The Stacks project

Lemma 42.4.3. Let $(R, \mathfrak m)$ be a Noetherian local ring of dimension $1$. Let $a, b \in R$ be nonzerodivisors with $a \in \mathfrak m$. There exists an integer $n = n(R, a, b)$ such that for a finite ring extension $R \subset R'$ if $b = a^ m c$ for some $c \in R'$, then $m \leq n$.

Proof. Choose a minimal prime $\mathfrak q \subset R$. Observe that $\dim (R/\mathfrak q) = 1$, in particular $R/\mathfrak q$ is not a field. We can choose a discrete valuation ring $A$ dominating $R/\mathfrak q$ with the same fraction field, see Algebra, Lemma 10.119.1. Observe that $a$ and $b$ map to nonzero elements of $A$ as nonzerodivisors in $R$ are not contained in $\mathfrak q$. Let $v$ be the discrete valuation on $A$. Then $v(a) > 0$ as $a \in \mathfrak m$. We claim $n = v(b)/v(a)$ works.

Let $R \subset R'$ be given. Set $A' = A \otimes _ R R'$. Since $\mathop{\mathrm{Spec}}(R') \to \mathop{\mathrm{Spec}}(R)$ is surjective (Algebra, Lemma 10.36.17) also $\mathop{\mathrm{Spec}}(A') \to \mathop{\mathrm{Spec}}(A)$ is surjective (Algebra, Lemma 10.30.3). Pick a prime $\mathfrak q' \subset A'$ lying over $(0) \subset A$. Then $A \subset A'' = A'/\mathfrak q'$ is a finite extension of rings (again inducing a surjection on spectra). Pick a maximal ideal $\mathfrak m'' \subset A''$ lying over the maximal ideal of $A$ and a discrete valuation ring $A'''$ dominating $A''_{\mathfrak m''}$ (see lemma cited above). Then $A \to A'''$ is an extension of discrete valuation rings and we have $b = a^ m c$ in $A'''$. Thus $v'''(b) \geq mv'''(a)$. Since $v''' = ev$ where $e$ is the ramification index of $A'''/A$, we find that $m \leq n$ as desired. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 42.4: Preparation for tame symbols

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Your email address will not be published.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