# A weird Dedekind domain

Here is a random question that I was wondering about at some point (just out of curiosity):

Question. Does there exist a Dedekind domain that has infinitely many points whose residue field has characteristic and finitely many points (at least one) whose residue field has positive characteristic?

I don’t fully recall why this question came up, but it had something to do with a similar property that was satisfied by an object involved in the definition of the Farguesâ€“Fontaine curve. However, we don’t need such deep theory to discuss this elementary commutative algebra question.

Lemma. Such a Dedekind domain exists.

Proof. We will construct as a localisation of . Recall that prime ideals of come in four types:

• The generic point , of height 0;
• Height 1 primes for every prime ;
• Height 1 primes for every irreducible polynomial ;
• Height 2 closed points for a prime and a polynomial whose reduction is irreducible.

We first localise at ; then the only primes we have left are the ones contained in . This is the generic point, the height 1 prime , the height 1 primes where is an irreducible polynomial whose constant coefficient is divisible by (e.g. ), and exactly one height 2 prime .

Next, we invert ; denoting the resulting ring by . This gets rid of all prime ideals containing , which are and . In particular, there are no more height 2 primes, so is 1-dimensional. It is a normal Noetherian domain because it a localisation of a normal Noetherian domain. Therefore, is a Dedekind domain.

The primes of are with residue field ; the prime with residue field ; and the prime ideals with a polynomial whose constant coefficient is divisible by , whose residue field is a finite extension of .

Remark. The ring we constructed is essentially of finite type over (a localisation of a finite type -algebra). There are no examples of finite type over , because by Chevalley’s theorem the image of would be constructible. However, no set of the form for finite is constructible. (Alternatively, the weak Nullstellensatz implies that every closed point of a finite type -algebra has residue characteristic .)

I was a little surprised that we can make examples when we drop the finite type assumption. I don’t know if this type of ring has ever been used for anything.