The Stacks project

Lemma 20.42.4. Let $(X, \mathcal{O}_ X)$ be a ringed space. The bifunctor $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (- , -)$ transforms distinguished triangles into distinguished triangles in both variables.

Proof. This follows from the observation that the assignment

\[ (\mathcal{L}^\bullet , \mathcal{M}^\bullet ) \longmapsto \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{M}^\bullet ) \]

transforms a termwise split short exact sequences of complexes in either variable into a termwise split short exact sequence. Details omitted. $\square$

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