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.

1. 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

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

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

4. The sequence $(R^ iF, \delta )$ forms a universal $\delta$-functor (see Homology, Definition 12.12.3) from $\mathcal{A}$ to $\mathcal{B}$.

Proof. This lemma simply reviews some of the results obtained so far. Note that by Lemma 13.20.2 $RF$ is everywhere defined. Here are some references:

1. This follows from Lemma 13.20.3 part (3) combined with the long exact cohomology sequence (13.11.1.1) for $D^{+}(\mathcal{B})$.

2. This is Lemma 13.16.3.

3. This is the fact that injective objects are acyclic.

4. This is Lemma 13.16.6.

$\square$

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