Suppose $X \to k$ is a scheme over a field. A point $x$ is defined to be rational(by Hartshorne) if $k(x) = \mathcal{O}_{X,x}/{\mathfrak{m}_x}$ is isomorphic to $k$.
My questions are:
1) Does he mean that the canonical map($k \to \mathcal{O}_{X,x} \to \mathcal{O}_{X,x}/{\mathfrak{m}_x}$) is an isomorphism?(or will any isomorphism do?)
2) Does the fact that they are isomorphic imply that the canonical map is an isomorphism?
3) If 2) does not hold in general does it hold if $X$ is locally of finite type over $k$?
Best Answer
From the comments:
I dug out my copy of Hartshorne to check exactly what he said, and the first and only time he defines a rational point is on page 80 (exercise II.2.8, final two lines), and he says that a rational point is one so that "$k(x)=k$". So the canonical map should be the isomorphism involved. This is also true if one requires that the isomorphism be as $k$-algebras, which one should do if one is working in the category of schemes over a field. Basically, these definitions are/should be constructed to prohibit the issues you're considering.