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

  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 $Y$.

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. $\square$

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