The Stacks project

Lemma 13.16.6. Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor between abelian categories and assume $RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ is everywhere defined.

  1. The functors $R^ iF$, $i \geq 0$ come equipped with a canonical structure of a $\delta $-functor from $\mathcal{A} \to \mathcal{B}$, see Homology, Definition 12.12.1.

  2. If every object of $\mathcal{A}$ is a subobject of a right acyclic object for $F$, then $\{ R^ iF, \delta \} _{i \geq 0}$ is a universal $\delta $-functor, see Homology, Definition 12.12.3.

Proof. The functor $\mathcal{A} \to \text{Comp}^{+}(\mathcal{A})$, $A \mapsto A[0]$ is exact. The functor $\text{Comp}^{+}(\mathcal{A}) \to D^{+}(\mathcal{A})$ is a $\delta $-functor, see Lemma 13.12.1. The functor $RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ is exact. Finally, the functor $H^0 : D^{+}(\mathcal{B}) \to \mathcal{B}$ is a homological functor, see Definition 13.11.3. Hence we get the structure of a $\delta $-functor from Lemma 13.4.22 and Lemma 13.4.21. Part (2) follows from Homology, Lemma 12.12.4 and the description of acyclics in Lemma 13.16.4. $\square$

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