nLab geometric point

Idea

The notion of geometric point refers to a certain kind of morphism between schemes in algebraic geometry; that is, to a specialized sort of generalized element of a scheme.

While a point in a topological space, $X$, can be thought of as a continuous function from a singleton space to $X$, in algebraic geometry the ‘spaces’ come with more structure as they are schemes and singletons correspond to the spectra of fields. Category-theoretically, one may think of any morphism into $X$ as a generalized point of $X$, but when doing geometry it is often appropriate to restrict to a subclass of these to consider as the (less generalized) “points”.

Definition

Suppose a scheme $S$ is defined over a field $k$, so is equipped with a morphism to $Spec (k)$.

Definition

A geometric point $\xi$ in $S$ is a morphism from the spectrum $Spec(K)$ to $S$ where $K$ is an algebraic closure/separable closure of $k$.

Remark

In general the set of geometric points of a scheme is different from the set of ordinary points of its underlying topological space.

Related concepts

• singular point of an algebraic variety

• étale fundamental group

• generalized element

• geometric fiber

• section conjecture

References

• EGA

• SGA1

• James Milne, section 4 of Lectures on Étale Cohomology