Algebra Lineal OEA 5 by Orlando E. Villamayor

By Orlando E. Villamayor

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11. Under the partial order , the Gauss point ζGauss is the unique maximal point of D(0, 1), and the points of type I and type IV are the minimal points. Let (T, ≤) be a partially ordered set satisfying the following two axioms: (P1) T has a unique maximal element ζ, called the root of T . (P2) For each x ∈ T , the set Sx = {z ∈ T : z ≥ x} is totally ordered. We say that T is a parametrized rooted tree if there is a function α : T → R≥0 with values in the nonnegative reals such that: (P3) α(ζ) = 0.

Let F0 be the collection of all finite R-trees of the form ΓS as above. We have already noted that F0 is a directed set under inclusion, and we write Γ ≤ Γ if Γ ⊆ Γ as subsets of D(0, 1). Thus whenever Γ ≤ Γ , there is an inclusion map iΓ,Γ : Γ → Γ . The set Λ is the direct limit of ΓS over F0 with respect to the maps iΓ,Γ . There is also a retraction map rΓ ,Γ : Γ → Γ defined whenever Γ ≤ Γ . This is a general property of R-trees: since there is a unique path between any two points of Γ , if x ∈ Γ we can define rΓ ,Γ (x) to be, for any y ∈ Γ, the first point where the unique path in Γ from x to y intersects Γ.

We define maps in each direction between P1Berk \{∞} and A1Berk as follows. Given z ∈ A1Berk , define a seminorm on K[X, Y ] by P1Berk \{0} [[G]]χ∞ (z) = [G(T, 1)]z for all G ∈ K[X, Y ]. It is easy to see that [[ ]]χ∞ (z) is an element of S, with [[Y ]]χ∞ (z) = 0. By abuse of notation, let χ∞ : A1Berk → P1Berk \{∞} be the map which takes z to the equivalence class of [[ ]]χ∞ (z) . Define ψ∞ : P1Berk \{∞} → A1Berk by the formula [g]ψ∞ (z) = [[G]]z /[[Y ]]dz for each g ∈ K[T ], where G ∈ K[X, Y ] is homogeneous of degree d and satisfies G(T, 1) = g(T ), and [[ ]]z is any representative of the class z.

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