The Stacks project

Lemma 67.22.1. Let $\mathcal{P}$ be a property of morphisms of schemes which is étale local on the source-and-target. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Consider commutative diagrams

\[ \xymatrix{ U \ar[d]_ a \ar[r]_ h & V \ar[d]^ b \\ X \ar[r]^ f & Y } \]

where $U$ and $V$ are schemes and the vertical arrows are étale. The following are equivalent

  1. for any diagram as above the morphism $h$ has property $\mathcal{P}$, and

  2. for some diagram as above with $a : U \to X$ surjective the morphism $h$ has property $\mathcal{P}$.

If $X$ and $Y$ are representable, then this is also equivalent to $f$ (as a morphism of schemes) having property $\mathcal{P}$. If $\mathcal{P}$ is also preserved under any base change, and fppf local on the base, then for representable morphisms $f$ this is also equivalent to $f$ having property $\mathcal{P}$ in the sense of Section 67.3.

Proof. Let us prove the equivalence of (1) and (2). The implication (1) $\Rightarrow $ (2) is immediate (taking into account Spaces, Lemma 65.11.6). Assume

\[ \xymatrix{ U \ar[d] \ar[r]_ h & V \ar[d] \\ X \ar[r]^ f & Y } \quad \quad \xymatrix{ U' \ar[d] \ar[r]_{h'} & V' \ar[d] \\ X \ar[r]^ f & Y } \]

are two diagrams as in the lemma. Assume $U \to X$ is surjective and $h$ has property $\mathcal{P}$. To show that (2) implies (1) we have to prove that $h'$ has $\mathcal{P}$. To do this consider the diagram

\[ \xymatrix{ U \ar[d]_ h & U \times _ X U' \ar[l] \ar[d]^{(h, h')} \ar[r] & U' \ar[d]^{h'} \\ V & V \times _ Y V' \ar[l] \ar[r] & V' } \]

By Descent, Lemma 35.32.5 we see that $h$ has $\mathcal{P}$ implies $(h, h')$ has $\mathcal{P}$ and since $U \times _ X U' \to U'$ is surjective this implies (by the same lemma) that $h'$ has $\mathcal{P}$.

If $X$ and $Y$ are representable, then Descent, Lemma 35.32.5 applies which shows that (1) and (2) are equivalent to $f$ having $\mathcal{P}$.

Finally, suppose $f$ is representable, and $U, V, a, b, h$ are as in part (2) of the lemma, and that $\mathcal{P}$ is preserved under arbitrary base change. We have to show that for any scheme $Z$ and morphism $Z \to X$ the base change $Z \times _ Y X \to Z$ has property $\mathcal{P}$. Consider the diagram

\[ \xymatrix{ Z \times _ Y U \ar[d] \ar[r] & Z \times _ Y V \ar[d] \\ Z \times _ Y X \ar[r] & Z } \]

Note that the top horizontal arrow is a base change of $h$ and hence has property $\mathcal{P}$. The left vertical arrow is étale and surjective and the right vertical arrow is étale. Thus Descent, Lemma 35.32.5 once again kicks in and shows that $Z \times _ Y X \to Z$ has property $\mathcal{P}$. $\square$

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