The Stacks project

Lemma 66.46.1. Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces over $S$. Then $f$ is finite locally free (in the sense of Section 66.3) if and only if $f$ is affine and the sheaf $f_*\mathcal{O}_ X$ is a finite locally free $\mathcal{O}_ Y$-module.

Proof. Assume $f$ is finite locally free (as defined in Section 66.3). This means that for every morphism $V \to Y$ whose source is a scheme the base change $f' : V \times _ Y X \to V$ is a finite locally free morphism of schemes. This in turn means (by the definition of a finite locally free morphism of schemes) that $f'_*\mathcal{O}_{V \times _ Y X}$ is a finite locally free $\mathcal{O}_ V$-module. We may choose $V \to Y$ to be surjective and ├ętale. By Properties of Spaces, Lemma 65.26.2 we conclude the restriction of $f_*\mathcal{O}_ X$ to $V$ is finite locally free. Hence by Modules on Sites, Lemma 18.23.3 applied to the sheaf $f_*\mathcal{O}_ X$ on $Y_{spaces, {\acute{e}tale}}$ we conclude that $f_*\mathcal{O}_ X$ is finite locally free.

Conversely, assume $f$ is affine and that $f_*\mathcal{O}_ X$ is a finite locally free $\mathcal{O}_ Y$-module. Let $V$ be a scheme, and let $V \to Y$ be a surjective ├ętale morphism. Again by Properties of Spaces, Lemma 65.26.2 we see that $f'_*\mathcal{O}_{V \times _ Y X}$ is finite locally free. Hence $f' : V \times _ Y X \to V$ is finite locally free (as it is also affine). By Spaces, Lemma 64.11.5 we conclude that $f$ is finite locally free (use Morphisms, Lemma 29.48.4 Descent, Lemmas 35.23.30 and 35.37.1). Thus we win. $\square$


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