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Valuations and alternating pairs.

Let $K$ and $F$ be two fields. An alternating pair is a pair of homomorphisms $f,g : K^\times \to F$ such that

$$ f(u) \cdot g(v) = f(v) \cdot g(u) $$

whenever $u,v \in K^\times$ satisfy $u + v = 1$ in $K$.

This repository contains a formalization of the relationship between such alternating pairs and valuation rings.

Valuations yield alternating pairs.

First we discuss the "trivial" direction, saying that alternating pairs arise naturally in the presence of valuations as a relatively easy consequence of the ultrametric inequality. The formal statement that we prove along these lines can be found in the file src/main_converse.lean. We have reproduced the main result from this file (along with some "setup" code) in the following code block.

-- We are given two fields, `K` and `F`
variables {K F : Type*} [field K] [field F] 

open module finite_dimensional 
open_locale tensor_product

/-
NOTE: This introduces notation `[a]ₘ` for `a : Kˣ`, where `[a]ₘ` is the element of
the base-change `F ⊗[ℤ] (additive Kˣ)` corresponding to `a`. 
-/
notation `[`:max a`]ₘ`:max := 1 ⊗ₜ (additive.of_mul a)

lemma one_tmul_mul (a b : Kˣ) : ([a * b]ₘ : F ⊗[ℤ] additive Kˣ) = 
  [a]ₘ + [b]ₘ := 
tensor_product.tmul_add _ _ _

lemma one_tmul_inv (a : Kˣ) : ([a⁻¹]ₘ : F ⊗[ℤ] additive Kˣ) = - [a]ₘ :=
tensor_product.tmul_neg _ _

/-
We consider the weak topology on `dual F (F ⊗[ℤ] additive Kˣ)`. 
This is just the pointwise convergence topology, i.e. the topology
induced by the product topology on the type of functions `F ⊗[ℤ] additive Kˣ → F` 
where `F` is given the discrete topology.
-/
def module.dual.weak_topology : 
  topological_space (dual F (F ⊗[ℤ] additive Kˣ)) := 
topological_space.induced (λ e a, e a) $ 
(@Pi.topological_space (F ⊗[ℤ] additive Kˣ) (λ _, F) $ λ a, ⊥)

/-
We only activate this topological space instance for this file.
-/
local attribute [instance] 
  module.dual.weak_topology

/- 
The converse to the main theorem about alternating pairs. 
This is a simple result, and we prove it without many dependencies from the imports.
-/
theorem valuation_implies_alternating
  -- Given submodules `I` and `D` of `dual F (F ⊗[ℤ] additive Kˣ)` 
  (D I : submodule F (dual F (F ⊗[ℤ] additive Kˣ))) 
  -- which are closed with respect to the topology mentioned above,
  (hDclosed : is_closed (D : set (dual F (F ⊗[ℤ] additive Kˣ))))
  (hIclosed : is_closed (I : set (dual F (F ⊗[ℤ] additive Kˣ))))
  -- and a valuation ring of `K`
  (R : valuation_subring K)
  -- satisfying (1) `I ≤ D`;
  (le : I ≤ D)
  -- (2) the elements of `D` act trivially on `-1 : Kˣ`;
  (hnegone : ∀ (f : dual F (F ⊗[ℤ] additive Kˣ)) (hf : f ∈ D), f [-1]ₘ = 0) 
  -- (3) the elements of `I` act trivially on the units of `R`;
  (units : ∀ (u : Kˣ) (hu : u ∈ R.unit_group) 
    (f : dual F (F ⊗[ℤ] additive Kˣ))
    (hf : f ∈ I), f [u]ₘ = 0)
  -- (4) the elements of `D` act trivially on the principal units of `R`;
  (punits : ∀ (u : Kˣ) (hu : u ∈ R.principal_unit_group) 
    (f : dual F (F ⊗[ℤ] additive Kˣ))
    (hf : f ∈ D), f [u]ₘ = 0)
  -- (5) `D/I` is finite dimensional;
  (fd : finite_dimensional F (↥D ⧸ I.comap D.subtype))
  -- (6) and `I` has codimension at most `1` in `D`,
  (codim : finrank F (↥D ⧸ I.comap D.subtype) ≤ 1) :
  -- then any pair of elements of `D` satisfies the alternating condition.
  ∀ (u v : Kˣ) (huv : (u : K) + v = 1) 
    (f g : dual F (F ⊗[ℤ] additive Kˣ))
    (hf : f ∈ D) (hg : g ∈ D), 
    f [u]ₘ * g [v]ₘ = f [v]ₘ * g [u]ₘ := 
-- the proof...

Alternating pairs yield valuations

The primary goal of this work is the formalization of converses to the above theorem, which shows the existence of valuations in the presence of alternating pairs. We do this in two cases:

  1. In the case where F is a prime field.
  2. In the case where K has positive characteristic p and F has characteristic not dividing 2 * p.

The first case

The first case appears in src/main_theorem.lean. The relevant formal statement is reproduced below (the "setup" portions appearing above have been omitted).

-- We now assume that `F` is a prime field.
-- The is defined as saying that every element `a : F` can be expressed as 
-- `m/n` for some `m : ℤ` and some `n : ℕ` such that `(n : F) ≠ 0`.
variable [is_prime_field F]

example : is_prime_field ℚ := infer_instance
example (p : ℕ) [fact (nat.prime p)] : is_prime_field (zmod p) := infer_instance

/- The main theorem of alternating pairs (prime field case). -/
theorem main_alternating_theorem_of_prime_field
  -- Given a submodule `D` of `dual F (F ⊗[ℤ] additive Kˣ)`,
  (D : submodule F (dual F (F ⊗[ℤ] additive Kˣ))) 
  -- which is: (1) closed with respect to the topology introduced above; 
  (h1 : is_closed (D : set (dual F (F ⊗[ℤ] additive Kˣ))))
  -- (2) every element of `D` maps `[(-1 : Kˣ)]ₘ` to zero;
  (h2 : ∀ (f : dual F (F ⊗[ℤ] additive Kˣ)) (hf : f ∈ D), f [-1]ₘ = 0) 
  -- (3) satisfies the alternating condition, i.e. whenever `u v : Kˣ` satisfy
  -- `(u : K) + v = 1`, then `f [u]ₘ * g [v]ₘ = f [v]ₘ * g [u]ₘ`.
  (h3 : ∀ (u v : Kˣ) (huv : (u : K) + v = 1) 
    (f g : dual F (F ⊗[ℤ] additive Kˣ))
    (hf : f ∈ D) (hg : g ∈ D), 
    f [u]ₘ * g [v]ₘ = f [v]ₘ * g [u]ₘ) : 
  -- Then there exists a valuation subring `R` of `K`, 
  ∃ (R : valuation_subring K)
  -- and another submodule `I` of `dual F (F ⊗[ℤ] additive Kˣ)` 
    (I : submodule F (dual F (F ⊗[ℤ] additive Kˣ)))
    -- which is closed, and such that the following hold:
    (Iclosed : is_closed (I : set (dual F (F ⊗[ℤ] additive Kˣ))))
    -- (1) `I` is contained in `D`;
    (le : I ≤ D)
    -- (2) the elements `f` of `I` satisfy `f [u]ₘ = 0` for `R`-units;
    (units : ∀ (u : Kˣ) (hu : u ∈ R.unit_group) 
      (f : dual F (F ⊗[ℤ] additive Kˣ))
      (hf : f ∈ I), f [u]ₘ = 0)
    -- (3) the elements `f` of `D` satisfy `f [u]ₘ = 0` for `R`-principal-units;
    (punits : ∀ (u : Kˣ) (hu : u ∈ R.principal_unit_group) 
      (f : dual F (F ⊗[ℤ] additive Kˣ))
      (hf : f ∈ D), f [u]ₘ = 0)
    -- (4) the quotient `D / I` is finite dimensional;
    (fd : finite_dimensional F (↥D ⧸ I.comap D.subtype)),
    -- and `I` has codimension at most one in `D`.
    finrank F (↥D ⧸ I.comap D.subtype) ≤ 1 := 
-- the proof...

The second case

The second case appears in src/main_theorem_char.lean. The relevant formal statement is reproduced below (the "setup" portions appearing above have been omitted).

-- We give ourselves to natural numbers, `p` and `ℓ`, with `p` being prime.
variables (p ℓ : ℕ) [fact (nat.prime p)]
-- Assume that `K` has characteristic `p`.
variable [char_p K p]
-- Assume that `F` satisfies `[char_p F ℓ]`.
-- NB: If `ℓ = 0`, this is *weaker* than the assumption that `F` has characteristic zero.
-- See the docstring for the `char_p` for more information.
variable [char_p F ℓ]

/- The main theorem of alternating pairs (positive characteristic case). -/
theorem main_alternating_theorem_pos_char 
  -- Assume that `p` and `ℓ` are different
  (HH : p ≠ ℓ)
  -- and that 2 is invertible in `F`.
  (htwo : (2 : F) ≠ 0)
  -- Given a submodule `D` of `dual F (F ⊗[ℤ] additive Kˣ)`,
  (D : submodule F (dual F (F ⊗[ℤ] additive Kˣ))) 
  -- which is: (1) closed with respect to the topology introduced above; 
  (h1 : is_closed (D : set (dual F (F ⊗[ℤ] additive Kˣ))))
  -- (2) every element of `D` maps `[(-1 : Kˣ)]ₘ` to zero;
  (h2 : ∀ (f : dual F (F ⊗[ℤ] additive Kˣ)) (hf : f ∈ D), f [-1]ₘ = 0) 
  -- (3) satisfies the alternating condition, i.e. whenever `u v : Kˣ` satisfy
  -- `(u : K) + v = 1`, then `f [u]ₘ * g [v]ₘ = f [v]ₘ * g [u]ₘ`.
  (h3 : ∀ (u v : Kˣ) (huv : (u : K) + v = 1) 
    (f g : dual F (F ⊗[ℤ] additive Kˣ))
    (hf : f ∈ D) (hg : g ∈ D), 
    f [u]ₘ * g [v]ₘ = f [v]ₘ * g [u]ₘ) : 
  -- Then there exists a valuation subring `R` of `K`, 
  ∃ (R : valuation_subring K)
  -- and another submodule `I` of `dual F (F ⊗[ℤ] additive Kˣ)` 
    (I : submodule F (dual F (F ⊗[ℤ] additive Kˣ)))
    -- which is closed, and such that the following hold:
    (Iclosed : is_closed (I : set (dual F (F ⊗[ℤ] additive Kˣ))))
    -- (1) `I` is contained in `D`;
    (le : I ≤ D)
    -- (2) the elements `f` of `I` satisfy `f [u]ₘ = 0` for `R`-units;
    (units : ∀ (u : Kˣ) (hu : u ∈ R.unit_group) 
      (f : dual F (F ⊗[ℤ] additive Kˣ))
      (hf : f ∈ I), f [u]ₘ = 0)
    -- (3) the elements `f` of `D` satisfy `f [u]ₘ = 0` for `R`-principal-units;
    (punits : ∀ (u : Kˣ) (hu : u ∈ R.principal_unit_group) 
      (f : dual F (F ⊗[ℤ] additive Kˣ))
      (hf : f ∈ D), f [u]ₘ = 0)
    -- (4) the quotient `D / I` is finite dimensional;
    (fd : finite_dimensional F (↥D ⧸ I.comap D.subtype)),
    -- and `I` has codimension at most one in `D`.
    finrank F (↥D ⧸ I.comap D.subtype) ≤ 1 := 
-- the proof...

References

The arguments formalized in this repository are based on the following references.

  • Arason, R. Elman, and B. Jacob, Rigid elements, valuations, and realization of Witt rings, J. Algebra 110 (1987), no. 2, 449–467.
  • F. A. Bogomolov and Y. Tschinkel, Commuting elements of Galois groups of function fields, Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), 2002, pp. 75–120.
  • I. Efrat, Construction of valuations from K-theory, Math. Res. Lett. 6 (1999), no. 3-4, 335–343.
  • J. Koenigsmann, From p-rigid elements to valuations (with a Galois-characterization of p-adic fields), J. Reine Angew. Math. 465 (1995), 165–182.
  • A. Topaz, Commuting-liftable subgroups of Galois groups II, J. Reine Angew. Math. 730 (2017), 65–133.