Valued Modules Over Skew Polynomial Rings I

Abstract

Following our first article, we continue to investigate ultrametric modules over a ring of twisted polynomials of the form [K; ϕ], where ϕ is a ring endomorphism of the field K. The main motivation comes from the the theory of valued difference fields (including characteristic p > 0 valued fields equipped with the Frobenius endomorphism). We introduce the class of modules, that we call, affinely maximal and residually divisible and we prove (relative -) quantifier elimination results. Ax-Kochen & Erhov type theorems follow. As an application, we axiomatize, as a valued module, the ultraproduct of algebraically closed valued fields (F p n (t) alg ) n∈N , of fixed characteristic p > 0, each equipped with the morphism x → x p n and with the t-adic valuation.MSC : Primary 03C60; Secondary 03C10, 14G17, 12J20, 16W80, 13J15During this section, we let the pair (K, ϕ) range over difference fields, that is K is a field together with an ring endomorphism ϕ and we let R := K[t; ϕ].Definition 2.1 (K-chains). For a ∈ K, let ·a denotes a unary function symbol (acting on the right). A K-chain, is a structure of the form (∆, <, ∞, (·a) a∈K ) where (∆, <, ∞) is a linear order with a top element ∞ and such that for all non zero a, b ∈ K and γ, δ ∈ ∆ \ {∞}, we have:Remark 2.2. If a ∈ K, and γ is not infinity, then γ · a = ∞ implies a = 0.Notation. In the rest of this paper, whenever (M, v) is a valued structure, a valued field, valued module, valued abelian group etc., for a subset A ⊆ M we denote by vA, the set {v(a) | a ∈ A}.Example 2.3. If (K, v) is a valued field then letting, for λ ∈ K, v(λ) · a := v(λa) = v(λ) + v(a) makes the ordered set vK a K-chain.Conversely we have the following.Proposition 2.4. If ∆ is an infinite K-chain thenis a valuation ring of K.Proof. Same proof as the one given in [8], Proposition 10.Notation. When the K-chain ∆ is clear from the context, we denote by v K the valuation on K given by O ∆ .Theorem 2.5. Set the language L := {<, ·a (a ∈ K), c, ∞}. Then theory of dense K-chains such that ∆ \ {∞} is without end points, together with the axiom ∞ = c is complete and eliminates quantifiers in the language L. Proof. Follows immediately by Théorème 6, and Théorème 13 in [8].Notation. For r ∈ R, let ·r be a unary function symbol. We set L V := {<, ·r r∈R , ∞}, that we call the language of R-chains.Recall that any r ∈ R can be uniquely written as i t i a i , with a i = 0. We call a term t i a i in this expression a monomial of r.A potential jump of an R-chain ∆, is a potential jump for some non zero r. We denote by Jump ∆ (R) the set of all potential jumps in ∆.Remark 2.13.(1) Jump ∆ (R) is an L V -substructure of ∆.(2) Potential jumps of in ∆, can also be defined as the set of ∞ = γ ∈ ∆ such that γ · m 1 = γ · m 2 for all monomials m 1 , m 2 satisfying 0 deg(m 1 ) < deg(m 2 ).(3) Let dcl ∆ denotes the definable closure operator in an R-chain ∆. Then Jump ∆ (R) ⊆ dcl ∆ (∅). Corollary 2.14. Let ∆, ∆ ′ be R-chains such that Jump ∆ (R) ≡ Jump ∆ ′ (R) as L V -structures. Then Jump ∆ (R) and Jump ∆ ′ (R) are isom...

Valuations and orderings on the real Weyl algebra

“…Granja, Martínez, and Rodríguez have shown in [6] that the set of all real valuations extending to the skew polynomial ring has the structure of a parameterized complete non-metric tree. Further recent progress on valuations on Ore extensions is given by Onay in [18] and Rohwer in his PhD thesis [20].…”