Lemma 48.17.6. In Situation 48.16.1 let f : X \to Y be a morphism of \textit{FTS}_ S. Then f^! maps D_{\textit{Coh}}^+(\mathcal{O}_ Y) into D_{\textit{Coh}}^+(\mathcal{O}_ X).
Proof. The question is local on X hence we may assume that X and Y are affine schemes. In this case we can factor f : X \to Y as
X \xrightarrow {i} \mathbf{A}^ n_ Y \to \mathbf{A}^{n - 1}_ Y \to \ldots \to \mathbf{A}^1_ Y \to Y
where i is a closed immersion. The lemma follows from By Lemmas 48.17.3 and 48.9.6 and Dualizing Complexes, Lemma 47.15.10 and induction. \square
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