Classical left derived functors

Remark 6.6.20. Let \(F\colon \Aa \to \Bb \) be a right exact functor between abelian categories. For an object \(A \in \Aa \), viewed as a complex concentrated in degree zero, we may define functors \(\bL ^nF \colon \Aa \to \Bb \) by \[ \bL ^nF(A) \quad := \quad H_n(\bL F(A)) \] for \(n \geq 0\). They may be computed as the homology groups of \(F(P_{\bullet })\), where \(P_\bullet \to A\) is any choice of projective resolution of \(A\). Moreover, applying \(\bL F\) to a short exact sequence \(0 \to A \hookrightarrow B \twoheadrightarrow C \to 0\) in \(\Aa \) yields a cofiber sequence \(\bL F(A) \to \bL F(B) \to \bL F(C)\) in \(\D ^-(\Bb )\) by Lemma 6.6.19; its long exact sequence on homology is the classical long exact sequence relating the derived functors \(\bL ^n F\), and in particular the connecting homomorphisms of the latter arise precisely from the exactness of \(\bL F\). In classical treatments of homological algebra, one often merely considers the functors \(\bL ^nF\) rather than the entire derived functor \(\bL F\).