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ring_theory.witt_vector.domain

Witt vectors over a domain #

This file builds to the proof witt_vector.is_domain, an instance that says if R is an integral domain, then so is π•Ž R. It depends on the API around iterated applications of witt_vector.verschiebung and witt_vector.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 (witt_vector.verschiebung_nonzero). Known identities (witt_vector.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 #

The shift operator #

source
def witt_vector.shift {p : β„•} {R : Type u_1} (x : witt_vector p R) (n : β„•) :
witt_vector p R

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

Equations
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theorem witt_vector.shift_coeff {p : β„•} {R : Type u_1} (x : witt_vector p R) (n k : β„•) :
(x.shift n).coeff k = x.coeff (n + k)
source
theorem witt_vector.verschiebung_shift {p : β„•} {R : Type u_1} [hp : fact (nat.prime p)] [comm_ring R] (x : witt_vector p R) (k : β„•) (h : βˆ€ (i : β„•), i < k + 1 β†’ x.coeff i = 0) :
⇑witt_vector.verschiebung (x.shift k.succ) = x.shift k
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theorem witt_vector.eq_iterate_verschiebung {p : β„•} {R : Type u_1} [hp : fact (nat.prime p)] [comm_ring R] {x : witt_vector p R} {n : β„•} (h : βˆ€ (i : β„•), i < n β†’ x.coeff i = 0) :
x = ⇑witt_vector.verschiebung^[n] (x.shift n)
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theorem witt_vector.verschiebung_nonzero {p : β„•} {R : Type u_1} [hp : fact (nat.prime p)] [comm_ring R] {x : witt_vector p R} (hx : x β‰  0) :
βˆƒ (n : β„•) (x' : witt_vector p R), x'.coeff 0 β‰  0 ∧ x = ⇑witt_vector.verschiebung^[n] x'

Witt vectors over a domain #

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

source
@[protected, instance]
def witt_vector.no_zero_divisors {p : β„•} {R : Type u_1} [hp : fact (nat.prime p)] [comm_ring R] [char_p R p] [no_zero_divisors R] :
no_zero_divisors (witt_vector p R)
source
@[protected, instance]
def witt_vector.is_domain {p : β„•} {R : Type u_1} [hp : fact (nat.prime p)] [comm_ring R] [char_p R p] [is_domain R] :
is_domain (witt_vector p R)