Lemma 5.20.3. Let $X$ be a topological space. Let $\delta$, $\delta '$ be two dimension functions on $X$. If $X$ is locally Noetherian and sober then $\delta - \delta '$ is locally constant on $X$.

Proof. Let $x \in X$ be a point. We will show that $\delta - \delta '$ is constant in a neighbourhood of $x$. We may replace $X$ by an open neighbourhood of $x$ in $X$ which is Noetherian. Hence we may assume $X$ is Noetherian and sober. Let $Z_1, \ldots , Z_ r$ be the irreducible components of $X$ passing through $x$. (There are finitely many as $X$ is Noetherian, see Lemma 5.9.2.) Let $\xi _ i \in Z_ i$ be the generic point. Note $Z_1 \cup \ldots \cup Z_ r$ is a neighbourhood of $x$ in $X$ (not necessarily closed). We claim that $\delta - \delta '$ is constant on $Z_1 \cup \ldots \cup Z_ r$. Namely, if $y \in Z_ i$, then

$\delta (x) - \delta (y) = \delta (x) - \delta (\xi _ i) + \delta (\xi _ i) - \delta (y) = - \text{codim}(\overline{\{ x\} }, Z_ i) + \text{codim}(\overline{\{ y\} }, Z_ i)$

by Lemma 5.20.2. Similarly for $\delta '$. Whence the result. $\square$

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