I have some basic questions on the rings of Witt vectors. The first example one looks at is $W(\mathbb{F}_{p})= \mathbb{Z}_{p}$. Is it known if $W(\mathbb{F}_{p}[x]/(x^{n})) = \mathbb{Z}_{p}[x]/(x^{n})$ as a $\mathbb{Z}_{p}$module or even as a ring? If not, is there some kind of deformation that relates the two structures, or some other relationship? Does the Witt construction commute with filtered limits of rings and can something be said about the relationship between $W(\mathbb{F}_{p}[[x]])$ and $\mathbb{Z}_{p}[[x]]$ ? Is there some kind of deformation of ring schemes (or in some other sense) between the ring scheme $\mathbb{A}^{n}_{\mathbb{Z}}$ and the truncated Witt scheme? After the initial question, I am also adding that I am also interested in any possible relationship between $W(\mathbb{F}_{p}[[x^{\frac{1}{p^{\infty}}}]])$ and $\mathbb{Z}_{p}[[x^{\frac{1}{p^{\infty}}}]]$.

$\begingroup$ By the way, what is the definition of $\mathbb{F}_p[[x^{\frac{1}{p^{\infty}}}]]$ and $\mathbb{Z}_p[[x^{\frac{1}{p^{\infty}}}]]$? Is it $\bigcup_{n=0}^{\infty}\mathbb{F}_p[[x^{\frac{1}{p^{n}}}]]$, etc? $\endgroup$– neanderNov 2, 2021 at 8:01
1 Answer
The ring $W(\Bbb F_p[x]/x^n)$ is definitely not the one you describe. For example, if $n=2$ consider the map $\Bbb F_p[x]/x^2 \to \Bbb F_p$ that sends $x$ to zero. This induces a surjection $W(\Bbb F_p[x]/x^2) \to W(\Bbb F_p) = \Bbb Z_p$ whose kernel is an ideal $I$ with the following properties:
The unit makes it split off, so $W(\Bbb F_p[x]/x^2) \cong \Bbb Z_p \oplus I$.
As an abelian group, it is the product $\prod_{k \geq 0} \Bbb F_p$. In particular, all elements in $I$ are $p$torsion.
It is squarezero: $ab = 0$ for any $a$, $b \in I$.
You can verify these directly by the formulas: for example, $p \cdot (a_0,a_1,\dots) = (a_0^p, a_1^p, \dots)$ for rings of characteristic $p$.
(Similar things can be said if $n \leq p1$. This is connected to the fact that, in that case, the ideal $(x) \subset \Bbb F_p[x]/x^n$ has a divided power structure, and it's important for deformation theory.)
As a result, the ring of Witt vectors satisfies very few "nice" properties in general.
As far as commuting with filtered limits: The underlying set of $W(R)$ is naturally $\prod_{k \geq 0} R$, and the forgetful functor from rings to sets creates limits. Since the underlying setvalued functor of $W(R)$ preserves all limits, so does $W(R)$.

$\begingroup$ Thanks very much, I am still wondering about $W(\mathbb{F}_{p}[[x]])$ and $\mathbb{Z}_{p}[[x]]$ and in fact, the relationship between $W(\mathbb{F}_{p}[[x^{\frac{1}{p^{\infty}}}]])$ and $\mathbb{Z}_{p}[[x^{\frac{1}{p^{\infty}}}]]$. But Tyler's answer shows that it can't be very simple. $\endgroup$ May 8, 2015 at 16:23