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Simple derivations of differentiably simple Noetherian commutative rings in prime characteristic

Authors:	V V Bavula

Institution:	Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom

Abstract:	Let be a differentiably simple Noetherian commutative ring of characteristic (then $(R, \mathfrak{m})$ is local with $n:= {\rm emdim} (R)<\infty$ ). 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 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 ${\rm Aut}(R)$ of ring automorphisms of . Let $\operatorname{nsder}(R)$ be the set of all such derivations. Then $\operatorname{nsder} (R)\simeq {\rm Aut}(R)/{\rm Aut}(R/\mathfrak{m})$ . 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 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 $k':= {\rm ker } (\delta)\simeq R/ \mathfrak{m}$ .

Keywords:	Simple derivation iterative $\delta $-descent differentiably simple ring differential ideal coefficient field

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