The Stacks project

Remark 37.51.5. The proof of Lemma 37.51.4 actually shows that there exists a sequence of specializations

\[ x \leadsto x_1 \leadsto x_2 \leadsto \ldots \leadsto x_ d \leadsto x' \]

where all $x_ i$ are in the fibre $X_ s$, each specialization is immediate, and $x_ d$ is a closed point of $X_ s$. The integer $d = \text{trdeg}_{\kappa (s)}(\kappa (x)) = \dim (\overline{\{ x\} })$ where the closure is taken in $X_ s$. Moreover, the points $x_ i$ can be chosen to avoid any closed subset of $X_ s$ which does not contain the point $x$.

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