The Stacks project

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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