The Stacks project

Lemma 10.123.3. Let $R$ be a ring. Let $\varphi : R[x] \to S$ be a ring map. Let $t \in S$. Assume $t$ is integral over $R[x]$. Let $p \in R[x]$, $p = a_0 + a_1x + \ldots + a_ k x^ k$ such that $t \varphi (p) \in \mathop{\mathrm{Im}}(\varphi )$. Then there exists a $q \in R[x]$ and $n \geq 0$ such that $\varphi (a_ k)^ n t - \varphi (q)$ is integral over $R$.

Proof. Let $R'$ and $S'$ be the localization of $R$ and $S$ at the element $a_ k$. Let $\varphi ' : R'[x] \to S'$ be the localization of $\varphi $. Let $t' \in S'$ be the image of $t$. Set $p' = p/a_ k \in R'[x]$. Then $t' \varphi '(p') \in \mathop{\mathrm{Im}}(\varphi ')$ since $t \varphi (p) \in \mathop{\mathrm{Im}}(\varphi )$. As $p'$ is monic, by Lemma 10.123.2 there exists a $q' \in R'[x]$ such that $t' - \varphi '(q')$ is integral over $R'$. We may choose an $n \geq 0$ and an element $q \in R[x]$ such that $a_ k^ n q'$ is the image of $q$. Then $\varphi (a_ k)^ n t - \varphi (q)$ is an element of $S$ whose image in $S'$ is integral over $R'$. By Lemma 10.36.11 there exists an $m \geq 0$ such that $\varphi (a_ k)^ m(\varphi (a_ k)^ n t - \varphi (q))$ is integral over $R$. Thus $\varphi (a_ k)^{m + n}t - \varphi (a_ k^ m q)$ is integral over $R$ as desired. $\square$

Comments (2)

Comment #3631 by Brian Conrad on

Why not localize at throughout from the start (harmless for the proof), to instantly reduce to the case handled by the Lemma invoked near the end of the argument and thereby avoid all of the futzing around here?

Comment #3730 by on

Great! This also means I can move Lemma 113.5.1 to the obsolte chapter as it is now no longer used. Changes are here.

There are also:

  • 3 comment(s) on Section 10.123: Zariski's Main Theorem

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