Nonarchimedean and Tropical Geometry by Matthew Baker & Sam Payne

Author:Matthew Baker & Sam Payne
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Proof.

Density of the image follows from Corollary 4.2.9. Set for shortness. To prove that ψ L∕K∕A is an isometry we recall that Coker(χ) is a vector space and Ker(χ) is divisible by Lemma 5.4.7. Let χ tor and χ tf be the maps χ induces between the torsion submodules and the torsion free quotients of its arguments. In particular, χ tf is the map ϕ from diagram (2). The snake lemma yields an exact sequence

Since Coker(χ) is torsion free and Coker(χ tor) is torsion, α = 0 and so Coker(ϕ) = Coker(χ) is a vector space. In addition, the torsion free group Ker(ϕ) is an extension of the divisible group Im(β) by the torsion group Coker(χ tor). It follows easily that, in fact, Ker(ϕ) = Im(β). Thus, Ker(ϕ) is divisible and hence ψ L∕K∕A is an isometry by Lemma 5.6.2. □

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