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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Simple derivations of differentiably simple Noetherian commutative rings in prime characteristic


Author: V. V. Bavula
Journal: Trans. Amer. Math. Soc. 360 (2008), 4007-4027
MSC (2000): Primary 13N15, 13A35, 16W25
Published electronically: March 20, 2008
MathSciNet review: 2395162
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Abstract: Let $ R$ be a differentiably simple Noetherian commutative ring of characteristic $ p>0$ (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 $ 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 $ {\rm Aut}(R)$ of ring automorphisms of $ R$. 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 $ 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':= {\rm ker } (\delta)\simeq R/ \mathfrak{m}$.


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Additional Information

V. V. Bavula
Affiliation: Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom
Email: v.bavula@sheffield.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04567-4
PII: S 0002-9947(08)04567-4
Keywords: Simple derivation, iterative $\delta $-descent, differentiably simple ring, differential ideal, coefficient field.
Received by editor(s): February 27, 2006
Published electronically: March 20, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.