The Stacks project

Lemma 10.36.14. Let $R \to S$ be a ring map. Let $f_1, \ldots , f_ n \in R$ generate the unit ideal.

  1. If each $R_{f_ i} \to S_{f_ i}$ is integral, so is $R \to S$.

  2. If each $R_{f_ i} \to S_{f_ i}$ is finite, so is $R \to S$.

Proof. Proof of (1). Let $s \in S$. Consider the ideal $I \subset R[x]$ of polynomials $P$ such that $P(s) = 0$. Let $J \subset R$ denote the ideal (!) of leading coefficients of elements of $I$. By assumption and clearing denominators we see that $f_ i^{n_ i} \in J$ for all $i$ and certain $n_ i \geq 0$. Hence $J$ contains $1$ and we see $s$ is integral over $R$. Proof of (2) omitted. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 10.36: Finite and integral ring extensions

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Your email address will not be published.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 02JL. Beware of the difference between the letter 'O' and the digit '0'.