Lemma 15.51.2. Let $R \to \Lambda $ be a homomorphism of Noetherian rings. Assume $P$ has property (B). The following are equivalent

the fibres of $R \to \Lambda $ have $P$,

the fibres of $R_\mathfrak p \to \Lambda _\mathfrak q$ have $P$ for all $\mathfrak q \subset \Lambda $ lying over $\mathfrak p \subset R$, and

the fibres of $R_\mathfrak m \to \Lambda _{\mathfrak m'}$ have $P$ for all maximal ideals $\mathfrak m' \subset \Lambda $ lying over $\mathfrak m$ in $R$.

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