Lemma 6.3.7. Let \((C_{\geq 0}, C_{\leq 0})\) be a t-structure on \(C\), and let \(n \in \Z \).
- (1)
-
The inclusion \(C_{\geq n} \hookrightarrow C\) admits a right adjoint \(\tau _{\geq n}\colon C \to C_{\geq n}\).
- (2)
-
The inclusion \(C_{\leq n} \hookrightarrow C\) admits a left adjoint \(\tau _{\leq n}\colon C \to C_{\leq n}\).
Proof. We prove (1); the proof of (2) is dual. By shifting, we may assume that \(n = 0\); in general we may then set \(\tau _{\geq n}(X) := \tau _{\geq 0}(X[-n])[n]\). Using the pointwise formula for adjunctions, it suffices to show that for every object \(X \in C\) there exists some \(\tau _{\geq 0}X\) equipped with a morphism \(\tau _{\geq 0}X \to X\) inducing equivalences \(\Hom _{C}(Y, \tau _{\geq 0}X) \iso \Hom _{C}(Y,X)\) for every \(Y \in C_{\geq 0}\). For this, we consider the exact sequence \(\tau _{\geq 0}X \to X \to \tau _{\leq -1}X\) provided by axiom (3). This gives rise to a long exact sequence \[ \dots \to \pi _{n+1}\Hom _C(Y,\tau _{\leq -1} X) \to \pi _n\Hom _C(Y,\tau _{\geq 0}X) \to \pi _n \Hom _C(Y,X) \to \pi _n\Hom _C(Y,\tau _{\leq -1} X) \to \dots \] Since \(\Hom _C(Y,\tau _{\leq -1} X) = 0\) by axiom (2), this implies that the map \(\Hom _{C}(Y, \tau _{\geq 0}X) \to \Hom _{C}(Y,X)\) induces isomorphisms on all homotopy groups, hence is an equivalence. □
Generated from the authoritative LaTeX source.