The Stacks project

Lemma 10.125.2. Let $R \to S$ be a finite type ring map. Let $\mathfrak q \subset S$ be a prime. Suppose that $\dim _{\mathfrak q}(S/R) = n$. There exists a $g \in S$, $g \not\in \mathfrak q$ such that $S_ g$ is quasi-finite over a polynomial algebra $R[t_1, \ldots , t_ n]$.

Proof. The ring $\overline{S} = S \otimes _ R \kappa (\mathfrak p)$ is of finite type over $\kappa (\mathfrak p)$. Let $\overline{\mathfrak q}$ be the prime of $\overline{S}$ corresponding to $\mathfrak q$. By definition of the dimension of a topological space at a point there exists an open $U \subset \mathop{\mathrm{Spec}}(\overline{S})$ with $\overline{q} \in U$ and $\dim (U) = n$. Since the topology on $\mathop{\mathrm{Spec}}(\overline{S})$ is induced from the topology on $\mathop{\mathrm{Spec}}(S)$ (see Remark 10.17.8), we can find a $g \in S$, $g \not\in \mathfrak q$ with image $\overline{g} \in \overline{S}$ such that $D(\overline{g}) \subset U$. Thus after replacing $S$ by $S_ g$ we see that $\dim (\overline{S}) = n$.

Next, choose generators $x_1, \ldots , x_ N$ for $S$ as an $R$-algebra. By Lemma 10.115.4 there exist elements $y_1, \ldots , y_ n$ in the $\mathbf{Z}$-subalgebra of $S$ generated by $x_1, \ldots , x_ N$ such that the map $R[t_1, \ldots , t_ n] \to S$, $t_ i \mapsto y_ i$ has the property that $\kappa (\mathfrak p)[t_1\ldots , t_ n] \to \overline{S}$ is finite. In particular, $S$ is quasi-finite over $R[t_1, \ldots , t_ n]$ at $\mathfrak q$. Hence, by Lemma 10.123.13 we may replace $S$ by $S_ g$ for some $g\in S$, $g \not\in \mathfrak q$ such that $R[t_1, \ldots , t_ n] \to S$ is quasi-finite. $\square$

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Your email address will not be published.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 00QE. Beware of the difference between the letter 'O' and the digit '0'.