Lemma 15.127.3. In Situation 15.127.1 let $x_1, \ldots , x_ n \in \Omega$ be pairwise distinct. Let $v_ i \in V(x_ i)$. Then there exists an $s \in M$ such that $s(x_ i)$ maps to $v_ i$ for $i = 1, \ldots , n$.

Proof. Since $x_ i$ is a maximal ideal of $R$ we may use Algebra, Lemma 10.15.4 to see that $M(x_1) \oplus \ldots \oplus M(x_ n)$ is a quotient of $M$. $\square$

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).