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

 

Liftable derivations for generically separably algebraic morphisms of schemes


Author: Rolf Källström
Journal: Trans. Amer. Math. Soc. 361 (2009), 495-523
MSC (2000): Primary 14E22, 13N15; Secondary 14Axx, 13B22, 16W60
Published electronically: June 26, 2008
MathSciNet review: 2439414
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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

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04534-0
PII: S 0002-9947(08)04534-0
Received by editor(s): November 22, 2006
Received by editor(s) in revised form: April 13, 2007
Published electronically: June 26, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.