Lemma 8.2.3. Let \(C\) be a stably symmetric monoidal \(\infty \)-category equipped with a t-structure for which the symmetric monoidal structure is connectivity-preserving, and let \(R\in \Alg (C_{\geq 0})\). Then \(\LMod _R(C)\) admits a t-structure in which a left \(R\)-module is connective or coconnective precisely when its underlying object in \(C\) is so. The truncation functors are computed on underlying objects.
Proof. The adjunction between the symmetric monoidal inclusion \(C_{\geq 0}\hookrightarrow C\) and its lax symmetric monoidal right adjoint \(\tau _{\geq 0}\) induces, after passing to algebras and modules and taking the fiber over \(R\), an adjunction \[ \LMod _R(C_{\geq 0})\rightleftarrows \LMod _R(C). \] Its right adjoint is computed on underlying objects by \(\tau _{\geq 0}\). Finite limits and colimits of modules are computed on underlying objects by Corollary 19.1.17, so it remains to check the orthogonality and decomposition axioms for a t-structure. If \(X\) is connective and \(Y\) is coconnective, adjunction gives \[ \Hom _{\LMod _R(C)}(X,Y[-1]) \simeq \Hom _{\LMod _R(C_{\geq 0})}(X,\tau _{\geq 0}(Y[-1])) \simeq *. \] Finally, for an arbitrary \(R\)-module \(X\), the counit \(\tau _{\geq 0}X\to X\) in modules has connective source, and its cofiber is coconnective because this is true after forgetting to \(C\). This supplies the required decomposition. □
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