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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