Let $R$ be a perfect $\mathbb{F}_p$-algebra and write $W(R)$ for the Witt ring [i.e., ring of Witt vectors — PLC] on $R$. I want to know how much we can deduce about $\text{Spec } W(R)$ from knowledge of $\text{Spec } R$. I am especially interested in the case that $R$ is a domain.
Suppose that $R$ is a field. Then it's not hard to see that $W(R)$ is a DVR, so in this case the answer is completely known. For a general $R$, surjections $R \rightarrow R/\mathfrak{p}$ lift to surjections $W(R) \rightarrow W(R/\mathfrak{p})$, and as the latter is a domain, we get a copy of the spectrum of $R$ inside the spectrum of $W(R)$.
Sometimes, though, there are extra primes in the spectrum of $W(R)$! Here's the case that motivated my question. (These rings come up in the study of $(\phi, \Gamma)$-modules and in the construction of Fontaine's rings of periods.) Let $R$ be the "perfection" of $\mathcal{O}_{\mathbb{C}_p}/p$. That is, $R$ is the set of sequences $(x_i)$, $i \geq 0$, where each $x_i \in \mathcal{O}_{\mathbb{C}_p}/p$ and $x_i^{p} = x_{i-1}$. Note that $R$ is a (non-discrete!) valuation ring; reducing mod the maximal ideal of $\mathcal{O}_{\mathbb{C}_p}$ and projecting to the first component gives a surjection to $\overline{\mathbb{F}_p}$. It's not hard to show $R$ is a domain.
There's three obvious prime ideals. First, as before, there's the prime ideal $(p)$, the kernel of the surjection $W(R) \rightarrow R$. Further, the universal property of Witt vectors gives a map $W(R) \rightarrow W(\overline{\mathbb{F}_p}) = \widehat{\mathcal{O}_{\mathbb{Q}_p^{ur}}}$ whose kernel is a prime. Finally there's the maximal ideal, which is the kernel of $W(R) \rightarrow R \rightarrow \overline{\mathbb{F}_p}$. We've used only abstract reasoning about Witt vectors to find these.
But there's a fourth prime ideal! It turns out that there's a surjection $\theta: W(R) \rightarrow \mathcal{O}_{\mathbb{C}_p}$, which is critical in building Fontaine's rings. The existence of $\theta$ can't be deduced from the universal property of Witt vectors, since $\mathcal{O}_{\mathbb{C}_p}$ is not a strict $p$-ring. In the literature, all proofs that $\theta$ is a homomorphism "look under the hood" and actually think about the addition and multiplication of Witt vectors. Can this be abstracted? That is, is there a way to know which "extra" primes we'll get in $W(R)$, just from knowing $R$?
Best Answer
This might not be the answer you expect, but there is a result saying that when $R$ is a non-discrete valuation ring, like the $\mathcal{O}_\mathbb{C_p}^\flat$ you mentioned, then there are infinitely many prime ideals in $W(R)$. The following two papers show that the Krull dimension of $W(R)$ is infinite, moreover, at least the cardinality of the continuum.
Lang and Ludwig - $\mathbb A_{\text{inf}}$ is infinite dimensional
Du - $\mathbb A_{\text{inf}}$ has uncountable Krull dimension