Residue field of a point

Remarks

Example: for $X = \operatorname{Spec} A$ and a prime ideal $\mathfrak{p}$ of $A$, the residue field is $\kappa(\mathfrak{p}) = A_{\mathfrak p} / \mathfrak{p} A_{\mathfrak p}$. It is isomorphic to the field of fractions of $A / \mathfrak{p}$. For $A = \Z$ this gives $\kappa((p)) = \F_p$ for a prime $p$ and $\kappa((0)) = \Q$. For $A = \R[x]$ and $\mathfrak{p} = (x^2 + 1)$ it gives $\R[x]/(x^2+1) \cong \C$, so this point has residue field larger than $\R$. Non-example: the stalk $\mathcal{O}_{X,x}$ itself is in general not a field; for $\Z$ at $(p)$ it is $\Z_{(p)}$, which has the non-unit $p \ne 0$. For a field $K$, morphisms $\operatorname{Spec} K \to X$ correspond to pairs of a point $x \in X$ and a field embedding $\kappa(x) \to K$.