Spahn Witt vectors

Motivation in terms of number theory

In an expansion of a $p$-adic number $a=\Sigma a_i p^i$ the $a^i$ are called digits. Usually these digits are defined to be taken elements of the set $\{0,1,\dots,p-1\}$.

Equivalently the digits can be defined to be taken from the set $T_p:=\{x|x^{p-1}=1\}\cup \{0\}$. Elements from this set are called Teichmüller digits or Teichmüller representatives.

The set $T$ is in bijection with the finite field? $F_p$. The set $W(F_p)$ of (countably) infinite sequences of elements in $F_p$ hence is in bijection to the set $\mathbb{Z}_p$ of $p$-adic integers. There is a ring structure on $W(F_p)$ called Witt ring structure such that all ‘’truncated expansion polynomials’‘ $\Phi_n=X^{p^n}+pX^{p^{n-1}}+p^2X^{p^{n-2}}+\dots +p^n X$ called Witt polynomials are morphisms

$$\Phi_n:W(F_p)\to \mathbb{Z}_p$$

of groups.

For algebraists

The construction of Witt vectors gives a functorial way to lift a commutative ring $A$ of prime characteristic $p$ to a commutative ring $W(A)$ of characteristic 0. Since this construction is functorial, it can be applied to the structure sheaf of an algebraic variety. In interesting special cases the resulting ring $W(A)$ has even more desirable properties: If $A$ is perfect $W(A)$ is a discrete valuation ring. This is mainly due to the fact that the construction of $W(A)$ involves a ring of power series and a ring of power series over a field is always a discrete valuation ring.

$$R^\mathbb{N}\to \{1+ R[ [t] ]\}$$

$$O_k^\mathbb{N}\to (\Lambda_k: R\mapsto \{1+ R[ [t] ]\})$$

For category theorists

The functor of forming Witt rings (modulo some details) is a Lambda ring? it can be defined to be the right adjoint to the forgetful functor? forgetting the $\lambda$-structure.