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 $R$?

  • $\begingroup$ No thoughts on how to answer your question, but as I think I know what brought it up let me ask: is there any ergodic reasons behind this question? $\endgroup$
    – Ben Weiss
    Mar 7 '10 at 6:13
  • $\begingroup$ No, I don't think so. $\endgroup$ Mar 7 '10 at 16:11
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    $\begingroup$ This is not an answer to your question but just a comment on your comments about $\theta$. From my point of view, a better definition of $W(R)$, where $R$ is the perfection of $O_{\mathbf{C}_p}$ is the inverse limit of $W(O_{\mathbf{C}_p})$ with respect to the Frobenius maps $F$. (It's then an exercise to prove that this agrees with $W(R)$.) The map to $O_{\mathbf{C}_p}$ is then just the projection onto the zeroth component of the zeroth Witt vector in the system.It's surjective because for any $x\in O_{\mathbf{C}_p}$, the sequence of Teichm\"uller elements $[x],[x^{1/p}],\dots$ is in it. $\endgroup$
    – JBorger
    Apr 24 '10 at 9:18
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    $\begingroup$ Wait, wait -- you mean the "ring of Witt vectors", not the "Witt ring in the sense of quadratic forms", as became clear when I read farther. Mea culpa. (Witt has too many rings!) $\endgroup$ Aug 25 '10 at 9:47
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    $\begingroup$ Thanks for the edit! There should really be a more succinct name for "the Witt vector ring over R," given that brevity is the soul of Witt. $\endgroup$ Aug 25 '10 at 14:44

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.


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