Lemma 48.17.7. In Situation 48.16.1 let f : X \to Y be a morphism of \textit{FTS}_ S. If K is a dualizing complex for Y, then f^!K is a dualizing complex for 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. By Lemma 48.17.3 and Dualizing Complexes, Lemma 47.15.10 and induction we see that the p^!K is a dualizing complex on \mathbf{A}^ n_ Y where p : \mathbf{A}^ n_ Y \to Y is the projection. Similarly, by Dualizing Complexes, Lemma 47.15.9 and Lemmas 48.9.5 and 48.17.4 we see that i^! transforms dualizing complexes into dualizing complexes. \square
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