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The Stacks project

Lemma 48.21.5. 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. Assume

  1. \mathcal{O}_{Y, y} is Cohen-Macaulay, and

  2. \text{trdeg}_{\kappa (f(\xi ))}(\kappa (\xi )) \leq r for any generic point \xi of an irreducible component of X containing x.

Then

H^ i(f^!\mathcal{O}_ Y)_ x \not= 0 \Rightarrow - r \leq i

and the stalk H^{-r}(f^!\mathcal{O}_ Y)_ x is (S_2) as an \mathcal{O}_{X, x}-module.

Proof. After replacing X by an open neighbourhood of x, we may assume every irreducible component of X passes through x. Then arguing exactly as in the proof of Lemma 48.21.3 this follows from Dualizing Complexes, Lemma 47.25.9. \square

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