Lemma 75.26.2 (0E0R)—The Stacks project

Lemma 75.26.2. Let $Y$ be a scheme and let $X$ be an algebraic space over $Y$ such that the structure morphism $f : X \to Y$ is flat, proper, and of finite presentation. Let $\mathcal{F}$ be an $\mathcal{O}_ X$ -module of finite presentation, flat over $Y$ . For fixed $i \in \mathbf{Z}$ consider the function

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

Then we have

formation of $\beta _ i$ commutes with arbitrary base change,
the functions $\beta _ i$ are upper semi-continuous, and
the level sets of $\beta _ i$ are locally constructible in $Y$ .

Proof. By cohomology and base change (more precisely by Lemma 75.25.4) the object $K = Rf_*\mathcal{F}$ is a perfect object of the derived category of $Y$ whose formation commutes with arbitrary base change. In particular we have

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

Thus the lemma follows from Lemma 75.26.1. $\square$

The Stacks project

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