On the structure of the graded algebra associated to a valuation

Author(s)

Barnabe, M. S. ^[1] ; Novacoski, J. ^[1] ; Spivakovsky, M. ^{[2, 3, 4]}

Affiliation

^[1] Dept Matemat UFSCar, Rodovia Washington Luis 235, BR-13565905 Sao Carlos, SP - Brazil

^[2] CNRS, UMR 5219, 118 Rte Narbonne, F-31062 Toulouse 9 - France

^[3] Inst Math Toulouse, 118 Rte Narbonne, F-31062 Toulouse 9 - France

^[4] Univ Nacl Autonoma Mexico, CNRS, UMI 2001, Inst Matemat, Unidad Cuernavaca LaSol, Av Univ S-N Col Lomas Chamilpa, Cuernavaca 62210, Morelos - Mexico

• Document type: Journal article
• Source: Journal of Algebra; v. 560, p. 667-679, OCT 15 2020.
• Web of Science Citations: 0

Abstract

The main goal of this paper is to study the structure of the graded algebra associated to a valuation. More specifically, we prove that the associated graded algebra gr(v) (R) of a subring (R, m) of a valuation ring O-v, for which Kv := O-v/m(v) = R/m, is isomorphic to Kv {[}t(v(R))], where the multiplication is given by a twisting. We show that this twisted multiplication can be chosen to be the usual one in the cases where the value group is free or the residue field is closed by radicals. We also present an example that shows that the isomorphism (with the trivial twisting) does not have to exist. (C) 2020 Elsevier Inc. All rights reserved. (AU)

• FAPESP's process: 17/17835-9 - The relation between toric geometry, theory of local blow-ups and ramification theory and their applications in valuation theory
• Grantee: Josnei Antonio Novacoski
• Support Opportunities: Research Grants - Young Investigators Grants