Simple derivations of differentiably simple Noetherian commutative rings in prime characteristic

Details Simple derivations of differentiably simple Noetherian commutative rings in prime characteristic Simple derivations of differentiably simple Noetherian commutative rings in prime characteristic

2006-02-27

arxiv.org/abs/math/0602632v2

Mathematics

Rings and Algebras

17 pages

Scientific paper

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

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