Simple derivations of differentiably simple Noetherian commutative rings in prime characteristic

Abstract:

Let $R$ be a differentiably simple Noetherian commutative ring of characteristic $p>0$ (then $(R, \mathfrak {m})$ is local with $n:= \textrm {emdim} (R)<\infty$). A short proof is given of the Theorem of Harper (1961) on classification of differentiably simple Noetherian commutative rings in prime characteristic. The main result of the paper is that there exists a nilpotent simple derivation $\delta$ of the ring $R$ such that if $\delta ^{p^i}\neq 0$, then $\delta ^{p^i}(x_i)=1$ for some $x_i\in \mathfrak {m}$. The derivation $\delta$ is given explicitly, and it is unique up to the action of the group $\textrm {Aut}(R)$ of ring automorphisms of $R$. Let $\operatorname {nsder}(R)$ be the set of all such derivations. Then $\operatorname {nsder} (R)\simeq \textrm {Aut}(R)/\textrm {Aut}(R/\mathfrak {m})$. The proof is based on existence and uniqueness of an iterative $\delta$-descent (for each $\delta \in \operatorname {nsder}(R)$), i.e., a sequence $\{ y^{[i]}, 0\leq i<p^n\}$ in $R$ such that $y^{[0]}:=1$, $\delta (y^{[i]})=y^{[i-1]}$ and $y^{[i]}y^{[j]}={i+j\choose i} y^{[i+j]}$ for all $0\leq i,j<p^n$. For each $\delta \in \operatorname {nsder}(R)$, $\operatorname {Der}_{k’}(R)=\bigoplus _{i=0}^{n-1}R\delta ^{p^i}$ and $k’:= \textrm {ker } (\delta )\simeq R/ \mathfrak {m}$.