Corollary 8.2.5. Let \(R\) be a connective associative ring spectrum. Then \(\LMod _R\) admits a t-structure given by the connective and coconnective \(R\)-modules. This t-structure is both left and right complete, and every left \(R\)-module \(M\) satisfies \[ M \quad \simeq \quad \colim (\tau _{\geq 0}M \to \tau _{\geq -1}M \to \dots ) \qquadtext {and} M \quad \simeq \quad \lim (\dots \to \tau _{\leq 1}M \to \tau _{\leq 0}M). \] The same statements hold for \(\RMod _R\).
Proof. The existence of the t-structure is, in light of the previous lemma, an instance of Lemma 8.2.3. By that same lemma, an \(R\)-module is connective or coconnective precisely when its underlying spectrum is, so an infinitely connective or infinitely coconnective \(R\)-module has vanishing homotopy groups and is zero. Thus the t-structure is left and right separated. The forgetful functor \(\LMod _R \to \Sp \) preserves limits and colimits, and \(\Sp _{\geq 0}\) and \(\Sp _{\leq 0}\) are closed under countable products and countable coproducts, respectively. Lemma 6.3.31 therefore upgrades separatedness to left and right completeness. The two displayed formulas express these completeness properties. □
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