The Stacks project

Lemma 36.32.2. Let $f : X \to S$ be a flat, proper morphism of finite presentation. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module of finite presentation, flat over $S$. The function

\[ s \longmapsto \chi (X_ s, \mathcal{F}_ s) \]

is locally constant on $S$. Formation of this function commutes with base change.

Proof. By cohomology and base change (more precisely by Lemma 36.30.4) the object $K = Rf_*\mathcal{F}$ is a perfect object of the derived category of $S$ whose formation commutes with arbitrary base change. Thus we have to show the map

\[ s \longmapsto \sum (-1)^ i \dim _{\kappa (s)} H^ i(K \otimes ^\mathbf {L}_{\mathcal{O}_ S} \kappa (s)) \]

is locally constant on $S$. This is Lemma 36.31.2. $\square$

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