Lemma 33.19.1. Let $f : X \to Y$ be locally of finite type. Let $x \in X$ be a point with image $y \in Y$ such that $\mathcal{O}_{Y, y}$ is Noetherian of dimension $\leq 1$. Let $d \geq 0$ be an integer such that for every generic point $\eta $ of an irreducible component of $X$ which contains $x$, we have $\dim _\eta (X_{f(\eta )}) = d$. Then $\dim _ x(X_ y) = d$.

**Proof.**
Recall that $\mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y})$ is the set of points of $Y$ specializing to $y$, see Schemes, Lemma 26.13.2. Thus we may replace $Y$ by $\mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y})$ and assume $Y = \mathop{\mathrm{Spec}}(B)$ where $B$ is a Noetherian local ring of dimension $\leq 1$ and $y$ is the closed point. We may also replace $X$ by an affine neighbourhood of $x$.

Let $X = \bigcup X_ i$ be the irreducible components of $X$ viewed as reduced closed subschemes. If we can show each fibre $X_{i, y}$ has dimension $d$, then $X_ y = \bigcup X_{i, y}$ has dimension $d$ as well. Thus we may assume $X$ is an integral scheme.

If $X \to Y$ maps the generic point $\eta $ of $X$ to $y$, then $X$ is a scheme over $\kappa (y)$ and the result is true by assumption. Assume that $X$ maps $\eta $ to a point $\xi \in Y$ corresponding to a minimal prime $\mathfrak q$ of $B$ different from $\mathfrak m_ B$. We obtain a factorization $X \to \mathop{\mathrm{Spec}}(B/\mathfrak q) \to \mathop{\mathrm{Spec}}(B)$. By the dimension formula (Morphisms, Lemma 29.52.1) we have

We have $\dim (B/\mathfrak q) = 1$. We have $\text{trdeg}_{\kappa (\mathfrak q)}(R(X)) = d$ by our assumption that $\dim _\eta (X_\xi ) = d$, see Morphisms, Lemma 29.28.1. Since $\mathcal{O}_{X, x} \to \mathcal{O}_{X_ s, x}$ has a kernel (as $\eta \mapsto \xi \not= y$) and since $\mathcal{O}_{X, x}$ is a Noetherian domain we see that $\dim (\mathcal{O}_{X, x}) > \dim (\mathcal{O}_{X_ y, x})$. We conclude that

(Morphisms, Lemma 29.28.1). On the other hand, we have $\dim _ x(X_ s) \geq \dim _\eta (X_{f(\eta )}) = d$ by Morphisms, Lemma 29.28.4. $\square$

## Post a comment

Your email address will not be published.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)