Lemma 48.21.6 (0BV7)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 48: Duality for Schemes
Section 48.21: Dimension functions
Lemma 48.21.6 (cite)

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Lemma 48.21.6. In Situation 48.16.1 let $f : X \to Y$ be a morphism of $\textit{FTS}_ S$ . If $f$ is flat and quasi-finite, then

$f^!\mathcal{O}_ Y = \omega _{X/Y}[0]$

for some coherent $\mathcal{O}_ X$ -module $\omega _{X/Y}$ flat over $Y$ .

Proof. Consequence of Lemma 48.21.4 and the fact that the cohomology sheaves of $f^!\mathcal{O}_ Y$ are coherent by Lemma 48.17.6. $\square$

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