Lemma 29.41.3. Let $f : X \to S$ be a morphism of schemes. The following are equivalent:

The morphism $f$ is proper.

There exists an open covering $S = \bigcup V_ j$ such that $f^{-1}(V_ j) \to V_ j$ is proper for all indices $j$.

Lemma 29.41.3. Let $f : X \to S$ be a morphism of schemes. The following are equivalent:

The morphism $f$ is proper.

There exists an open covering $S = \bigcup V_ j$ such that $f^{-1}(V_ j) \to V_ j$ is proper for all indices $j$.

**Proof.**
Omitted.
$\square$

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