Lemma 48.21.4. In Situation 48.16.1 let $f : X \to Y$ be a morphism of $\textit{FTS}_ S$. Let $x \in X$ with image $y \in Y$. If $f$ is flat, then

$H^ i(f^!\mathcal{O}_ Y)_ x \not= 0 \Rightarrow - \dim _ x(X_ y) \leq i \leq 0.$

In fact, if all fibres of $f$ have dimension $\leq d$, then $f^!\mathcal{O}_ Y$ has tor-amplitude in $[-d, 0]$ as an object of $D(X, f^{-1}\mathcal{O}_ Y)$.

Proof. Arguing exactly as in the proof of Lemma 48.21.3 this follows from Dualizing Complexes, Lemma 47.25.8. $\square$

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