[[!redirects Witt vectors and the Tate module]] [[!redirects Prüfer group, Witt vectors and the Tate module]] ## 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.