# Does the category of algebraically closed fields of characteristic $p$ change when $p$ changes?

EDIT I've now posted this question on mathoverflow. It probably makes sense to post answers over there, unless someone prefers posting here.

Let $\mathrm{ACF}_p$ denote the category of algebraically closed fields of characteristic $p$, with all homomorphisms as morphisms. The question is: when is there an equivalence of categories between $\mathrm{ACF}_p$ and $\mathrm{ACF}_l$ (with the expected answer being: only when $p=l$)?

Here are some easy observations:

1. First, any equivalence of categories must do the obvious thing on objects, preserving transcendence degree because $K$ has a smaller transcendence degree than $L$ if and only if there is a morphism $K \to L$ but not $L \to K$.

2. We can distinguish the case $p=0$ from the case $p \neq 0$ because $\mathrm{Gal}(\mathbb{Q}) \not \cong \mathrm{Gal}(\mathbb{F}_p)$. But $\mathrm{Gal}(\mathbb{F}_p) \cong \hat{\mathbb{Z}}$ for any prime $p$. So we can't distinguish $\mathrm{ACF}_p$ from $\mathrm{ACF}_l$ for different primes $p \neq l$ in such a simple-minded way. So for the rest of this post, let $p,l$ be distinct primes.

3. The next guess is that maybe we can distinguish $\mathrm{ACF}_p$ from $\mathrm{ACF}_l$ by seeing that $\mathrm{Aut}(\overline{\mathbb{F}_p(t)}) \not \cong \mathrm{Aut}(\overline{\mathbb{F}_l(t)})$. To this end, note that there is a tower $\mathbb{F}_p \subset \overline{\mathbb{F}_p} \subset \overline{\mathbb{F}_p}(t) \subset \overline{\mathbb{F}_p(t)}$. The automorphism groups of these intermediate extensions are respectively $\hat{\mathbb{Z}}$, $\mathrm{PGL}_2(\overline{\mathbb{F}_p})$, and a free profinite group (the last one is according to wikipedia).

From (3), there is at least a subquotient $\mathrm{PGL}_2(\overline{\mathbb{F}_p})$ which looks different for different primes. But I don't see how to turn this observation into a proof that $\mathrm{Aut}(\overline{\mathbb{F}_p(t)}) \not \cong \mathrm{Aut}(\overline{\mathbb{F}_l(t)})$ specifically, or that $\mathrm{ACF}_p \not \simeq \mathrm{ACF}_l$ more generally.

Note that if we change "algebraically closed fields of characteristic $p$" to "fields of characteristic $p$", then its easy to distinguish these categories because their subcategories of finite fields look very different.

Also, there is a natural topological enrichment of $\mathrm{ACF}_p$ where one gives the homset the topology of pointwise convergence. I'd be interested to hear of a way to distinguish these topologically-enriched categories.

• Note that if we change "algebraically closed fields of characteristic $p$" to "fields of characteristic $p$", then its easy to distinguish these categories because their subcategories of finite fields look very different. Mar 29, 2016 at 17:51
• I don't agree. The subcategories of finite fields are all equivalent to the poset $(\Bbb{N} , \mbox{divides})$. Mar 29, 2016 at 17:58
• I'm taking all field homomorphisms as morphisms. So the subcategory of finite fields includes the automorphism group of each finite field. But this... actually is not different! So I think you're right to disagree after all! Mar 29, 2016 at 18:00
• No, not the same morphisms (even though it's true that the categories of finite fields for all $p$ are equivalent). Mar 29, 2016 at 18:00
• @egreg Both these automorphism groups are cyclic of order 2, generated by a Frobenius element, right? Mar 29, 2016 at 18:11