The Stacks project

Situation 78.5.6. Here $S$ is a base scheme, $f : Y \to X$ is a quasi-compact and quasi-separated morphism of algebraic spaces over $S$, and $Z \to X$ is a closed immersion of finite presentation. We assume that $f^{-1}(Z) \to Z$ is an isomorphism and that $f$ is flat in every point $x \in |f^{-1}Z|$. We set $U = X \setminus Z$ and $V = Y \setminus f^{-1}(Z)$. Picture

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

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