mathlib documentation

ring_theory.witt_vector.truncated

Truncated Witt vectors #

The ring of truncated Witt vectors (of length n) is a quotient of the ring of Witt vectors. It retains the first n coefficients of each Witt vector. In this file, we set up the basic quotient API for this ring.

The ring of Witt vectors is the projective limit of all the rings of truncated Witt vectors.

Main declarations #

References #

@[nolint]
def truncated_witt_vector (p n : ) (R : Type u_1) :
Type u_1

A truncated Witt vector over R is a vector of elements of R, i.e., the first n coefficients of a Witt vector. We will define operations on this type that are compatible with the (untruncated) Witt vector operations.

truncated_witt_vector p n R takes a parameter p : ℕ that is not used in the definition. In practice, this number p is assumed to be a prime number, and under this assumption we construct a ring structure on truncated_witt_vector p n R. (truncated_witt_vector p₁ n R and truncated_witt_vector p₂ n R are definitionally equal as types but will have different ring operations.)

Equations
@[instance]
Equations
def truncated_witt_vector.mk (p : ) {n : } {R : Type u_1} (x : fin n → R) :

Create a truncated_witt_vector from a vector x.

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def truncated_witt_vector.coeff {p n : } {R : Type u_1} (i : fin n) (x : truncated_witt_vector p n R) :
R

x.coeff i is the ith entry of x.

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@[ext]
theorem truncated_witt_vector.ext {p n : } {R : Type u_1} {x y : truncated_witt_vector p n R} (h : ∀ (i : fin n), truncated_witt_vector.coeff i x = truncated_witt_vector.coeff i y) :
x = y
theorem truncated_witt_vector.ext_iff {p n : } {R : Type u_1} {x y : truncated_witt_vector p n R} :
@[simp]
theorem truncated_witt_vector.coeff_mk {p n : } {R : Type u_1} (x : fin n → R) (i : fin n) :
@[simp]
theorem truncated_witt_vector.mk_coeff {p n : } {R : Type u_1} (x : truncated_witt_vector p n R) :
def truncated_witt_vector.out {p n : } {R : Type u_1} [comm_ring R] (x : truncated_witt_vector p n R) :

We can turn a truncated Witt vector x into a Witt vector by setting all coefficients after x to be 0.

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@[simp]
theorem truncated_witt_vector.coeff_out {p n : } {R : Type u_1} [comm_ring R] (x : truncated_witt_vector p n R) (i : fin n) :
def witt_vector.truncate_fun {p : } (n : ) {R : Type u_1} (x : witt_vector p R) :

truncate_fun n x uses the first n entries of x to construct a truncated_witt_vector, which has the same base p as x. This function is bundled into a ring homomorphism in witt_vector.truncate

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@[simp]
theorem witt_vector.coeff_truncate_fun {p n : } {R : Type u_1} (x : witt_vector p R) (i : fin n) :
@[simp]
theorem witt_vector.out_truncate_fun {p n : } {R : Type u_1} [comm_ring R] (x : witt_vector p R) :
@[simp]
@[instance]
def truncated_witt_vector.has_zero (p : ) [hp : fact (nat.prime p)] (n : ) (R : Type u_1) [comm_ring R] :
Equations
@[instance]
def truncated_witt_vector.has_one (p : ) [hp : fact (nat.prime p)] (n : ) (R : Type u_1) [comm_ring R] :
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@[instance]
def truncated_witt_vector.has_add (p : ) [hp : fact (nat.prime p)] (n : ) (R : Type u_1) [comm_ring R] :
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@[instance]
def truncated_witt_vector.has_mul (p : ) [hp : fact (nat.prime p)] (n : ) (R : Type u_1) [comm_ring R] :
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@[instance]
def truncated_witt_vector.has_neg (p : ) [hp : fact (nat.prime p)] (n : ) (R : Type u_1) [comm_ring R] :
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@[instance]
def truncated_witt_vector.has_sub (p : ) [hp : fact (nat.prime p)] (n : ) (R : Type u_1) [comm_ring R] :
Equations
@[simp]
theorem truncated_witt_vector.coeff_zero (p : ) [hp : fact (nat.prime p)] (n : ) (R : Type u_1) [comm_ring R] (i : fin n) :

A macro tactic used to prove that truncate_fun respects ring operations.

@[simp]
theorem witt_vector.truncate_fun_zero (p : ) [hp : fact (nat.prime p)] (n : ) (R : Type u_1) [comm_ring R] :
@[simp]
theorem witt_vector.truncate_fun_one (p : ) [hp : fact (nat.prime p)] (n : ) (R : Type u_1) [comm_ring R] :
@[simp]
theorem witt_vector.truncate_fun_add {p : } [hp : fact (nat.prime p)] (n : ) {R : Type u_1} [comm_ring R] (x y : witt_vector p R) :
@[simp]
theorem witt_vector.truncate_fun_mul {p : } [hp : fact (nat.prime p)] (n : ) {R : Type u_1} [comm_ring R] (x y : witt_vector p R) :
theorem witt_vector.truncate_fun_neg {p : } [hp : fact (nat.prime p)] (n : ) {R : Type u_1} [comm_ring R] (x : witt_vector p R) :
def witt_vector.truncate {p : } [hp : fact (nat.prime p)] (n : ) {R : Type u_1} [comm_ring R] :

truncate n is a ring homomorphism that truncates x to its first n entries to obtain a truncated_witt_vector, which has the same base p as x.

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@[simp]
theorem witt_vector.coeff_truncate {p : } [hp : fact (nat.prime p)] {n : } {R : Type u_1} [comm_ring R] (x : witt_vector p R) (i : fin n) :
theorem witt_vector.mem_ker_truncate {p : } [hp : fact (nat.prime p)] (n : ) {R : Type u_1} [comm_ring R] (x : witt_vector p R) :
x (witt_vector.truncate n).ker ∀ (i : ), i < nx.coeff i = 0
@[simp]
theorem witt_vector.truncate_mk (p : ) [hp : fact (nat.prime p)] (n : ) {R : Type u_1} [comm_ring R] (f : → R) :
def truncated_witt_vector.truncate {p : } [hp : fact (nat.prime p)] {n : } {R : Type u_1} [comm_ring R] {m : } (hm : n m) :

A ring homomorphism that truncates a truncated Witt vector of length m to a truncated Witt vector of length n, for n ≤ m.

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@[simp]
@[simp]
theorem truncated_witt_vector.truncate_truncate {p : } [hp : fact (nat.prime p)] {R : Type u_1} [comm_ring R] {n₁ n₂ n₃ : } (h1 : n₁ n₂) (h2 : n₂ n₃) (x : truncated_witt_vector p n₃ R) :
@[simp]
theorem truncated_witt_vector.truncate_comp {p : } [hp : fact (nat.prime p)] {R : Type u_1} [comm_ring R] {n₁ n₂ n₃ : } (h1 : n₁ n₂) (h2 : n₂ n₃) :
@[simp]
theorem truncated_witt_vector.card (p n : ) {R : Type u_1} [fintype R] :
theorem truncated_witt_vector.infi_ker_truncate {p : } [hp : fact (nat.prime p)] {R : Type u_1} [comm_ring R] :
(⨅ (i : ), (witt_vector.truncate i).ker) =
def witt_vector.lift_fun {p : } [hp : fact (nat.prime p)] {R : Type u_1} [comm_ring R] {S : Type u_2} [semiring S] (f : Π (k : ), S →+* truncated_witt_vector p k R) (s : S) :

Given a family fₖ : S → truncated_witt_vector p k R and s : S, we produce a Witt vector by defining the kth entry to be the final entry of fₖ s.

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@[simp]
theorem witt_vector.truncate_lift_fun {p : } [hp : fact (nat.prime p)] (n : ) {R : Type u_1} [comm_ring R] {S : Type u_2} [semiring S] {f : Π (k : ), S →+* truncated_witt_vector p k R} (f_compat : ∀ (k₁ k₂ : ) (hk : k₁ k₂), (truncated_witt_vector.truncate hk).comp (f k₂) = f k₁) (s : S) :
def witt_vector.lift {p : } [hp : fact (nat.prime p)] {R : Type u_1} [comm_ring R] {S : Type u_2} [semiring S] (f : Π (k : ), S →+* truncated_witt_vector p k R) (f_compat : ∀ (k₁ k₂ : ) (hk : k₁ k₂), (truncated_witt_vector.truncate hk).comp (f k₂) = f k₁) :

Given compatible ring homs from S into truncated_witt_vector n for each n, we can lift these to a ring hom S → 𝕎 R.

lift defines the universal property of 𝕎 R as the inverse limit of truncated_witt_vector n.

Equations
@[simp]
theorem witt_vector.truncate_lift {p : } [hp : fact (nat.prime p)] (n : ) {R : Type u_1} [comm_ring R] {S : Type u_2} [semiring S] {f : Π (k : ), S →+* truncated_witt_vector p k R} (f_compat : ∀ (k₁ k₂ : ) (hk : k₁ k₂), (truncated_witt_vector.truncate hk).comp (f k₂) = f k₁) (s : S) :
(witt_vector.truncate n) ((witt_vector.lift (λ (k₂ : ), f k₂) f_compat) s) = (f n) s
@[simp]
theorem witt_vector.truncate_comp_lift {p : } [hp : fact (nat.prime p)] (n : ) {R : Type u_1} [comm_ring R] {S : Type u_2} [semiring S] {f : Π (k : ), S →+* truncated_witt_vector p k R} (f_compat : ∀ (k₁ k₂ : ) (hk : k₁ k₂), (truncated_witt_vector.truncate hk).comp (f k₂) = f k₁) :
(witt_vector.truncate n).comp (witt_vector.lift (λ (k₂ : ), f k₂) f_compat) = f n
theorem witt_vector.lift_unique {p : } [hp : fact (nat.prime p)] {R : Type u_1} [comm_ring R] {S : Type u_2} [semiring S] {f : Π (k : ), S →+* truncated_witt_vector p k R} (f_compat : ∀ (k₁ k₂ : ) (hk : k₁ k₂), (truncated_witt_vector.truncate hk).comp (f k₂) = f k₁) (g : S →+* witt_vector p R) (g_compat : ∀ (k : ), (witt_vector.truncate k).comp g = f k) :
witt_vector.lift (λ (k₂ : ), f k₂) f_compat = g

The uniqueness part of the universal property of 𝕎 R.

def witt_vector.lift_equiv {p : } [hp : fact (nat.prime p)] {R : Type u_1} [comm_ring R] {S : Type u_2} [semiring S] :
{f // ∀ (k₁ k₂ : ) (hk : k₁ k₂), (truncated_witt_vector.truncate hk).comp (f k₂) = f k₁} (S →+* witt_vector p R)

The universal property of 𝕎 R as projective limit of truncated Witt vector rings.

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@[simp]
theorem witt_vector.lift_equiv_symm_apply_coe {p : } [hp : fact (nat.prime p)] {R : Type u_1} [comm_ring R] {S : Type u_2} [semiring S] (g : S →+* witt_vector p R) (k : ) :
@[simp]
theorem witt_vector.lift_equiv_apply {p : } [hp : fact (nat.prime p)] {R : Type u_1} [comm_ring R] {S : Type u_2} [semiring S] (f : {f // ∀ (k₁ k₂ : ) (hk : k₁ k₂), (truncated_witt_vector.truncate hk).comp (f k₂) = f k₁}) :
theorem witt_vector.hom_ext {p : } [hp : fact (nat.prime p)] {R : Type u_1} [comm_ring R] {S : Type u_2} [semiring S] (g₁ g₂ : S →+* witt_vector p R) (h : ∀ (k : ), (witt_vector.truncate k).comp g₁ = (witt_vector.truncate k).comp g₂) :
g₁ = g₂