Are there $k$-rational points that are not closed point

Question

I know for a scheme $X$ locally of finite type over a field $k$, $k$-rational points are closed ponits.

If we remove the assumption that $X$ is locally of finite type over $k$, are there some $k$-rational points which are not closed point?

Answer 1

Reducing to the affine case, your question comes down to the following: if $A$ is an algebra over a field $k$, not necessarily finitely generated, and $\mathfrak p$ is a prime of $A$ such that $k \rightarrow A_{\mathfrak p}/\mathfrak p A_{\mathfrak p}$ is an isomorphism, is it possible for $\mathfrak p$ to be a nonmaximal ideal? The answer is no.

By hypothesis, the composition $k \xrightarrow{i} A \xrightarrow{\pi} A/\mathfrak p \xrightarrow{h} \operatorname{Quot}(A/\mathfrak p) = A_{\mathfrak p}/\mathfrak p A_{\mathfrak p}$ is an isomorphism. Since

$$h \circ (\pi \circ i)$$

is a bijection, it is in particular surjective, which implies that $h$ must be surjective. The surjectivity of

$$A/\mathfrak p \rightarrow \operatorname{Quot}(A/\mathfrak p)$$ implies that $A/\mathfrak p$ is a field, i.e. $\mathfrak p$ is maximal.