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Mathlib.RingTheory.WittVector.DiscreteValuationRing

Witt vectors over a perfect ring #

This file establishes that Witt vectors over a perfect field are a discrete valuation ring. When k is a perfect ring, a nonzero a : 𝕎 k can be written as p^m * b for some m : ℕ and b : 𝕎 k with nonzero 0th coefficient. When k is also a field, this b can be chosen to be a unit of 𝕎 k.

Main declarations #

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def WittVector.succNthValUnits {p : ℕ} [hp : Fact (Nat.Prime p)] {k : Type u_1} [CommRing k] [CharP k p] (n : ℕ) (a : kˣ) (A : WittVector p k) (bs : Fin (n + 1) → k) :
k

This is the n+1st coefficient of our inverse.

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      @[irreducible]
      noncomputable def WittVector.inverseCoeff {p : ℕ} [hp : Fact (Nat.Prime p)] {k : Type u_1} [CommRing k] [CharP k p] (a : kˣ) (A : WittVector p k) :
      ℕ → k

      Recursively defines the sequence of coefficients for the inverse to a Witt vector whose first entry is a unit.

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          def WittVector.mkUnit {p : ℕ} [hp : Fact (Nat.Prime p)] {k : Type u_1} [CommRing k] [CharP k p] {a : kˣ} {A : WittVector p k} (hA : A.coeff 0 = ↑a) :
          (WittVector p k)ˣ

          Upgrade a Witt vector A whose first entry A.coeff 0 is a unit to be, itself, a unit in 𝕎 k.

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              @[simp]
              theorem WittVector.coe_mkUnit {p : ℕ} [hp : Fact (Nat.Prime p)] {k : Type u_1} [CommRing k] [CharP k p] {a : kˣ} {A : WittVector p k} (hA : A.coeff 0 = ↑a) :
              ↑(mkUnit hA) = A
              source
              theorem WittVector.isUnit_of_coeff_zero_ne_zero {p : ℕ} [hp : Fact (Nat.Prime p)] {k : Type u_1} [Field k] [CharP k p] (x : WittVector p k) (hx : x.coeff 0 ≠ 0) :
              IsUnit x
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              theorem WittVector.irreducible (p : ℕ) [hp : Fact (Nat.Prime p)] {k : Type u_1} [Field k] [CharP k p] :
              Irreducible ↑p
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              theorem WittVector.exists_eq_pow_p_mul {p : ℕ} [hp : Fact (Nat.Prime p)] {k : Type u_1} [CommRing k] [CharP k p] [PerfectRing k p] (a : WittVector p k) (ha : a ≠ 0) :
              ∃ (m : ℕ) (b : WittVector p k), b.coeff 0 ≠ 0 ∧ a = ↑p ^ m * b
              source
              theorem WittVector.exists_eq_pow_p_mul' {p : ℕ} [hp : Fact (Nat.Prime p)] {k : Type u_1} [Field k] [CharP k p] [PerfectRing k p] (a : WittVector p k) (ha : a ≠ 0) :
              ∃ (m : ℕ) (b : (WittVector p k)ˣ), a = ↑p ^ m * ↑b
              source
              theorem WittVector.isDiscreteValuationRing {p : ℕ} [hp : Fact (Nat.Prime p)] {k : Type u_1} [Field k] [CharP k p] [PerfectRing k p] :
              IsDiscreteValuationRing (WittVector p k)

              The ring of Witt Vectors of a perfect field of positive characteristic is a DVR.