Exact sequence of derived truncations

Proof. For a chain complex \(C_{\bullet } \in \Ch (\Aa )\), put \(Q_{\bullet }:=C_{\bullet }/\tau _{\geq n}C_{\bullet }\). There is a functorial short exact sequence \[ 0 \to \tau _{\geq n} C_{\bullet } \hookrightarrow C_{\bullet } \twoheadrightarrow Q_{\bullet } \to 0. \] The natural map \(Q_{\bullet }\to \tau _{\leq n-1}C_{\bullet }\), given in degree \(n-1\) by the quotient \(C_{n-1}\to \coker (d_n)\) and by the identity below that degree, is a quasi-isomorphism. The theorem therefore turns the displayed short exact sequence into an exact sequence \(\gamma \tau _{\geq n} \to \gamma \to \gamma \tau _{\leq n-1}\) in \(\Fun (\Ch (\Aa ),\D (\Aa ))\). By the universal property of \(\gamma \colon \Ch (\Aa )\to \D (\Aa )\), this descends to the desired exact sequence \(\tau _{\geq n}\to \id \to \tau _{\leq n-1}\). □