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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Simple derivations of differentiably simple Noetherian commutative rings in prime characteristic
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by V. V. Bavula PDF
Trans. Amer. Math. Soc. 360 (2008), 4007-4027 Request permission

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 $k’:= \textrm {ker } (\delta )\simeq R/ \mathfrak {m}$.
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Additional Information
  • V. V. Bavula
  • Affiliation: Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom
  • MR Author ID: 293812
  • Email: v.bavula@sheffield.ac.uk
  • Received by editor(s): February 27, 2006
  • Published electronically: March 20, 2008
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 4007-4027
  • MSC (2000): Primary 13N15, 13A35, 16W25
  • DOI: https://doi.org/10.1090/S0002-9947-08-04567-4
  • MathSciNet review: 2395162