Liftable derivations for generically separably algebraic morphisms of schemes
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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.References
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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