The Stacks project

Proposition 66.32.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent sheaf of $\mathcal{O}_ X$-modules. Assume

  1. $Y$ is reduced,

  2. $f$ is of finite type, and

  3. $\mathcal{F}$ is a finite type $\mathcal{O}_ X$-module.

Then there exists an open dense subspace $W \subset Y$ such that the base change $X_ W \to W$ of $f$ is flat, locally of finite presentation, and quasi-compact and such that $\mathcal{F}|_{X_ W}$ is flat over $W$ and of finite presentation over $\mathcal{O}_{X_ W}$.

Proof. Let $V$ be a scheme and let $V \to Y$ be a surjective étale morphism. Let $X_ V = V \times _ Y X$ and let $\mathcal{F}_ V$ be the restriction of $\mathcal{F}$ to $X_ V$. Suppose that the result holds for the morphism $X_ V \to V$ and the sheaf $\mathcal{F}_ V$. Then there exists an open subscheme $V' \subset V$ such that $X_{V'} \to V'$ is flat and of finite presentation and $\mathcal{F}_{V'}$ is an $\mathcal{O}_{X_{V'}}$-module of finite presentation flat over $V'$. Let $W \subset Y$ be the image of the étale morphism $V' \to Y$, see Properties of Spaces, Lemma 65.4.10. Then $V' \to W$ is a surjective étale morphism, hence we see that $X_ W \to W$ is flat, locally of finite presentation, and quasi-compact by Lemmas 66.28.4, 66.30.5, and 66.8.8. By the discussion in Properties of Spaces, Section 65.30 we see that $\mathcal{F}_ W$ is of finite presentation as a $\mathcal{O}_{X_ W}$-module and by Lemma 66.31.3 we see that $\mathcal{F}_ W$ is flat over $W$. This argument reduces the proposition to the case where $Y$ is a scheme.

Suppose we can prove the proposition when $Y$ is an affine scheme. Let $f : X \to Y$ be a finite type morphism of algebraic spaces over $S$ with $Y$ a scheme, and let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_ X$-module. Choose an affine open covering $Y = \bigcup V_ j$. By assumption we can find dense open $W_ j \subset V_ j$ such that $X_{W_ j} \to W_ j$ is flat, locally of finite presentation, and quasi-compact and such that $\mathcal{F}|_{X_{W_ j}}$ is flat over $W_ j$ and of finite presentation as an $\mathcal{O}_{X_{W_ j}}$-module. In this situation we simply take $W = \bigcup W_ j$ and we win. Hence we reduce the proposition to the case where $Y$ is an affine scheme.

Let $Y$ be an affine scheme over $S$, let $f : X \to Y$ be a finite type morphism of algebraic spaces over $S$, and let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_ X$-module. Since $f$ is of finite type it is quasi-compact, hence $X$ is quasi-compact. Thus we can find an affine scheme $U$ and a surjective étale morphism $U \to X$, see Properties of Spaces, Lemma 65.6.3. Note that $U \to Y$ is of finite type (this is what it means for $f$ to be of finite type in this case). Hence we can apply Morphisms, Proposition 29.27.2 to see that there exists a dense open $W \subset Y$ such that $U_ W \to W$ is flat and of finite presentation and such that $\mathcal{F}|_{U_ W}$ is flat over $W$ and of finite presentation as an $\mathcal{O}_{U_ W}$-module. According to our definitions this means that the base change $X_ W \to W$ of $f$ is flat, locally of finite presentation, and quasi-compact and $\mathcal{F}|_{X_ W}$ is flat over $W$ and of finite presentation over $\mathcal{O}_{X_ W}$. $\square$


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