# 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 [Haz09]
• WittVector.instIsDomain

## The shift operator #

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

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
• x.shift n = { coeff := fun (i : ) => x.coeff (n + i) }
Instances For
theorem WittVector.shift_coeff {p : } {R : Type u_1} (x : ) (n : ) (k : ) :
(x.shift n).coeff k = x.coeff (n + k)
theorem WittVector.verschiebung_shift {p : } {R : Type u_1} [hp : Fact ()] [] (x : ) (k : ) (h : i < k + 1, x.coeff i = 0) :
WittVector.verschiebung (x.shift k.succ) = x.shift k
theorem WittVector.eq_iterate_verschiebung {p : } {R : Type u_1} [hp : Fact ()] [] {x : } {n : } (h : i < n, x.coeff i = 0) :
x = (WittVector.verschiebung)^[n] (x.shift n)
theorem WittVector.verschiebung_nonzero {p : } {R : Type u_1} [hp : Fact ()] [] {x : } (hx : x 0) :
∃ (n : ) (x' : ), x'.coeff 0 0 x = (WittVector.verschiebung)^[n] x'

## Witt vectors over a domain #

If R is an integral domain, then so is 𝕎 R. This argument is adapted from https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723.

instance WittVector.instNoZeroDivisorsOfCharP {p : } {R : Type u_1} [hp : Fact ()] [] [CharP R p] [] :
Equations
• =
instance WittVector.instIsDomain {p : } {R : Type u_1} [hp : Fact ()] [] [CharP R p] [] :
Equations
• =