In this post, we’ll explore a particularly nice situation in which the set of closed points of an affine scheme is dense. Those affine schemes are closer to our geometric intuition than other types of schemes: that’s because we expect points in a space to be “atomic”, i.e. they make up everything, and so they are everywhere (they are “dense”).
Proposition. Let be an algebra of finite type over a field . In this situation, the closed points are a dense subset of .
Proof. Because the distinguished open sets form a basis for the topology on , it suffices to show that any nonempty contains a closed point.
Let be an element of that is not nilpotent. Because is of finite type, the localization is also of finite type (just add to some finite set of generators for ). Therefore, the canonical localization homomorphism induces an inclusion map which sends closed points to closed points. This inclusion map is an homeomorphism on its image, which is . Hence the image of any closed point in is a closed point in , and there exists at least one closed point in since is not the zero ring ( is not nilpotent).
Here’s a fun algebraic application of this schematic fact! We can use it to show that the algebra is not finitely generated. By the proposition, it suffices to show that the set of closed points in is not dense. This space is very simple: it only has two points, the generic point and the closed point . But since is closed, that means is an open point. In particular, there exists an open set, namely , which contains no closed points. Therefore, the set of closed points, namely , is not dense in the space, which shows cannot be of finite type. Pretty neat!