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The Stacks project

Remark 20.42.13. Let h : X \to Y be a morphism of ringed spaces. Let K, L be objects of D(\mathcal{O}_ Y). We claim there is a canonical map

Lh^*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (Lh^*K, Lh^*L)

in D(\mathcal{O}_ X). Namely, by (20.42.0.1) proved in Lemma 20.42.2 such a map is the same thing as a map

Lh^*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \otimes ^\mathbf {L} Lh^*K \longrightarrow Lh^*L

The source of this arrow is Lh^*(\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \otimes ^\mathbf {L} K) by Lemma 20.27.3 hence it suffices to construct a canonical map

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

For this we take the arrow corresponding to

\text{id} : R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)

via (20.42.0.1).


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