The Stacks project

Lemma 48.21.3. In Situation 48.16.1 let $f : X \to Y$ be a morphism of $\textit{FTS}_ S$. Let $x \in X$ with image $y \in Y$. Then

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

Proof. Since the statement is local on $X$ we may assume $X$ and $Y$ are affine schemes. Write $X = \mathop{\mathrm{Spec}}(A)$ and $Y = \mathop{\mathrm{Spec}}(R)$. Then $f^!\mathcal{O}_ Y$ corresponds to the relative dualizing complex $\omega _{A/R}^\bullet $ of Dualizing Complexes, Section 47.25 by Remark 48.17.5. Thus the lemma follows from Dualizing Complexes, Lemma 47.25.7. $\square$

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