The Stacks project

Lemma 10.114.3. Let $k$ be a field. Let $\mathfrak p \subset \mathfrak q \subset k[x_1, \ldots , x_ n]$ be a pair of primes. Any maximal chain of primes between $\mathfrak p$ and $\mathfrak q$ has length $\text{height}(\mathfrak q) - \text{height}(\mathfrak p)$.

Proof. By Proposition 10.114.2 any local ring of $k[x_1, \ldots , x_ n]$ is regular. Hence all local rings are Cohen-Macaulay, see Lemma 10.106.3. The local rings at maximal ideals have dimension $n$ hence every maximal chain of primes in $k[x_1, \ldots , x_ n]$ has length $n$, see Lemma 10.104.3. Hence every maximal chain of primes between $(0)$ and $\mathfrak p$ has length $\text{height}(\mathfrak p)$, see Lemma 10.104.4 for example. Putting these together leads to the assertion of the lemma. $\square$

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