Lemma 48.21.4 (0BV6)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 48: Duality for Schemes
Section 48.21: Dimension functions
Lemma 48.21.4 (cite)

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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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