Lemma 13.20.4. Let $\mathcal{A}$ be an abelian category with enough injectives. Let $F : \mathcal{A} \to \mathcal{B}$ be a left exact functor.

For any short exact sequence $0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0$ of complexes in $\text{Comp}^{+}(\mathcal{A})$ there is an associated long exact sequence

\[ \ldots \to H^ i(RF(A^\bullet )) \to H^ i(RF(B^\bullet )) \to H^ i(RF(C^\bullet )) \to H^{i + 1}(RF(A^\bullet )) \to \ldots \]The functors $R^ iF : \mathcal{A} \to \mathcal{B}$ are zero for $i < 0$. Also $R^0F = F : \mathcal{A} \to \mathcal{B}$.

We have $R^ iF(I) = 0$ for $i > 0$ and $I$ injective.

The sequence $(R^ iF, \delta )$ forms a universal $\delta $-functor (see Homology, Definition 12.11.3) from $\mathcal{A}$ to $\mathcal{B}$.

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