Lemma 36.32.1. 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 $S$. For fixed $i \in \mathbf{Z}$ consider the function

$\beta _ i : S \to \{ 0, 1, 2, \ldots \} ,\quad s \longmapsto \dim _{\kappa (s)} H^ i(X_ s, \mathcal{F}_ s)$

Then we have

1. formation of $\beta _ i$ commutes with arbitrary base change,

2. the functions $\beta _ i$ are upper semi-continuous, and

3. the level sets of $\beta _ i$ are locally constructible in $S$.

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. In particular we have

$H^ i(X_ s, \mathcal{F}_ s) = H^ i(K \otimes _{\mathcal{O}_ S}^\mathbf {L} \kappa (s))$

Thus the lemma follows from Lemma 36.31.1. $\square$

Comment #3592 by Zhiyu on

Maybe a typo, $\beta_i$ is a function on $s \in S$ rather than on $X$. Then the result follows from Lemma 0BDI.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).