Essentially finite generation of valuation rings in terms of classical invariants

Abstract

The main goal of this paper is to study some properties of an extension of valuations from classical invariants. More specifically, we consider a valued field and an extension ω of ν to a finite extension L of K. Then we study when the valuation ring of ω is essentially finitely generated over the valuation ring of ν. We present a necessary condition in terms of classic invariants of the extension by Hagen Knaf and show that in some particular cases, this condition is also sufficient. We also study when the corresponding extension of graded algebras is finitely generated. For this problem we present an equivalent condition (which is weaker than the one for the finite generation of the valuation rings).

Local Uniformization of Abhyankar Valuations

“…If e(ω/ν) = 1, δ(ω/ν) = 1 and V ω /m ω is separable over V ν /m ν , then the only if direction of the question is true, as is proven in [20]. Also, the only if direction of the question is true if L/K is normal or ω is the unique extension of ν to L by [10,Corollary 2.2].…”

“…Knaf proved the implies direction of his question; his proof is reproduced in [10,Theorem 4.1]. If e(ω/ν) = 1, δ(ω/ν) = 1 and V ω /m ω is separable over V ν /m ν , then the only if direction of the question is true, as is proven in [20].…”

“…Essentially finitely generated extensions of valuation rings. In this subsection, we discuss a very interesting question proposed by Hagen Knaf, and give an application of our local uniformization theorem to improve a positive result on this question from [10].…”

“…The only if direction of the question is proven when K is the quotient field of an excellent two-dimensional excellent local domain and ν dominates R in [10,Theorem 1.4]. The proof of [10,Theorem 1.4] uses the existence of a resolution of excellent surface singularities ( [21] or [4]) and local monomialization of defectless extensions of two dimensional excellent local domains ([7, Theorem 3.7] and [11,Theorem 7.3]).…”

“…Theorem 1.7 is proven in [10, Theorem 1.5], with the additional assumption that V ω /m ω is separable over k. To prove the stronger Theorem 1.7, we must only modify the proof of [10,Theorem 1.1] by observing that the statement of [10, Theorem 7.2] is true without the assumption that V ν /m ν is separable over k, using Theorems 1.1 and 1.2 of this paper in place of [17, Theorem 1.1].…”