Lemma 10.106.8. If $k \to S$ is an epimorphism and $k$ is a field, then $S = k$ or $S = 0$.
This is clear from the result of Lemma 10.106.7 (as any nonzero algebra over $k$ is faithfully flat), or by arguing directly that $R \to R \otimes _ k R$ cannot be surjective unless $\dim _ k(R) \leq 1$.
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).