The Stacks project

Lemma 15.9.9. Let $I$ be an ideal of a ring $A$. Let $A \to B$ be an integral ring map. Let $b \in B$ map to an idempotent in $B/IB$. Then there exists a monic $f \in A[x]$ with $f(b) = 0$ and $f \bmod I = x^ d(x - 1)^ d$ for some $d \geq 1$.

Proof. Observe that $z = b^2 - b$ is an element of $IB$. By Algebra, Lemma 10.38.4 there exist a monic polynomial $g(x) = x^ d + \sum a_ j x^ j$ of degree $d$ with $a_ j \in I$ such that $g(z) = 0$ in $B$. Hence $f(x) = g(x^2 - x) \in A[x]$ is a monic polynomial such that $f(x) \equiv x^ d(x - 1)^ d \bmod I$ and such that $f(b) = 0$ in $B$. $\square$

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Your email address will not be published.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 09XG. Beware of the difference between the letter 'O' and the digit '0'.