The Stacks project

Remark 48.32.5. In Situation 48.16.1 let $f : X \to Y$ and $g : Y \to Z$ be composable morphisms of $\textit{FTS}_ S$. Let us define the composition

\[ Rg_! \circ Rf_! : D^ b_{\textit{Coh}}(\mathcal{O}_ X) \longrightarrow \text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Z) \]

Namely, by the very construction of $Rf_!$ for $K$ in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ the output $Rf_!K$ is the pro-isomorphism class of an inverse system $(M_ n)$ in $D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$. Then, since $Rg_!$ is constructed similarly, we see that

\[ \ldots \to Rg_!M_3 \to Rg_!M_2 \to Rg_!M_1 \]

is an inverse system of $\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$. By the discussion in Remark 48.32.4 there is a unique pro-isomorphism class, which we will denote $Rg_! Rf_! K$, of inverse systems in $D^ b_{\textit{Coh}}(\mathcal{O}_ Z)$ such that

\[ \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Z)}(Rg_!Rf_!K, L) = \mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Z)}(Rg_!M_ n, L) \]

We omit the discussion necessary to see that this construction is functorial in $K$ as it will immediately follow from the next lemma.

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