How can I show that for a field $K,$ $K(X)$ is not faithfully flat as $K[X]$ module.
$K(X)$ is a localization of $K[X],$ it is flat $K[X]$ module. But I cannot prove that tensor by $K(X)$ is not a faithful functor. I need some help.
2026-03-25 06:24:29.1774419869
$K(X)$ is not faithfully flat $K[X]$ module.
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More generally, if $R \subset S$ is an extension of integral domains having the same field of fractions and if $S$ is faithfully flat over $R$, then $R=S$.
For a domain $D$ that is not a field, this shows that the field of fractions of $D$ is never faithfully flat over $D$.
A proof of this fact, which appears as an exercise in Matsumura, is found here