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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