# Counting points on a scheme of finite type over an infinite field

Let $$k$$ be an infinite field. Let $$f:X\rightarrow \mathrm{Spec}\:k$$ be a morphism of finite type. Assume that $$X$$ is not the empty scheme and that $$f$$ is not of relative dimension $$\leq 0$$ (definition).

Do the following sets have equal cardinalities:

• the set of elements of $$k$$
• the set of closed points of $$X$$
• the set of all points of $$X$$?

Remark: I am not sure if this is non-trivial enough to be on MO, feel free to downvote/vote to delete/transfer to MSE if you think this is trivial. I tried to write an obnoxiously detailed answer, primarily for my own understanding. Please point out if the argument is non-optimal in some places.

Throughout the answer, $$k$$ is a fixed infinite field. Since closed immersions are of finite type and $$X_{red}\rightarrow X$$ induces a bijection on the sets of all points/closed points, we can replace $$X$$ by $$X_{red}$$.

First, note the following.

Let $$k$$ be an infinite field. Let $$d$$ be a positive integer. Then the cardinality of the set of maximal ideals of $$k[x_1, \dots, x_d]$$ is equal to the cardinality of the set of prime ideals of $$k[x_1, \dots, x_d]$$ and is equal to the cardinality of $$k$$.

For an integral affine scheme of finite type over $$k$$ and of relative dimension $$d>0$$, the set of closed points and the set of all points have cardinality equal to that of $$k$$. To see this, note that by Noether normalization there is a finite surjective morphism $$X\rightarrow \mathbb{A}_k^d$$.

By definition, the image of a closed point under a finite morphism between arbitrary schemes is a closed point. I also claim that the preimage of a closed point under a finite morphism between schemes of finite type over $$k$$ consists of closed points. This is because Nullstellensatz says that a point on such a scheme is closed iff the residue field is a finite extension of $$k$$, and for finite $$f$$ the field extension $$k(f(x))\subset k(x)$$ is finite. The second claim is probably true without any assumptions on the schemes but I don't know how to prove it.

Therefore, the cardinality of the set of closed points of $$X$$ is at least the cardinality of the set of closed points of $$\mathbb{A}^k_d$$ and at most $$\aleph_0$$ times the cardinality of the set of closed points of $$\mathbb{A}^k_d$$ (since finite morphisms are quasi-finite). The same is true verbatim for the set of all points so we are done by the boxed statement.

By the hypothesis on relative dimension, we can an irreducible component $$Y\subset X$$ such that $$f|_Y$$ is not of relative dimension $$\leq 0$$. Closed immersions are of finite type, so $$Y$$ (with its reduced induced scheme structure) is an integral scheme of finite type over $$k$$. Since $$\mathrm{Spec}\:k$$ has only one point, it follows that $$f|_Y$$ has a well-defined relative dimension $$d>0$$. The closed immersion $$Y\rightarrow X$$ gives us an injection from the set of points of $$Y$$ into the set of points of $$X$$, and an injection from the set of closed points of $$Y$$ into the set of the closed points of $$X$$.

Now consider any non-empty affine open $$U\subset Y$$ (whose structure morphism is automatically of relative dimension $$d$$). For schemes of finite type over a field, a point is closed iff it is closed in some affine open. Thus the open immersion gives us an injection from the set of points of $$U$$ into the set of points of $$Y$$, and an injection from the set of closed points of $$U$$ into the set of the closed points of $$Y$$. So the cardinality of the set of closed points/all points of $$X$$ is at least the cardinality of $$k$$.

Now, let's go in the converse direction. Since $$X$$ is of finite type over a Noetherian scheme ($$\mathrm{Spec}\:k$$), it is Noetherian. In particular, it has finitely many irreducible components $$C_i$$ (which, being closed subschemes of a Noetherian scheme, are Noetherian). Cover each of $$C_i$$ by finitely many non-empty affine opens $$U_{ij}$$. Then each $$U_{ij}$$ is an integral scheme of finite type over a field (integral because it is a non-empty open subscheme of an integral scheme, of finite type because open immersions into a Noetherian scheme are of finite type and closed immersions are of finite type).

We have a surjective morphism $$\sqcup U_{ij}\rightarrow X$$ which sends closed points to closed points (combine the fact that a point is closed in a closed subset iff it is closed in the entire space and the second link above) and under which the preimage of a closed point consists of closed points. Thus the cardinality of the set of closed points of $$X$$ is at most $$\aleph_0$$ times the cardinality of $$k$$ and the same applies to the set of all points.