The Stacks project

Lemma 67.22.5. Let $\mathcal{Q}$ be a property of morphisms of germs 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 $S$. Let $x \in |X|$ be a point of $X$. Consider the diagrams

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

where $U$ and $V$ are schemes, $a, b$ are étale, and $u, v, x, y$ are points of the corresponding spaces. The following are equivalent

  1. for any diagram as above we have $\mathcal{Q}((U, u) \to (V, v))$, and

  2. for some diagram as above we have $\mathcal{Q}((U, u) \to (V, v))$.

If $X$ and $Y$ are representable, then this is also equivalent to $\mathcal{Q}((X, x) \to (Y, y))$.

Proof. Omitted. Hint: Very similar to the proof of Lemma 67.22.1. $\square$

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