Scheme point

A scheme point is a point of the underlying topological space of a scheme, together with the local algebra supplied by its structure sheaf. On an affine scheme $\mathrm{Spec}R$, points correspond to prime ideals $\mathfrak{p}\subset R$, and the local ring at the point is $R_{\mathfrak{p}}$. Closed points correspond to maximal ideals; schemes also have nonclosed points.

For $\mathrm{Spec}\mathbb{Z}$, each prime number $p$ gives a closed point $(p)$ with residue field ${\mathbb{F}}_p$. The prime ideal $(0)$ is a nonclosed generic point with residue field $\mathbb{Q}$. Its closure is the whole spectrum, so it records the common generic arithmetic rather than one reduction modulo a prime.

More generally, the point for $\mathfrak{q}$ is a specialization of the point for $\mathfrak{p}$ exactly when $\mathfrak{p}\subseteq \mathfrak{q}$. Thus incidence in the prime-ideal ordering becomes topology. Treating every scheme point as an ordinary coordinate tuple would discard this generic and specialization information.