Lemma 29.50.4. Let $f : X \to Y$ be a morphism of schemes. Assume that

$Y$ is locally Noetherian,

$X$ and $Y$ are integral schemes,

$f$ is dominant, and

$f$ is locally of finite type.

Then we have

If $f$ is closed^{1} then equality holds.

Lemma 29.50.4. Let $f : X \to Y$ be a morphism of schemes. Assume that

$Y$ is locally Noetherian,

$X$ and $Y$ are integral schemes,

$f$ is dominant, and

$f$ is locally of finite type.

Then we have

\[ \dim (X) \leq \dim (Y) + \text{trdeg}_{R(Y)} R(X). \]

If $f$ is closed^{1} then equality holds.

**Proof.**
Let $f : X \to Y$ be as in the lemma. Let $\xi _0 \leadsto \xi _1 \leadsto \ldots \leadsto \xi _ e$ be a sequence of specializations in $X$. Set $x = \xi _ e$ and $y = f(x)$. Observe that $e \leq \dim (\mathcal{O}_{X, x})$ as the given specializations occur in the spectrum of $\mathcal{O}_{X, x}$, see Schemes, Lemma 26.13.2. By the dimension formula, Lemma 29.50.1, we see that

\begin{align*} e & \leq \dim (\mathcal{O}_{X, x}) \\ & \leq \dim (\mathcal{O}_{Y, y}) + \text{trdeg}_{R(Y)} R(X) - \text{trdeg}_{\kappa (y)} \kappa (x) \\ & \leq \dim (\mathcal{O}_{Y, y}) + \text{trdeg}_{R(Y)} R(X) \end{align*}

Hence we conclude that $e \leq \dim (Y) + \text{trdeg}_{R(Y)} R(X)$ as desired.

Next, assume $f$ is also closed. Say $\overline{\xi }_0 \leadsto \overline{\xi }_1 \leadsto \ldots \leadsto \overline{\xi }_ d$ is a sequence of specializations in $Y$. We want to show that $\dim (X) \geq d + r$. We may assume that $\overline{\xi }_0 = \eta $ is the generic point of $Y$. The generic fibre $X_\eta $ is a scheme locally of finite type over $\kappa (\eta ) = R(Y)$. It is nonempty as $f$ is dominant. Hence by Lemma 29.15.10 it is a Jacobson scheme. Thus by Lemma 29.15.8 we can find a closed point $\xi _0 \in X_\eta $ and the extension $\kappa (\eta ) \subset \kappa (\xi _0)$ is a finite extension. Note that $\mathcal{O}_{X, \xi _0} = \mathcal{O}_{X_\eta , \xi _0}$ because $\eta $ is the generic point of $Y$. Hence we see that $\dim (\mathcal{O}_{X, \xi _0}) = r$ by Lemma 29.50.1 applied to the scheme $X_\eta $ over the universally catenary scheme $\mathop{\mathrm{Spec}}(\kappa (\eta ))$ (see Lemma 29.16.4) and the point $\xi _0$. This means that we can find $\xi _{-r} \leadsto \ldots \leadsto \xi _{-1} \leadsto \xi _0$ in $X$. On the other hand, as $f$ is closed specializations lift along $f$, see Topology, Lemma 5.19.7. Thus, as $\xi _0$ lies over $\eta = \overline{\xi }_0$ we can find specializations $\xi _0 \leadsto \xi _1 \leadsto \ldots \leadsto \xi _ d$ lying over $\overline{\xi }_0 \leadsto \overline{\xi }_1 \leadsto \ldots \leadsto \overline{\xi }_ d$. In other words we have

\[ \xi _{-r} \leadsto \ldots \leadsto \xi _{-1} \leadsto \xi _0 \leadsto \xi _1 \leadsto \ldots \leadsto \xi _ d \]

which means that $\dim (X) \geq d + r$ as desired. $\square$

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: