The Stacks project

Lemma 33.19.3. 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. 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 $f(\eta ) \not= y$ and $\dim _\eta (X_{f(\eta )}) = d$. Then $\dim _ x(X_ y) \leq d + \dim (\mathcal{O}_{Y, y}) - 1$.

Proof. Exactly as in the proof of Lemma 33.19.1 we reduce to the case $X = \mathop{\mathrm{Spec}}(A)$ with $A$ a domain and $Y = \mathop{\mathrm{Spec}}(B)$ where $B$ is a Noetherian local ring whose maximal ideal corresponds to $y$. After replacing $B$ by $B/\mathop{\mathrm{Ker}}(B \to A)$ we may assume that $B$ is a domain and that $B \subset A$. Then we use the dimension formula (Morphisms, Lemma 29.52.1) to get

\[ \dim (\mathcal{O}_{X, x}) + \text{trdeg}_{\kappa (y)} \kappa (x) \leq \dim (B) + \text{trdeg}_ B(A) \]

We have $\text{trdeg}_ B(A) = d$ 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 $f(\eta ) \not= y$) 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

\[ \dim _ x(X_ s) = \dim (\mathcal{O}_{X_ s, x}) + \text{trdeg}_{\kappa (y)} \kappa (x) < \dim (B) + d \]

(equality by Morphisms, Lemma 29.28.1) which proves what we want. $\square$


Comments (0)


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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0B2K. Beware of the difference between the letter 'O' and the digit '0'.