Identities between operations on the ring of Witt vectors

In this file we derive common identities between the Frobenius and Verschiebung operators.

Main declarations

• frobenius_verschiebung: the composition of Frobenius and Verschiebung is multiplication by p
• verschiebung_mul_frobenius: the “projection formula”: V(x * F y) = V x * y
• iterate_verschiebung_mul_coeff: an identity from [Haz09] 6.2

References

• Hazewinkel, Witt Vectors

• Commelin and Lewis, Formalizing the Ring of Witt Vectors

theorem WittVector.frobenius_verschiebung {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x : WittVector p R) : frobenius (verschiebung x) = x * ↑p

The composition of Frobenius and Verschiebung is multiplication by p.

theorem WittVector.verschiebung_zmod {p : ℕ} [hp : Fact (Nat.Prime p)] (x : WittVector p (ZMod p)) : verschiebung x = x * ↑p

Verschiebung is the same as multiplication by p on the ring of Witt vectors of ZMod p.

theorem WittVector.coeff_p_pow (p : ℕ) (R : Type u_1) [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] (i : ℕ) : (↑p ^ i).coeff i = 1

theorem WittVector.coeff_p_pow_eq_zero (p : ℕ) (R : Type u_1) [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] {i j : ℕ} (hj : j ≠ i) : (↑p ^ i).coeff j = 0

theorem WittVector.coeff_p (p : ℕ) (R : Type u_1) [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] (i : ℕ) : (↑p).coeff i = if i = 1 then 1 else 0

@[simp]

theorem WittVector.coeff_p_zero (p : ℕ) (R : Type u_1) [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] : (↑p).coeff 0 = 0

@[simp]

theorem WittVector.coeff_p_one (p : ℕ) (R : Type u_1) [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] : (↑p).coeff 1 = 1

theorem WittVector.p_nonzero (p : ℕ) (R : Type u_1) [hp : Fact (Nat.Prime p)] [CommRing R] [Nontrivial R] [CharP R p] : ↑p ≠ 0

theorem WittVector.FractionRing.p_nonzero (p : ℕ) (R : Type u_1) [hp : Fact (Nat.Prime p)] [CommRing R] [Nontrivial R] [CharP R p] : ↑p ≠ 0

theorem WittVector.verschiebung_mul_frobenius {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x y : WittVector p R) : verschiebung (x * frobenius y) = verschiebung x * y

The “projection formula” for Frobenius and Verschiebung.

theorem WittVector.mul_charP_coeff_zero {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] (x : WittVector p R) : (x * ↑p).coeff 0 = 0

theorem WittVector.mul_charP_coeff_succ {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] (x : WittVector p R) (i : ℕ) : (x * ↑p).coeff (i + 1) = x.coeff i ^ p

theorem WittVector.mul_pow_charP_coeff_zero {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] (x : WittVector p R) {m n : ℕ} (h : m < n) : (x * ↑p ^ n).coeff m = 0

theorem WittVector.mul_pow_charP_coeff_succ {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] (x : WittVector p R) {m n : ℕ} : (x * ↑p ^ n).coeff (m + n) = x.coeff m ^ p ^ n

theorem WittVector.verschiebung_frobenius {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] (x : WittVector p R) : verschiebung (frobenius x) = x * ↑p

theorem WittVector.verschiebung_frobenius_comm {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] : Function.Commute ⇑verschiebung ⇑frobenius

Iteration lemmas

theorem WittVector.iterate_verschiebung_coeff_eq_zero {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x : WittVector p R) {n m : ℕ} (h : m < n) : ((⇑verschiebung)^[n] x).coeff m = 0

theorem WittVector.iterate_verschiebung_coeff {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x : WittVector p R) (n k : ℕ) : ((⇑verschiebung)^[n] x).coeff (k + n) = x.coeff k

theorem WittVector.iterate_verschiebung_mul_left {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x y : WittVector p R) (i : ℕ) : (⇑verschiebung)^[i] x * y = (⇑verschiebung)^[i] (x * (⇑frobenius)^[i] y)

theorem WittVector.iterate_verschiebung_mul {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] (x y : WittVector p R) (i j : ℕ) : (⇑verschiebung)^[i] x * (⇑verschiebung)^[j] y = (⇑verschiebung)^[i + j] ((⇑frobenius)^[j] x * (⇑frobenius)^[i] y)

theorem WittVector.iterate_frobenius_coeff {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] (x : WittVector p R) (i k : ℕ) : ((⇑frobenius)^[i] x).coeff k = x.coeff k ^ p ^ i

theorem WittVector.iterate_verschiebung_mul_coeff {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] (x y : WittVector p R) (i j : ℕ) : ((⇑verschiebung)^[i] x * (⇑verschiebung)^[j] y).coeff (i + j) = x.coeff 0 ^ p ^ j * y.coeff 0 ^ p ^ i

This is a slightly specialized form of Hazewinkel, Witt Vectors 6.2 equation 5.

theorem WittVector.iterate_verschiebung_iterate_frobenius {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] (x : WittVector p R) (n : ℕ) : (⇑verschiebung)^[n] ((⇑frobenius)^[n] x) = x * ↑p ^ n