Let $R\in CRing$ a commutative unitary ring. Let $W(R)$ denote the ring of [[Witt vectors]] of $R$. Let $$w_n:\begin{cases} W(R)\to R \\ (x_0,\dots,x_i,\dots)\mapsto x_0^{p^n}+p x_1^{p^{n-1}}+ \dots + p^n x_n \end{cases}$$ denote the morphism of rings assigning to Witt vector its correspnding [[Witt polynomial]]. Let $$w_n:\begin{cases} W(R)\to W(R) \\ (x_0,\dots,x_i,\dots)\mapsto (0,x_0,\dots,x_i,\dots) \end{cases}$$ denote the [[Verschiebung morphism]] which is a morphism of the underlying additive groups. Let $p$ be a [[prime number]] and let $$F:W(R)\to W(R)$$ denote the [[Frobenius morphism]]. Then Frobenius, Verschiebung, and the Witt-polynomial morphism satisfy the ''$p$-adic Witt-Frobenius identities'': $$w_n(F(x))=w_{n+1}(x)\; n\ge 0$$ $$w_n(V(x))=w_{n-1}(x)\; n\gt 0$$ $$w_0(V(x))=0$$ $$FV=p$$ $$VF(xy)=xV(y)$$ ## References * T. Zink, the display of a formal p-divisible group, to appear in Asterisque, [pdf](http://www.math.uni-bielefeld.de/~zink/display.pdf) * T. Zink, Windows for displays of p-divisible groups. in:Moduli of Abelian Varieties, Progress in Mathematics 195, Birkhäuser 2001, [pdf](http://www.math.uni-bielefeld.de/~zink/Texel.pdf)