The Stacks project

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. $\square$

Comments (2)

Comment #3592 by Zhiyu on

Maybe a typo, is a function on rather than on . Then the result follows from Lemma 0BDI.

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