Lemma 15.104.20 (092U)—The Stacks project

Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.104: Weakly étale ring maps
Lemma 15.104.20 (cite)

Lemma 15.104.20. Let $A$ be a normal domain with fraction field $K$. There exists a cartesian diagram

\[ \xymatrix{ A \ar[d] \ar[r] & K \ar[d] \\ V \ar[r] & L } \]

of rings where $V$ has weak dimension at most $1$ and $V \to L$ is a flat, injective, epimorphism of rings.

Proof. For every $x \in K$, $x \not\in A$ pick $V_ x \subset K$ as in Algebra, Lemma 10.50.11. Set $V = \prod _{x \in K \setminus A} V_ x$ and $L = \prod _{x \in K \setminus A} K$. The ring $V$ has weak dimension at most $1$ by Lemma 15.104.19 which also shows that $V \to L$ is a localization. A localization is flat and an epimorphism, see Algebra, Lemmas 10.39.18 and 10.107.5. $\square$

Comments (2)

Comment #3759 by Laurent Moret-Bailly on October 30, 2018 at 05:47

Typo in proof: " $V\rightarrow K$ " should be " $V\rightarrow L$ ". Also, in the statement, why discard the extra information that $V\rightarrow L$ is a localization?

Comment #3892 by Johan on January 22, 2019 at 10:00

THanks and fixed here. We never use that the map is a localization.

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