Spahn Witt vectors (changes)

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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

of groups.

For category theorists

The construction functor of forming Witt vectors rings gives (modulo some details) is a functorial way to lift a commutative ring$A$Lambda ring? of it prime can characteristic be defined to be the right adjoint to the$p$forgetful functor? to forgetting a the commutative ring$\mathrm{<span><del\; class="diffmod">\; W</del><ins\; class="diffmod">\; \lambda </ins></span>}(A)$ W(A) \lambda -structure. 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.