _{\LMod _R}(N,M'') = 0\), as desired. □

Proof. By Proposition 8.4.17 there is an exact sequence \(M' \to M \to M''\) of left \(R\)-modules such that \(M'\) is \(S\)-nilpotent and \(M''\) is \(S\)-local. We claim that \(M''\) is an \(S\)-localization. Indeed, for any other \(S\)-local left \(R\)-module \(N\), mapping into \(N\) provides a fiber sequence of animae \[ \Hom _{\LMod _R}(M'',N) \to \Hom _{\LMod _R}(M,N) \to \Hom _{\LMod _R}(M',N). \] But since \(M'\) is \(S\)-nilpotent, the anima \(\Hom _{\LMod _R}(M',N)\) is contractible by part (3) of Proposition 8.4.16. It follows that the first map is an equivalence, as desired. □

Proof. (1) Write \(M' \to M \to S^{-1}M\) for the decomposition of Proposition 8.4.17. If \(M\) is \(S\)-nilpotent, then the counit \(M' = G(M) \to M\) of the coreflection is an isomorphism, so its cofiber \(S^{-1}M\) vanishes. Conversely, if \(S^{-1}M = 0\) then \(M' \to M\) is an isomorphism, so \(M\) is \(S\)-nilpotent.

(2) The functor \(S^{-1}(-)\) is a left adjoint between stable \(\infty \)-categories, hence preserves cofibers. Thus \(S^{-1}(f)\) is an isomorphism if and only if \(S^{-1}\cofib (f) \simeq \cofib (S^{-1}f)\) vanishes, which by (1) happens if and only if \(\cofib (f)\) is \(S\)-nilpotent.

(3) By (2) the map \(S^{-1}(f)\colon S^{-1}M \to S^{-1}T\) is an isomorphism, and the unit \(\eta _T\colon T \to S^{-1}T\) is an isomorphism because \(T\) is already \(S\)-local. Naturality of the unit gives \(\eta _T\circ f=S^{-1}(f)\circ \eta _M\). It follows that the isomorphism \(\eta _T^{-1}\circ S^{-1}(f)\colon S^{-1}M\to T\) identifies \(f\) with the localization map \(\eta _M\colon M\to S^{-1}M\). □