The Stacks project

Lemma 96.5.6. Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces as in Algebraic Stacks, Definition 93.10.1. If

  1. $f$ is representable by algebraic spaces,

  2. $\mathcal{Y} \to (\mathit{Sch}/S)_{fppf}$ is limit preserving on objects,

  3. for an affine scheme $V$ locally of finite presentation over $S$ and $y \in \mathcal{Y}_ V$ the resulting morphism of algebraic spaces $f_ y : F_ y \to V$, see Algebraic Stacks, Equation (93.9.1.1), has property $\mathcal{P}$.

Then $f$ has property $\mathcal{P}$.

Proof. Let $V$ be a scheme over $S$ and $y \in \mathcal{Y}_ V$. We have to show that $F_ y \to V$ has property $\mathcal{P}$. Since $\mathcal{P}$ is fppf local on the base we may assume that $V$ is an affine scheme which maps into an affine open $\mathop{\mathrm{Spec}}(\Lambda ) \subset S$. Thus we can write $V = \mathop{\mathrm{lim}}\nolimits V_ i$ with each $V_ i$ affine and of finite presentation over $\mathop{\mathrm{Spec}}(\Lambda )$, see Algebra, Lemma 10.127.2. Then $y$ comes from an object $y_ i$ over $V_ i$ for some $i$ by assumption (2). By assumption (3) the morphism $F_{y_ i} \to V_ i$ has property $\mathcal{P}$. As $\mathcal{P}$ is stable under arbitrary base change and since $F_ y = F_{y_ i} \times _{V_ i} V$ we conclude that $F_ y \to V$ has property $\mathcal{P}$ as desired. $\square$


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