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. $\square$

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