The Stacks project

Lemma 98.3.3. Let $P$ be a property of morphisms of algebraic spaces as above. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. Let $W$ be an algebraic space and let $W \to \mathcal{Y}$ be surjective, locally of finite presentation, and flat. Set $V = W \times _\mathcal {Y} \mathcal{X}$. Then

\[ (f\text{ has }P) \Leftrightarrow (\text{the projection }V \to W\text{ has }P). \]

Proof. The implication from left to right follows from Lemma 98.3.2. Assume $V \to W$ has $P$. Let $T$ be a scheme, and let $T \to \mathcal{Y}$ be a morphism. Consider the commutative diagram

\[ \xymatrix{ T \times _\mathcal {Y} \mathcal{X} \ar[d] & T \times _\mathcal {Y} W \ar[d] \ar[l] \ar[r] & W \ar[d] \\ T & T \times _\mathcal {Y} V \ar[l] \ar[r] & V } \]

of algebraic spaces. The squares are cartesian. The bottom left morphism is a surjective, flat morphism which is locally of finite presentation, hence $\{ T \times _\mathcal {Y} V \to T\} $ is an fppf covering. Hence the fact that the right vertical arrow has property $P$ implies that the left vertical arrow has property $P$. $\square$


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