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Let $A$ be a $\mathbb{F}_p$-algebra of finite presentation. Then we know that $W(A)$ is a $W(\mathbb{F}_p)=\mathbb{Z}_p$-algebra. My question is if there is a "good" presentation of $W(A)$ as $\mathbb{Z}_p$-algebra. If it is helpful for this, I'm perfectly happy to assume that $A$ is smooth or a hypersurface (by which I mean $A\cong \mathbb{F}_p[x_1,\ldots,x_n]/f$).

curious math guy
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No, there is not. Even in the simple case were $A=\mathbb{F}_{p}\left[X_{1},\ldots,X_{n}\right]$, then $W\left(\mathbb{F}_{p}\right)\to W\left(A\right)$ is not even of finite type.

It was first shown by Illusie (I think) that $W\left(\mathbb{F}_{p}\left[X\right]\right)$ is isomorphic to the ring of formal series: \begin{equation*} \left\{\sum_{k\in\mathbb{N}\left[\frac{1}{p}\right]}p^{\min\{-\operatorname{v}_{p}\left(k\right),0\}}a_{k}X^{k}\mid\forall k\in\mathbb{N}\left[\frac{1}{p}\right],\ a_{k}\in\mathbb{Z}_{p},\ p^{\min\{-\operatorname{v}_{p}\left(k\right),0\}}a_{k}\rightarrow0\right\}\text{.} \end{equation*}

To the best of my knowledge, nobody has an explicit description for the ring of Witt vectors of a smooth commutative ring. This is why authors always study the ring of Witt vectors locally on $\operatorname{Spec}\left(A\right)$, that is they instead suppose they have a finite étale morphism $\mathbb{F}_{p}\left[X_{1},\ldots,X_{n}\right]\to A$.

user242429
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