The Stacks project

Lemma 10.127.15. Suppose $R \to S$ is a ring map. Assume that $S$ is integral over $R$. Then there exists a directed set $(\Lambda , \leq )$, and a system of ring maps $R_\lambda \to S_\lambda $ such that

  1. The colimit of the system $R_\lambda \to S_\lambda $ is equal to $R \to S$.

  2. Each $R_\lambda $ is of finite type over $\mathbf{Z}$.

  3. Each $S_\lambda $ is finite over $R_\lambda $.

Proof. Consider the set $\Lambda $ of pairs $(E, F)$ where $E \subset R$ is a finite subset, $F \subset S$ is a finite subset, and every element $f \in F$ is the root of a monic $P(X) \in R[X]$ whose coefficients are in $E$. Say $(E, F) \leq (E', F')$ if $E \subset E'$ and $F \subset F'$. Given $\lambda = (E, F) \in \Lambda $ set $R_\lambda \subset R$ equal to the $\mathbf{Z}$-subalgebra of $R$ generated by $E$ and $S_\lambda \subset S$ equal to the $\mathbf{Z}$-subalgebra generated by $F$ and the image of $E$ in $S$. It is clear that $R = \mathop{\mathrm{colim}}\nolimits R_\lambda $. We have $S = \mathop{\mathrm{colim}}\nolimits S_\lambda $ as every element of $S$ is integral over $S$. The ring maps $R_\lambda \to S_\lambda $ are finite by Lemma 10.36.5 and the fact that $S_\lambda $ is generated over $R_\lambda $ by the elements of $F$ which are integral over $R_\lambda $ by our condition on the pairs $(E, F)$. The lemma follows. $\square$

Comments (4)

Comment #3269 by Dennis Keeler on

Part (3) is missing the word "type" in "finite type."

Comment #3270 by Dario WeiƟmann on

No, and are constructed sucht that is finite and not just of finite type. (We take the algebra generated by finitely many integral elements)

Comment #8110 by Owen on

it should say (3) Each is finite over , not 'of finite.'

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