Exactness and t-exactness of derived functors

Proof. For right t-exactness, let \(C_{\bullet }\) be concentrated in degrees \(\geq 0\). The construction in Proposition 6.6.13 gives a projective resolution \(P_{\bullet }\to C_{\bullet }\) which is also concentrated in degrees \(\geq 0\). Hence \(\bL F(C_{\bullet })\simeq F(P_{\bullet })\) is connective.

For exactness, it suffices to show that \(\bL F\) preserves cofiber sequences. Every morphism in \(\D ^-(\Aa )\) can be represented by a chain map \(f\colon P_{\bullet } \to Q_{\bullet }\) between bounded below projective complexes (by first replacing both source and target with projective resolutions). The cofiber in \(\D ^-(\Aa )\) is represented by the mapping cone \(\Cone (f)\), which is again a bounded below projective complex. Since \(F\) is additive, it preserves mapping cones: \(F(\Cone (f)) \cong \Cone (F(f))\). We then compute \[ \bL F(\cofib (f)) \,\simeq \, F(\Cone (f)) \,\cong \, \Cone (F(f)) \,\simeq \, \cofib (\bL F(f)), \] showing that \(\bL F\) preserves cofiber sequences. □