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ring_theory.witt_vector.discrete_valuation_ring

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 #

• witt_vector.exists_eq_pow_p_mul: the existence of this element b over a perfect ring
• witt_vector.exists_eq_pow_p_mul': the existence of this unit b over a perfect field
• witt_vector.discrete_valuation_ring: 𝕎 k is a discrete valuation ring if k is a perfect field
noncomputable def witt_vector.succ_nth_val_units {p : } [hp : fact (nat.prime p)] {k : Type u_1} [comm_ring k] [ p] (n : ) (a : kˣ) (A : k) (bs : fin (n + 1) → k) :
k

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

Equations
noncomputable def witt_vector.inverse_coeff {p : } [hp : fact (nat.prime p)] {k : Type u_1} [comm_ring k] [ p] (a : kˣ) (A : k) :
→ k

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

Equations
noncomputable def witt_vector.mk_unit {p : } [hp : fact (nat.prime p)] {k : Type u_1} [comm_ring k] [ p] {a : kˣ} {A : k} (hA : A.coeff 0 = a) :
k)ˣ

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

Equations
@[simp]
theorem witt_vector.coe_mk_unit {p : } [hp : fact (nat.prime p)] {k : Type u_1} [comm_ring k] [ p] {a : kˣ} {A : k} (hA : A.coeff 0 = a) :
= A
theorem witt_vector.is_unit_of_coeff_zero_ne_zero {p : } [hp : fact (nat.prime p)] {k : Type u_1} [field k] [ p] (x : k) (hx : x.coeff 0 0) :
theorem witt_vector.irreducible (p : ) [hp : fact (nat.prime p)] {k : Type u_1} [field k] [ p] :
theorem witt_vector.exists_eq_pow_p_mul {p : } [hp : fact (nat.prime p)] {k : Type u_1} [comm_ring k] [ p] [ p] (a : k) (ha : a 0) :
∃ (m : ) (b : k), b.coeff 0 0 a = p ^ m * b
theorem witt_vector.exists_eq_pow_p_mul' {p : } [hp : fact (nat.prime p)] {k : Type u_1} [field k] [ p] [ p] (a : k) (ha : a 0) :
∃ (m : ) (b : k)ˣ), a = p ^ m * b
theorem witt_vector.discrete_valuation_ring {p : } [hp : fact (nat.prime p)] {k : Type u_1} [field k] [ p] [ p] :

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