Return to Revisions

Let $\sigma$ denote the nontrivial automorphism of $\mathbb{F}_4$ and put $C=\{1,\sigma\}$. Let $\mathcal{R}$ denote the category of rings in which $a^4=a$, and let $\mathcal{X}$ denote the category of Stone spaces with an action of $C$. There is a functor $F\colon\mathcal{X}\to\mathcal{R}$ given by $F(X)=\text{Map}_C(X,\mathbb{F}_4)$, and there is a functor $G\colon\mathcal{R}\to\mathcal{X}$ given by $G(R)=\text{Rings}(R,\mathbb{F}_4)$ (with the obvious Zariski-type topology). There are evident natural maps $R\to FG(R)$ and $X\to GF(X)$. I have not checked that these are isomorphisms, but I would be surprised if they were not.