Extensions of Valuations to the Henselization and Completion

2017/12/22 by Cutkosky, Steven Dale
#11S15 #13B10 #14B05 #14B22 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1712.08562

Abstract

We show that if R is a local domain which is dominated by a valuation ν, then there does not always exist a regular local ring R' which birationally dominates R and is dominated by ν and an extension of ν to the Henselization (R')^h of R' such that the associated graded rings of R' and (R')^h along the valuations are equal. We also show that there does not always exist R', a prime ideal p of the completion \widehatR' such that p∩ R'=(0) and an extension of ν to \widehatR' such that the associated graded rings of R' and \widehatR'/p along the valuation are equal.

Related

• Valuations and henselization
• Henselian Elements
• Completions of valuation rings
• Uniformly defining p-henselian valuations
• On the construction of valuations and generating sequences on hypersurface singularities