Bounded subcategories of chain complexes

Notation 6.1.21. Given \(k \in \Z \), we denote by \(\Ch (\Aa )_{\geq k} \subseteq \Ch (\Aa )\) the full subcategory spanned by those chain complexes \(C_{\bullet }\) satisfying \(C_n = 0\) for \(n < k\). Similarly, we write \(\Ch (\Aa )_{\leq k}\) for the full subcategory on those chain complexes satisfying \(C_n = 0\) for \(n > k\). We denote by \[ \D (\Aa )_{\geq k} \, := \, \Ch (\Aa )_{\geq k}[\{\text {quasi-isos}\}^{-1}], \qquad \qquad \D (\Aa )_{\leq k} \, := \, \Ch (\Aa )_{\leq k}[\{\text {quasi-isos}\}^{-1}] \] the localizations of these subcategories at the quasi-isomorphisms.