Proposition 6.1.26. For every \(k \in \Z \), the composite \[ \Aa \xrightarrow {A \mapsto A[k]} \Ch (\Aa ) \to \D (\Aa ) \] is fully faithful, with essential image those complexes whose homology is concentrated in degree \(k\).
Proof. Let \(\Ch (\Aa )_{=k} \subseteq \Ch (\Aa )\) denote the subcategory of chain complexes \(C_{\bullet }\) satisfying \(C_n = 0\) for \(n \neq k\). Then the assignment \(A \mapsto A[k]\) induces an equivalence \(\Aa \iso \Ch (\Aa )_{=k}\). The adjunction \(\Ch (\Aa )_{\geq k} \rightleftarrows \Ch (\Aa )\) restricts to an adjunction \(\Aa \simeq \Ch (\Aa )_{=k} \rightleftarrows \Ch (\Aa )_{\leq k}\), and since both functors preserve quasi-isomorphisms, we get an induced adjunction \[ \Aa \rightleftarrows \D (\Aa )_{\leq k}, \] where the left adjoint is fully faithful as the unit is a natural isomorphism. The functor in question is now given by the composite \[ \Aa \hookrightarrow \D (\Aa )_{\leq k} \hookrightarrow \D (\Aa ), \] hence is fully faithful as well. □
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