# $p$-adic completeness of the ring of Witt vectors

Let $R$ be a ring that is $p$-adically complete for a prime $p$ and let $W(R)$ denote the ring of $p$-typical Witt vectors. Is it true that $W(R)$ is $p$-adically complete? (A ring $A$ is $p$-adically complete if the map $A \rightarrow \varprojlim(A/p^nA)$ is bijective.) A reference that contains a proof or a counterexample would suffice.

I am familiar with the case when $R$ is a perfect $\mathbb{F}_p$-algebra and am interested in the general case.

• I believe that the p-Witt vectors for $\mathbf{Z}_p$ is isomorphic, as an abelian group, to a product of countably infinitely many copies of $\mathbf{Z}_p$ (I believe this because if memory serves then Hendrik Lenstra told me this in 2002, and if it's wrong then it's my memory at fault), and the p-Witt vectors for $\mathbf{Z}/p^m$ is isomorphic as an abelian group to a product of $\mathbf{Z}_p$ and countably infinitely many copies of $\mathbf{Z}/p^{m-1}$ (with the same reference and caveat). So there are least are some examples when it's true.
– znt
Jun 4, 2016 at 12:18
• @znt if $G$ is a compact abelian group such that for all $x$, $p^nx\to 0$ when $n\to\infty$ and $G$ torsion-free, then $G$ is isomorphic to some power of $\mathbf{Z}_p$. This is easy to establish with Pontryagin duality (because the Pontryagin dual is a discrete, divisible abelian group in which every element is killed by some power of $p$, hence is a direct sum of copies of $\mathbf{Z}[1/p]/\mathbf{Z}$).
– YCor
Jun 4, 2016 at 13:17
• Perhaps you already know this, but page 11, proposition 3 (and the subsequent sub-lemmas) of Zink's "The display of a formal $p$-divisible group" looks relevant. Here's the address of the paper on his website... math.uni-bielefeld.de/~zink/display.pdf Jun 7, 2016 at 8:29
• @OliGregory: Thanks! You should post this as an answer: as far as I can tell the reference you give resolves the question. Jun 7, 2016 at 15:08

The answer is yes. A reference is $\S1$, proposition 3 of "The display of a formal $p$-divisible group" by Thomas Zink, published in Asterisque no. 278. Here is the address of the paper on Zink's website...