Lemma 10.59.11. Let $R$ be a Noetherian ring. Let $f_1, \ldots , f_ r \in R$.

If $\mathfrak p$ is minimal over $(f_1, \ldots , f_ r)$ then the height of $\mathfrak p$ is $\leq r$.

If $\mathfrak p, \mathfrak q \in \mathop{\mathrm{Spec}}(R)$ and $\mathfrak q$ is minimal over $(\mathfrak p, f_1, \ldots , f_ r)$, then every chain of primes between $\mathfrak p$ and $\mathfrak q$ has length at most $r$.

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