Universal valued fields and lifting points in local tropical varieties. Issue 2 (1st February 2017)

• Record Type: Journal Article
• Authors: Stepanov, D. A.
• Is Part Of: Communications in algebra. Volume 45:Issue 2(2017)
• Page Start: 469
• Page End: 480
• Publication Date: 2017-02-01
• Subjects: Local ring -- tropical variety -- valuation; Primary 14T05 -- Secondary 12J10; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2016.1175447
• Languages: English
• ISSNs: 0092-7872

Abstract

ABSTRACT: Let k be a field with a real valuation ν and R a k -algebra. We show that there exist a k -algebra K and a valuation μ on K extending ν such that any real valuation of R is induced by μ through some homomorphism from R to K . Let ν be trivial and R a complete local Noetherian ring with the residue field k . Let K be the ring k ̄ [ [ t ℝ ] ] of Hahn series with its natural valuation μ and coefficients in k ̄ . We prove the following weak universality property: For any local valuation v and a finite set of elements x 1, …, x n of R, there exists a homomorphism f : R → K such that v ( x i ) = μ ( f ( x i ) ), i = 1, …, n . This implies that if R = k [ [ x 1, …, x n ] ] ∕ I for an ideal I, then every point of the local tropicalization of I lifts to a K -point of R .