The Stacks project

Lemma 67.17.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{P} \in \{ (\beta ), decent, reasonable, very\ reasonable\} $. The following are equivalent

  1. $f$ is $\mathcal{P}$,

  2. for every affine scheme $Z$ and every morphism $Z \to Y$ the base change $Z \times _ Y X \to Z$ of $f$ is $\mathcal{P}$,

  3. for every affine scheme $Z$ and every morphism $Z \to Y$ the algebraic space $Z \times _ Y X$ is $\mathcal{P}$, and

  4. there exists a Zariski covering $Y = \bigcup Y_ i$ such that each morphism $f^{-1}(Y_ i) \to Y_ i$ has $\mathcal{P}$.

If $\mathcal{P} \in \{ (\beta ), decent, reasonable\} $, then this is also equivalent to

  1. there exists a scheme $V$ and a surjective étale morphism $V \to Y$ such that the base change $V \times _ Y X \to V$ has $\mathcal{P}$.

Proof. The implications (1) $\Rightarrow $ (2) $\Rightarrow $ (3) $\Rightarrow $ (4) are trivial. The implication (3) $\Rightarrow $ (1) can be seen as follows. Let $Z \to Y$ be a morphism whose source is a scheme over $S$. Consider the algebraic space $Z \times _ Y X$. If we assume (3), then for any affine open $W \subset Z$, the open subspace $W \times _ Y X$ of $Z \times _ Y X$ has property $\mathcal{P}$. Hence by Lemma 67.5.2 the space $Z \times _ Y X$ has property $\mathcal{P}$, i.e., (1) holds. A similar argument (omitted) shows that (4) implies (1).

The implication (1) $\Rightarrow $ (5) is trivial. Let $V \to Y$ be an étale morphism from a scheme as in (5). Let $Z$ be an affine scheme, and let $Z \to Y$ be a morphism. Consider the diagram

\[ \xymatrix{ Z \times _ Y V \ar[r]_ q \ar[d]_ p & V \ar[d] \\ Z \ar[r] & Y } \]

Since $p$ is étale, and hence open, we can choose finitely many affine open subschemes $W_ i \subset Z \times _ Y V$ such that $Z = \bigcup p(W_ i)$. Consider the commutative diagram

\[ \xymatrix{ V \times _ Y X \ar[d] & (\coprod W_ i) \times _ Y X \ar[l] \ar[d] \ar[r] & Z \times _ Y X \ar[d] \\ V & \coprod W_ i \ar[l] \ar[r] & Z } \]

We know $V \times _ Y X$ has property $\mathcal{P}$. By Lemma 67.5.3 we see that $(\coprod W_ i) \times _ Y X$ has property $\mathcal{P}$. Note that the morphism $(\coprod W_ i) \times _ Y X \to Z \times _ Y X$ is étale and quasi-compact as the base change of $\coprod W_ i \to Z$. Hence by Lemma 67.17.8 we conclude that $Z \times _ Y X$ has property $\mathcal{P}$. $\square$

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