The Stacks project

Lemma 20.14.1. Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{G}^\bullet $, resp. $\mathcal{F}^\bullet $ be a bounded below complex of $\mathcal{O}_ Y$-modules, resp. $\mathcal{O}_ X$-modules. Let $\varphi : \mathcal{G}^\bullet \to f_*\mathcal{F}^\bullet $ be a morphism of complexes. There is a canonical morphism

\[ \mathcal{G}^\bullet \longrightarrow Rf_*(\mathcal{F}^\bullet ) \]

in $D^{+}(Y)$. Moreover this construction is functorial in the triple $(\mathcal{G}^\bullet , \mathcal{F}^\bullet , \varphi )$.

Proof. Choose an injective resolution $\mathcal{F}^\bullet \to \mathcal{I}^\bullet $. By definition $Rf_*(\mathcal{F}^\bullet )$ is represented by $f_*\mathcal{I}^\bullet $ in $K^{+}(\mathcal{O}_ Y)$. The composition

\[ \mathcal{G}^\bullet \to f_*\mathcal{F}^\bullet \to f_*\mathcal{I}^\bullet \]

is a morphism in $K^{+}(Y)$ which turns into the morphism of the lemma upon applying the localization functor $j_ Y : K^{+}(Y) \to D^{+}(Y)$. $\square$

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