$\mathbb{Z}_p$, $\mathbb{Q}_p$, and the ring of Witt vectors

No.05:

From here on, we make use of several notions of category theory. Readers who are unfamiliar with the subject is advised to see a book such as [1] for basic definitions and first properties.

Let $p$ be a prime number. For any commutative ring $k$ of characteristic $p\neq 0$, we want to construct a ring $W(k)$ of characteristic 0 in such a way that:

1. $W(\mathbb{F}_p)=\mathbb{Z}_p$.
2. $W(\bullet)$ is a functor. That means,
   1. For any ring homomorphism $\varphi: k_1\to k_2$ between rings of characterisic $p$, there is given a unique ring homomorphism $W(\varphi):W(k_1)\to W(k_2)$.
   2. $W(\bullet)$ should furthermore commutes with compositions of homomorphisms.

Recent days, it gets easier for us on the net to i find some good articles concerning the ring of Witt vectors. The treatment here borrows some ideas from them. See for example the “comments” section in https://www.encyclopediaofmath.org/index.php/Witt_vector