Lemma 10.39.20 (090N)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.39: Flat modules and flat ring maps
Lemma 10.39.20 (cite)

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Lemma 10.39.20. Let $R$ be a ring. Let $\{ S_ i, \varphi _{ii'}\} $ be a directed system of faithfully flat $R$-algebras. Then $S = \mathop{\mathrm{colim}}\nolimits _ i S_ i$ is a faithfully flat $R$-algebra.

Proof. By Lemma 10.39.3 we see that $S$ is flat. Let $\mathfrak m \subset R$ be a maximal ideal. By Lemma 10.39.16 none of the rings $S_ i/\mathfrak m S_ i$ is zero. Hence $S/\mathfrak mS = \mathop{\mathrm{colim}}\nolimits S_ i/\mathfrak mS_ i$ is nonzero as well because $1$ is not equal to zero. Thus the image of $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ contains $\mathfrak m$ and we see that $R \to S$ is faithfully flat by Lemma 10.39.16. $\square$

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