The Stacks project

Lemma 75.26.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is proper, flat, and of finite presentation. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module of finite presentation, flat over $Y$. Fix $i, r \in \mathbf{Z}$. Then there exists an open subspace $V \subset Y$ with the following property: A morphism $T \to Y$ factors through $V$ if and only if $Rf_{T, *}\mathcal{F}_ T$ is isomorphic to a finite locally free module of rank $r$ placed in degree $i$.

Proof. By cohomology and base change ( 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. Thus this lemma follows immediately from Lemma 75.26.4. $\square$

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