Let $X$ be a scheme over a field $k$.
Consider the following definitions.
-
The residue field of a point $x\in X$ is $k(x)=\mathcal{O}_{X,x}/\mathfrak{m}_x$.
-
The $k$-point of $X$ is the morphism of schemes $\text{Spec}\,k\to X$ such that the composition with the structure morphism $X\to\text{Spec}\,k$ gives identity morphism $\text{Spec}\,k\to\text{Spec}\,k$.
I have some troubles with these definitions. Suppose we have a point $x\in X$ with the residue field $k(x)=k$. Is it then true that we have a $k$-point of $X$?
In other words, is there an example of a field $k$ and a scheme $X$ over $k$, such that $X$ doesn't have $k$-points but there exists a point with the residue field $k$?
Best Answer
Let $X$ be a $k$-scheme. Then the $k$-points of $X$ are exactly the scheme-theoretic points with residue field $k$.
Indeed, since $X$ is a $k$-scheme, the residue field of any scheme-theoretic point of $X$ must be a field extension of $k$, so the image of a $k$-morphism $\operatorname{Spec} k \to X$ must be a scheme-theoretic point with residue field $k$. The converse is easy: if $x$ is a scheme-theoretic point with residue field $K$, then it corresponds to a $k$-morphism $\operatorname{Spec} K \to X$.