Corollary 1

Tag 04PS, permanent. Cite it as [Bourbaki, Tag 04PS].

Commutative Algebra, chapter V, § 2. The lift of prime ideals

With the hypotheses and notation of Theorem 1, $\bar{h}^{-1}(pA') = p$ .

$pA' \subset p'$ and $\bar{h}(p') = p$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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Nothing.