Observation 8.4.20. Let \(R\) be an associative ring spectrum and let \(S \subseteq \pi _*(R)\) satisfy the left Ore condition. Then:
- (1)
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For \(M \in \LMod _R\) we have \(S^{-1}M = 0\) if and only if \(M\) is \(S\)-nilpotent.
- (2)
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A map \(f\colon M \to N\) of left \(R\)-modules becomes an isomorphism after \(S\)-localization if and only if \(\cofib (f)\) is \(S\)-nilpotent.
- (3)
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If \(T\) is an \(S\)-local module and \(f\colon M \to T\) is a map with \(S\)-nilpotent cofiber, then \(f\) exhibits \(T\) as the \(S\)-localization of \(M\).
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 \(\eta _M\colon M\to S^{-1}M\). □
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