Witt vectors over a domain #

This file builds to the proof WittVector.instIsDomain, an instance that says if R is an integral domain, then so is 𝕎 R. It depends on the API around iterated applications of WittVector.verschiebung and WittVector.frobenius found in Identities.lean.

The proof sketch goes as follows: any nonzero $x$ is an iterated application of $V$ to some vector $w_x$ whose 0th component is nonzero (WittVector.verschiebung_nonzero). Known identities (WittVector.iterate_verschiebung_mul) allow us to transform the product of two such $x$ and $y$ to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$, the 0th component of which must be nonzero.

Main declarations #

WittVector.iterate_verschiebung_mul_coeff : an identity from [Haze09]
WittVector.instIsDomain

The `shift` operator #

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def WittVector.shift {p : ℕ} {R : Type u_1} (x : WittVector p R) (n : ℕ) :

WittVector p R

WittVector.verschiebung translates the entries of a Witt vector upward, inserting 0s in the gaps. WittVector.shift does the opposite, removing the first entries. This is mainly useful as an auxiliary construction for WittVector.verschiebung_nonzero.

Equations

WittVector.shift x n = { coeff := fun (i : ℕ) => x.coeff (n + i) }

Instances For

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theorem WittVector.shift_coeff {p : ℕ} {R : Type u_1} (x : WittVector p R) (n : ℕ) (k : ℕ) :

(WittVector.shift x n).coeff k = x.coeff (n + k)

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theorem WittVector.verschiebung_shift {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x : WittVector p R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :

WittVector.verschiebung (WittVector.shift x (Nat.succ k)) = WittVector.shift x k

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theorem WittVector.eq_iterate_verschiebung {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] {x : WittVector p R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :

x = (⇑WittVector.verschiebung)^[n] (WittVector.shift x n)

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theorem WittVector.verschiebung_nonzero {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] {x : WittVector p R} (hx : x ≠ 0) :

∃ (n : ℕ) (x' : WittVector p R), x'.coeff 0 ≠ 0 ∧ x = (⇑WittVector.verschiebung)^[n] x'