MacLane-Vaquié chains of valuations on a polynomial ring

Abstract

Let (K, v) be a valued field. We review some results of MacLane and Vaquié on extensions of v to valuations on the polynomial ring K[x]. We introduce certain MacLane-Vaquié chains of residually transcendental valuations, and we prove that every valuation µ on K[x] is a limit of a finite or countably infinite MacLane-Vaquié chain. This chain underlying µ is essentially unique and contains arithmetic data yielding an explicit description of the graded algebra of µ as an algebra over the graded algebra of v.

Small extensions of abelian ordered groups

“…It is easy to check that the complementary subspaces cmpl Λ (H) := H ′ ⊕ cmpl(H ⊕ H ′ ) satisfy condition (2) as well. These complementary subspaces cmpl Λ (H) obviously satisfy (6).…”

“…By the last statement of Proposition 4.1, Γ ν ⊂ S∈Init(I) R I S lex . Let us describe the equivalence classes of the valuations of depth zero, in the terminology of [6].…”

“…The valuations for which equality holds in (13) are called valuation-transcendental in [5]. Equivalently, they are finite-depth valuations in the terminology of [6], or "bien specifiée" valuations in the terminology of [8].…”

Invariants of limit key polynomials

“…The main result of MacLane-Vaquié states that ν falls in one, and only one, of the following cases [7,Thm.4.8].…”

“…We say that µ has finite depth r, quasi-finite depth r, or infinite depth, respectively. The valuations of finite depth are characterized by the condition [7,Lem.-Def. 4.9] KP(ν) = ∅ or supp(ν) = 0.…”