The Stacks project

Lemma 48.32.3. In Situation 48.16.1 let $f : X \to Y$ be a morphism of $\textit{FTS}_ S$. Let

\[ K \to L \to M \to K[1] \]

be a distinguished triangle of $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$. Then there exists an inverse system of distinguished triangles

\[ K_ n \to L_ n \to M_ n \to K_ n[1] \]

in $D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$ such that the pro-systems $(K_ n)$, $(L_ n)$, and $(M_ n)$ give $Rf_!K$, $Rf_!L$, and $Rf_!M$.

Proof. Choose a compactification $j : X \to \overline{X}$ over $Y$ and denote $\overline{f} : \overline{X} \to Y$ the structure morphism. Choose an inverse system of distinguished triangles

\[ \overline{K}_ n \to \overline{L}_ n \to \overline{M}_ n \to \overline{K}_ n[1] \]

in $D^ b_{\textit{Coh}}(\mathcal{O}_{\overline{X}})$ as in Lemma 48.30.4 corresponding to the open immersion $j$ and the given distinguished triangle. Take $K_ n = R\overline{f}_*\overline{K}_ n$ and similarly for $L_ n$ and $M_ n$. This works by the very definition of $Rf_!$. $\square$

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