On the structure of the graded algebra associated to a valuation

2020/04/13 by Matheus dos S. Barnabe, Barnabé, M. S., Josnei Novacoski +3
Mathematics · #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2004.06161

openalex publication_date 2020/04/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \rm gr_v(R) of a subring (R,\mathfrakm) of a valuation ring O_v, for which Kv:=O_v / \mathfrakm_v=R / \mathfrakm, 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.

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