Composition on RSheafHom.

Lemma 21.34.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Given $K, L, M$ in $D(\mathcal{O})$ there is a canonical morphism

$R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M) \otimes _\mathcal {O}^\mathbf {L} R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, M)$

in $D(\mathcal{O})$.

Proof. Choose a K-injective complex $\mathcal{I}^\bullet$ representing $M$, a K-injective complex $\mathcal{J}^\bullet$ representing $L$, and any complex of $\mathcal{O}$-modules $\mathcal{K}^\bullet$ representing $K$. By Lemma 21.33.2 there is a map of complexes

$\text{Tot}\left( \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{J}^\bullet , \mathcal{I}^\bullet ) \otimes _\mathcal {O} \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathcal{J}^\bullet ) \right) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathcal{I}^\bullet )$

The complexes of $\mathcal{O}$-modules $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{J}^\bullet , \mathcal{I}^\bullet )$, $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathcal{J}^\bullet )$, and $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathcal{I}^\bullet )$ represent $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M)$, $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$, and $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, M)$. If we choose a K-flat complex $\mathcal{H}^\bullet$ and a quasi-isomorphism $\mathcal{H}^\bullet \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathcal{J}^\bullet )$, then there is a map

$\text{Tot}\left( \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{J}^\bullet , \mathcal{I}^\bullet ) \otimes _\mathcal {O} \mathcal{H}^\bullet \right) \longrightarrow \text{Tot}\left( \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{J}^\bullet , \mathcal{I}^\bullet ) \otimes _\mathcal {O} \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathcal{J}^\bullet ) \right)$

whose source represents $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M) \otimes _\mathcal {O}^\mathbf {L} R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$. Composing the two displayed arrows gives the desired map. We omit the proof that the construction is functorial. $\square$

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