The Stacks project

Lemma 115.5.1. Let $R$ be a ring and let $\varphi : R[x] \to S$ be a ring map. Let $t \in S$. If $t$ is integral over $R[x]$, then there exists an $\ell \geq 0$ such that for every $a \in R$ the element $\varphi (a)^\ell t$ is integral over $\varphi _ a : R[y] \to S$, defined by $y \mapsto \varphi (ax)$ and $r \mapsto \varphi (r)$ for $r\in R$.

Proof. Say $t^ d + \sum _{i < d} \varphi (f_ i)t^ i = 0$ with $f_ i \in R[x]$. Let $\ell $ be the maximum degree in $x$ of all the $f_ i$. Multiply the equation by $\varphi (a)^\ell $ to get $\varphi (a)^\ell t^ d + \sum _{i < d} \varphi (a^\ell f_ i)t^ i = 0$. Note that each $\varphi (a^\ell f_ i)$ is in the image of $\varphi _ a$. The result follows from Algebra, Lemma 10.123.1. $\square$

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Your email address will not be published.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