Simple derivations of differentiably simple Noetherian commutative rings in prime characteristic

Abstract

Let $R$ be a {\em differentiably simple Noetherian commutative} ring of characteristic $p>0$ (then $(R, \gm)$ is local with $n:= {\rm emdim} (R)<\infty$). A short proof is given of the Theorem of Harper \cite{Harper61} on classification of differentiably simple Noetherian commutative rings in prime characteristic. The main result of the paper is that there {\em exists} a {\em nilpotent simple} derivation of the ring $R$ such that if $\d^{p^i}\neq 0$ then $\d^{p^i}(x_i)=1$ for some $x_i\in \gm$. The derivation $\d $ is given {\em explicitly}, it is {\em unique} up to the action of the group ${\rm Aut}(R)$ of {\em ring} automorphisms of $R$. Let $\nsder (R)$ be the set of all such derivations. Then $\nsder (R)\simeq {\rm Aut}(R)/{\rm Aut}(R/\gm)$. The proof is based on {\em existence} and {\em uniqueness} of an {\em iterative} $\d$-{\em descent} (for each $\d \in \nsder (R)$), i.e. a sequence $\{y^{[i]}, 0\leq i<p^n\}$ in $R$ such that $y^{[0]}:=1$, $\d(y^{[i]})=y^{[i-1]}$ and $y^{[i]}y^{[j]}={i+j\choose i}y^{[i+j]}$ for all $0\leq i,j<p^n$. For each $\d\in \nsder (R)$, $\Der_{k&#39;}(R)=\oplus_{i=0}^{n-1}R\d^{p^i}$ and $k&#39;:= \ker (\d)\simeq R/ \gm$.