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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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Liftable derivations for generically separably algebraic morphisms of schemes
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by Rolf Källström PDF
Trans. Amer. Math. Soc. 361 (2009), 495-523 Request permission

Abstract:

We consider dominant, generically algebraic (e.g. generically finite), and tamely ramified (if the characteristic is positive) morphisms $\pi : X/S \to Y/S$ of $S$-schemes, where $Y,S$ are Nœtherian and integral and $X$ is a Krull scheme (e.g. normal Nœtherian), and study the sheaf of tangent vector fields on $Y$ that lift to tangent vector fields on $X$. We give an easily computable description of these vector fields using valuations along the critical locus. We apply this to answer the question when the liftable derivations can be defined by a tangency condition along the discriminant. In particular, if $\pi$ is a blow-up of a coherent ideal $I$, we show that tangent vector fields that preserve the Ratliff-Rush ideal (equals $[I^{n+1}:I^n]$ for high $n$) associated to $I$ are liftable, and that all liftable tangent vector fields preserve the integral closure of $I$. We also generalise in positive characteristic Seidenberg’s theorem that all tangent vector fields can be lifted to the normalisation, assuming tame ramification.
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Additional Information
  • Rolf Källström
  • Affiliation: Department of Mathematics, University of Gävle, 801 76 Gävle, Sweden
  • Email: rkm@hig.se
  • Received by editor(s): November 22, 2006
  • Received by editor(s) in revised form: April 13, 2007
  • Published electronically: June 26, 2008
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 495-523
  • MSC (2000): Primary 14E22, 13N15; Secondary 14Axx, 13B22, 16W60
  • DOI: https://doi.org/10.1090/S0002-9947-08-04534-0
  • MathSciNet review: 2439414