Lemma 6.3.10. Let \(C\) be a stable \(\infty \)-category equipped with a t-structure, and let \(n,m \in \Z \).
- (1)
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If \(X \in C_{\leq m}\), then \(\tau _{\geq n}X \in C_{\leq m}\).
- (2)
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If \(X \in C_{\geq n}\), then \(\tau _{\leq m}X \in C_{\geq n}\).
Proof. We prove (1) first. Let \(X \in C_{\leq m}\). If \(n>m\), then \(\Hom _C(Z,X)=0\) for every \(Z \in C_{\geq n}\) by orthogonality. By adjunction, this gives \(\Hom _C(Z,\tau _{\geq n}X)=0\) for every \(Z \in C_{\geq n}\). Taking \(Z=\tau _{\geq n}X\), we see that the identity of \(\tau _{\geq n}X\) is null, so \(\tau _{\geq n}X\) is zero.
Assume instead that \(n \leq m\). The truncation sequence \[ \tau _{\geq n}X \to X \to \tau _{\leq n-1}X \] exhibits \(\tau _{\geq n}X\) as the fiber of a morphism between objects of \(C_{\leq m}\), since \(C_{\leq n-1} \subseteq C_{\leq m}\). By Corollary 6.3.8, it follows that \(\tau _{\geq n}X \in C_{\leq m}\).
The proof of (2) is dual. □
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