Ring Perfection and Tilt

In this file we define the perfection of a ring of characteristic p, and the tilt of a field given a valuation to ℝ≥0.

TODO

Define the valuation on the tilt, and define a characteristic predicate for the tilt.

def Monoid.perfection (M : Type u₁) [CommMonoid M] (p : ℕ) : Submonoid (ℕ → M)

The perfection of a monoid M, defined to be the projective limit of M using the p-th power maps M → M indexed by the natural numbers, implemented as { f : ℕ → M | ∀ n, f (n + 1) ^ p = f n }.

Equations

• Monoid.perfection M p = { carrier := {f : ℕ → M | ∀ (n : ℕ), f (n + 1) ^ p = f n}, mul_mem' := ⋯, one_mem' := ⋯ }

def Ring.perfectionSubsemiring (R : Type u₁) [CommSemiring R] (p : ℕ) [hp : Fact (Nat.Prime p)] [CharP R p] : Subsemiring (ℕ → R)

The perfection of a ring R with characteristic p, as a subsemiring, defined to be the projective limit of R using the Frobenius maps R → R indexed by the natural numbers, implemented as { f : ℕ → R | ∀ n, f (n + 1) ^ p = f n }.

Equations

• Ring.perfectionSubsemiring R p = { toSubmonoid := Monoid.perfection R p, add_mem' := ⋯, zero_mem' := ⋯ }

def Ring.perfectionSubring (R : Type u₁) [CommRing R] (p : ℕ) [hp : Fact (Nat.Prime p)] [CharP R p] : Subring (ℕ → R)

The perfection of a ring R with characteristic p, as a subring, defined to be the projective limit of R using the Frobenius maps R → R indexed by the natural numbers, implemented as { f : ℕ → R | ∀ n, f (n + 1) ^ p = f n }.

Equations

• Ring.perfectionSubring R p = (Ring.perfectionSubsemiring R p).toSubring ⋯

def Ring.Perfection (R : Type u₁) [CommSemiring R] (p : ℕ) : Type u₁

The perfection of a ring R with characteristic p, defined to be the projective limit of R using the Frobenius maps R → R indexed by the natural numbers, implemented as {f : ℕ → R // ∀ n, f (n + 1) ^ p = f n}.

Equations

• Ring.Perfection R p = { f : ℕ → R // ∀ (n : ℕ), f (n + 1) ^ p = f n }

instance Perfection.commSemiring (R : Type u₁) [CommSemiring R] (p : ℕ) [hp : Fact (Nat.Prime p)] [CharP R p] : CommSemiring (Ring.Perfection R p)

Equations

• Perfection.commSemiring R p = (Ring.perfectionSubsemiring R p).toCommSemiring

instance Perfection.charP (R : Type u₁) [CommSemiring R] (p : ℕ) [hp : Fact (Nat.Prime p)] [CharP R p] : CharP (Ring.Perfection R p) p

instance Perfection.ring (p : ℕ) [hp : Fact (Nat.Prime p)] (R : Type u₁) [CommRing R] [CharP R p] : Ring (Ring.Perfection R p)

Equations

• Perfection.ring p R = (Ring.perfectionSubring R p).toRing

instance Perfection.commRing (p : ℕ) [hp : Fact (Nat.Prime p)] (R : Type u₁) [CommRing R] [CharP R p] : CommRing (Ring.Perfection R p)

Equations

• Perfection.commRing p R = (Ring.perfectionSubring R p).toCommRing

instance Perfection.instInhabitedPerfection (R : Type u₁) [CommSemiring R] (p : ℕ) [hp : Fact (Nat.Prime p)] [CharP R p] : Inhabited (Ring.Perfection R p)

Equations

• Perfection.instInhabitedPerfection R p = { default := 0 }

def Perfection.coeff (R : Type u₁) [CommSemiring R] (p : ℕ) [hp : Fact (Nat.Prime p)] [CharP R p] (n : ℕ) : Ring.Perfection R p →+* R

The n-th coefficient of an element of the perfection.

Equations

• Perfection.coeff R p n = { toFun := fun (f : Ring.Perfection R p) => ↑f n, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }

theorem Perfection.ext {R : Type u₁} [CommSemiring R] {p : ℕ} [hp : Fact (Nat.Prime p)] [CharP R p] {f g : Ring.Perfection R p} (h : ∀ (n : ℕ), (Perfection.coeff R p n) f = (Perfection.coeff R p n) g) : f = g

def Perfection.pthRoot (R : Type u₁) [CommSemiring R] (p : ℕ) [hp : Fact (Nat.Prime p)] [CharP R p] : Ring.Perfection R p →+* Ring.Perfection R p

The p-th root of an element of the perfection.

Equations

• Perfection.pthRoot R p = { toFun := fun (f : Ring.Perfection R p) => ⟨fun (n : ℕ) => (Perfection.coeff R p (n + 1)) f, ⋯⟩, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem Perfection.coeff_mk {R : Type u₁} [CommSemiring R] {p : ℕ} [hp : Fact (Nat.Prime p)] [CharP R p] (f : ℕ → R) (hf : ∀ (n : ℕ), f (n + 1) ^ p = f n) (n : ℕ) : (Perfection.coeff R p n) ⟨f, hf⟩ = f n

theorem Perfection.coeff_pthRoot {R : Type u₁} [CommSemiring R] {p : ℕ} [hp : Fact (Nat.Prime p)] [CharP R p] (f : Ring.Perfection R p) (n : ℕ) : (Perfection.coeff R p n) ((Perfection.pthRoot R p) f) = (Perfection.coeff R p (n + 1)) f

theorem Perfection.coeff_pow_p {R : Type u₁} [CommSemiring R] {p : ℕ} [hp : Fact (Nat.Prime p)] [CharP R p] (f : Ring.Perfection R p) (n : ℕ) : (Perfection.coeff R p (n + 1)) (f ^ p) = (Perfection.coeff R p n) f

theorem Perfection.coeff_pow_p' {R : Type u₁} [CommSemiring R] {p : ℕ} [hp : Fact (Nat.Prime p)] [CharP R p] (f : Ring.Perfection R p) (n : ℕ) : (Perfection.coeff R p (n + 1)) f ^ p = (Perfection.coeff R p n) f

theorem Perfection.coeff_frobenius {R : Type u₁} [CommSemiring R] {p : ℕ} [hp : Fact (Nat.Prime p)] [CharP R p] (f : Ring.Perfection R p) (n : ℕ) : (Perfection.coeff R p (n + 1)) ((frobenius (Ring.Perfection R p) p) f) = (Perfection.coeff R p n) f

theorem Perfection.coeff_iterate_frobenius {R : Type u₁} [CommSemiring R] {p : ℕ} [hp : Fact (Nat.Prime p)] [CharP R p] (f : Ring.Perfection R p) (n m : ℕ) : (Perfection.coeff R p (n + m)) ((⇑(frobenius (Ring.Perfection R p) p))^[m] f) = (Perfection.coeff R p n) f

theorem Perfection.coeff_iterate_frobenius' {R : Type u₁} [CommSemiring R] {p : ℕ} [hp : Fact (Nat.Prime p)] [CharP R p] (f : Ring.Perfection R p) (n m : ℕ) (hmn : m ≤ n) : (Perfection.coeff R p n) ((⇑(frobenius (Ring.Perfection R p) p))^[m] f) = (Perfection.coeff R p (n - m)) f

theorem Perfection.pthRoot_frobenius {R : Type u₁} [CommSemiring R] {p : ℕ} [hp : Fact (Nat.Prime p)] [CharP R p] : (Perfection.pthRoot R p).comp (frobenius (Ring.Perfection R p) p) = RingHom.id (Ring.Perfection R p)

theorem Perfection.frobenius_pthRoot {R : Type u₁} [CommSemiring R] {p : ℕ} [hp : Fact (Nat.Prime p)] [CharP R p] : (frobenius (Ring.Perfection R p) p).comp (Perfection.pthRoot R p) = RingHom.id (Ring.Perfection R p)

theorem Perfection.coeff_add_ne_zero {R : Type u₁} [CommSemiring R] {p : ℕ} [hp : Fact (Nat.Prime p)] [CharP R p] {f : Ring.Perfection R p} {n : ℕ} (hfn : (Perfection.coeff R p n) f ≠ 0) (k : ℕ) : (Perfection.coeff R p (n + k)) f ≠ 0

theorem Perfection.coeff_ne_zero_of_le {R : Type u₁} [CommSemiring R] {p : ℕ} [hp : Fact (Nat.Prime p)] [CharP R p] {f : Ring.Perfection R p} {m n : ℕ} (hfm : (Perfection.coeff R p m) f ≠ 0) (hmn : m ≤ n) : (Perfection.coeff R p n) f ≠ 0

instance Perfection.perfectRing (R : Type u₁) [CommSemiring R] (p : ℕ) [hp : Fact (Nat.Prime p)] [CharP R p] : PerfectRing (Ring.Perfection R p) p

noncomputable def Perfection.lift (p : ℕ) [hp : Fact (Nat.Prime p)] (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing R p] (S : Type u₂) [CommSemiring S] [CharP S p] : (R →+* S) ≃ (R →+* Ring.Perfection S p)

Given rings R and S of characteristic p, with R being perfect, any homomorphism R →+* S can be lifted to a homomorphism R →+* Perfection S p.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Perfection.lift_apply_apply_coe (p : ℕ) [hp : Fact (Nat.Prime p)] (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing R p] (S : Type u₂) [CommSemiring S] [CharP S p] (f : R →+* S) (r : R) (n : ℕ) : ↑(((Perfection.lift p R S) f) r) n = f ((⇑↑(frobeniusEquiv R p).symm)^[n] r)

@[simp]

theorem Perfection.lift_symm_apply (p : ℕ) [hp : Fact (Nat.Prime p)] (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing R p] (S : Type u₂) [CommSemiring S] [CharP S p] (f : R →+* Ring.Perfection S p) : (Perfection.lift p R S).symm f = (Perfection.coeff S p 0).comp f

theorem Perfection.hom_ext (p : ℕ) [hp : Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] [PerfectRing R p] {S : Type u₂} [CommSemiring S] [CharP S p] {f g : R →+* Ring.Perfection S p} (hfg : ∀ (x : R), (Perfection.coeff S p 0) (f x) = (Perfection.coeff S p 0) (g x)) : f = g

def Perfection.map {R : Type u₁} [CommSemiring R] (p : ℕ) [hp : Fact (Nat.Prime p)] [CharP R p] {S : Type u₂} [CommSemiring S] [CharP S p] (φ : R →+* S) : Ring.Perfection R p →+* Ring.Perfection S p

A ring homomorphism R →+* S induces Perfection R p →+* Perfection S p.

Equations

• Perfection.map p φ = { toFun := fun (f : Ring.Perfection R p) => ⟨fun (n : ℕ) => φ ((Perfection.coeff R p n) f), ⋯⟩, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem Perfection.map_apply_coe {R : Type u₁} [CommSemiring R] (p : ℕ) [hp : Fact (Nat.Prime p)] [CharP R p] {S : Type u₂} [CommSemiring S] [CharP S p] (φ : R →+* S) (f : Ring.Perfection R p) (n : ℕ) : ↑((Perfection.map p φ) f) n = φ ((Perfection.coeff R p n) f)

theorem Perfection.coeff_map {R : Type u₁} [CommSemiring R] (p : ℕ) [hp : Fact (Nat.Prime p)] [CharP R p] {S : Type u₂} [CommSemiring S] [CharP S p] (φ : R →+* S) (f : Ring.Perfection R p) (n : ℕ) : (Perfection.coeff S p n) ((Perfection.map p φ) f) = φ ((Perfection.coeff R p n) f)

structure PerfectionMap (p : ℕ) [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₂} [CommSemiring P] [CharP P p] [PerfectRing P p] (π : P →+* R) : Prop

A perfection map to a ring of characteristic p is a map that is isomorphic to its perfection.

• injective ⦃x y : P⦄ : (∀ (n : ℕ), π ((⇑(frobeniusEquiv P p).symm)^[n] x) = π ((⇑(frobeniusEquiv P p).symm)^[n] y)) → x = y
• surjective (f : ℕ → R) : (∀ (n : ℕ), f (n + 1) ^ p = f n) → ∃ (x : P), ∀ (n : ℕ), π ((⇑(frobeniusEquiv P p).symm)^[n] x) = f n

theorem PerfectionMap.mk' {p : ℕ} [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] {f : P →+* R} (g : P ≃+* Ring.Perfection R p) (hfg : (Perfection.lift p P R) f = ↑g) : PerfectionMap p f

Create a PerfectionMap from an isomorphism to the perfection.

theorem PerfectionMap.of (p : ℕ) [Fact (Nat.Prime p)] (R : Type u₁) [CommSemiring R] [CharP R p] : PerfectionMap p (Perfection.coeff R p 0)

The canonical perfection map from the perfection of a ring.

theorem PerfectionMap.id (p : ℕ) [Fact (Nat.Prime p)] (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing R p] : PerfectionMap p (RingHom.id R)

For a perfect ring, it itself is the perfection.

noncomputable def PerfectionMap.equiv {p : ℕ} [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] {π : P →+* R} (m : PerfectionMap p π) : P ≃+* Ring.Perfection R p

A perfection map induces an isomorphism to the perfection.

Equations

• m.equiv = RingEquiv.ofBijective ((Perfection.lift p P R) π) ⋯

theorem PerfectionMap.equiv_apply {p : ℕ} [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] {π : P →+* R} (m : PerfectionMap p π) (x : P) : m.equiv x = ((Perfection.lift p P R) π) x

theorem PerfectionMap.comp_equiv {p : ℕ} [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] {π : P →+* R} (m : PerfectionMap p π) (x : P) : (Perfection.coeff R p 0) (m.equiv x) = π x

theorem PerfectionMap.comp_equiv' {p : ℕ} [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] {π : P →+* R} (m : PerfectionMap p π) : (Perfection.coeff R p 0).comp ↑m.equiv = π

theorem PerfectionMap.comp_symm_equiv {p : ℕ} [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] {π : P →+* R} (m : PerfectionMap p π) (f : Ring.Perfection R p) : π (m.equiv.symm f) = (Perfection.coeff R p 0) f

theorem PerfectionMap.comp_symm_equiv' {p : ℕ} [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] {π : P →+* R} (m : PerfectionMap p π) : π.comp ↑m.equiv.symm = Perfection.coeff R p 0

noncomputable def PerfectionMap.lift (p : ℕ) [Fact (Nat.Prime p)] (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing R p] (S : Type u₂) [CommSemiring S] [CharP S p] (P : Type u₃) [CommSemiring P] [CharP P p] [PerfectRing P p] (π : P →+* S) (m : PerfectionMap p π) : (R →+* S) ≃ (R →+* P)

Given rings R and S of characteristic p, with R being perfect, any homomorphism R →+* S can be lifted to a homomorphism R →+* P, where P is any perfection of S.

Equations

• PerfectionMap.lift p R S P π m = { toFun := fun (f : R →+* S) => (↑m.equiv.symm).comp ((Perfection.lift p R S) f), invFun := fun (f : R →+* P) => π.comp f, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem PerfectionMap.lift_apply (p : ℕ) [Fact (Nat.Prime p)] (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing R p] (S : Type u₂) [CommSemiring S] [CharP S p] (P : Type u₃) [CommSemiring P] [CharP P p] [PerfectRing P p] (π : P →+* S) (m : PerfectionMap p π) (f : R →+* S) : (PerfectionMap.lift p R S P π m) f = (↑m.equiv.symm).comp ((Perfection.lift p R S) f)

@[simp]

theorem PerfectionMap.lift_symm_apply (p : ℕ) [Fact (Nat.Prime p)] (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing R p] (S : Type u₂) [CommSemiring S] [CharP S p] (P : Type u₃) [CommSemiring P] [CharP P p] [PerfectRing P p] (π : P →+* S) (m : PerfectionMap p π) (f : R →+* P) : (PerfectionMap.lift p R S P π m).symm f = π.comp f

theorem PerfectionMap.hom_ext {p : ℕ} [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] [PerfectRing R p] {S : Type u₂} [CommSemiring S] [CharP S p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] (π : P →+* S) (m : PerfectionMap p π) {f g : R →+* P} (hfg : ∀ (x : R), π (f x) = π (g x)) : f = g

noncomputable def PerfectionMap.map (p : ℕ) [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] {S : Type u₂} [CommSemiring S] [CharP S p] {Q : Type u₄} [CommSemiring Q] [CharP Q p] [PerfectRing Q p] {π : P →+* R} : PerfectionMap p π → {σ : Q →+* S} → (n : PerfectionMap p σ) → (φ : R →+* S) → P →+* Q

A ring homomorphism R →+* S induces P →+* Q, a map of the respective perfections.

Equations

• PerfectionMap.map p x✝ n φ = (PerfectionMap.lift p P S Q σ n) (φ.comp π)

theorem PerfectionMap.comp_map (p : ℕ) [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] {S : Type u₂} [CommSemiring S] [CharP S p] {Q : Type u₄} [CommSemiring Q] [CharP Q p] [PerfectRing Q p] {π : P →+* R} (m : PerfectionMap p π) {σ : Q →+* S} (n : PerfectionMap p σ) (φ : R →+* S) : σ.comp (PerfectionMap.map p m n φ) = φ.comp π

theorem PerfectionMap.map_map (p : ℕ) [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] {S : Type u₂} [CommSemiring S] [CharP S p] {Q : Type u₄} [CommSemiring Q] [CharP Q p] [PerfectRing Q p] {π : P →+* R} (m : PerfectionMap p π) {σ : Q →+* S} (n : PerfectionMap p σ) (φ : R →+* S) (x : P) : σ ((PerfectionMap.map p m n φ) x) = φ (π x)

theorem PerfectionMap.map_eq_map (p : ℕ) [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {S : Type u₂} [CommSemiring S] [CharP S p] (φ : R →+* S) : PerfectionMap.map p ⋯ ⋯ φ = Perfection.map p φ

def ModP (K : Type u₁) [Field K] (v : Valuation K NNReal) (O : Type u₂) [CommRing O] [Algebra O K] : v.Integers O → (p : ℕ) → Type u₂

O/(p) for O, ring of integers of K.

Equations

• ModP K v O x✝ p = (O ⧸ Ideal.span {↑p})

instance ModP.commRing (K : Type u₁) [Field K] (v : Valuation K NNReal) (O : Type u₂) [CommRing O] [Algebra O K] (hv : v.Integers O) (p : ℕ) : CommRing (ModP K v O hv p)

Equations

• ModP.commRing K v O hv p = Ideal.Quotient.commRing (Ideal.span {↑p})

instance ModP.charP (K : Type u₁) [Field K] (v : Valuation K NNReal) (O : Type u₂) [CommRing O] [Algebra O K] (hv : v.Integers O) (p : ℕ) [Fact (Nat.Prime p)] [hvp : Fact (v ↑p ≠ 1)] : CharP (ModP K v O hv p) p

instance ModP.instNontrivialOfPrimeOfFactNeNNRealCoeValuationCastOfNat (K : Type u₁) [Field K] (v : Valuation K NNReal) (O : Type u₂) [CommRing O] [Algebra O K] (hv : v.Integers O) (p : ℕ) [hp : Fact (Nat.Prime p)] [Fact (v ↑p ≠ 1)] : Nontrivial (ModP K v O hv p)

noncomputable def ModP.preVal (K : Type u₁) [Field K] (v : Valuation K NNReal) (O : Type u₂) [CommRing O] [Algebra O K] (hv : v.Integers O) (p : ℕ) (x : ModP K v O hv p) : NNReal

For a field K with valuation v : K → ℝ≥0 and ring of integers O, a function O/(p) → ℝ≥0 that sends 0 to 0 and x + (p) to v(x) as long as x ∉ (p).

Equations

• ModP.preVal K v O hv p x = if x = 0 then 0 else v ((algebraMap O K) (Quotient.out x))

theorem ModP.preVal_mk {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} {x : O} (hx : (Ideal.Quotient.mk (Ideal.span {↑p})) x ≠ 0) : ModP.preVal K v O hv p ((Ideal.Quotient.mk (Ideal.span {↑p})) x) = v ((algebraMap O K) x)

theorem ModP.preVal_zero {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} : ModP.preVal K v O hv p 0 = 0

theorem ModP.preVal_mul {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} {x y : ModP K v O hv p} (hxy0 : x * y ≠ 0) : ModP.preVal K v O hv p (x * y) = ModP.preVal K v O hv p x * ModP.preVal K v O hv p y

theorem ModP.preVal_add {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} (x y : ModP K v O hv p) : ModP.preVal K v O hv p (x + y) ≤ ModP.preVal K v O hv p x ⊔ ModP.preVal K v O hv p y

theorem ModP.v_p_lt_preVal {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} {x : ModP K v O hv p} : v ↑p < ModP.preVal K v O hv p x ↔ x ≠ 0

theorem ModP.preVal_eq_zero {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} {x : ModP K v O hv p} : ModP.preVal K v O hv p x = 0 ↔ x = 0

theorem ModP.v_p_lt_val {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] (hv : v.Integers O) {p : ℕ} {x : O} : v ↑p < v ((algebraMap O K) x) ↔ (Ideal.Quotient.mk (Ideal.span {↑p})) x ≠ 0

theorem ModP.mul_ne_zero_of_pow_p_ne_zero {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} [hp : Fact (Nat.Prime p)] {x y : ModP K v O hv p} (hx : x ^ p ≠ 0) (hy : y ^ p ≠ 0) : x * y ≠ 0

def PreTilt (K : Type u₁) [Field K] (v : Valuation K NNReal) (O : Type u₂) [CommRing O] [Algebra O K] (hv : v.Integers O) (p : ℕ) : Type u₂

Perfection of O/(p) where O is the ring of integers of K.

Equations

• PreTilt K v O hv p = Ring.Perfection (ModP K v O hv p) p

instance PreTilt.instCommRing (K : Type u₁) [Field K] (v : Valuation K NNReal) (O : Type u₂) [CommRing O] [Algebra O K] (hv : v.Integers O) (p : ℕ) [Fact (Nat.Prime p)] [Fact (v ↑p ≠ 1)] : CommRing (PreTilt K v O hv p)

Equations

• PreTilt.instCommRing K v O hv p = Perfection.commRing p (ModP K v O hv p)

instance PreTilt.instCharP (K : Type u₁) [Field K] (v : Valuation K NNReal) (O : Type u₂) [CommRing O] [Algebra O K] (hv : v.Integers O) (p : ℕ) [Fact (Nat.Prime p)] [Fact (v ↑p ≠ 1)] : CharP (PreTilt K v O hv p) p

noncomputable def PreTilt.valAux (K : Type u₁) [Field K] (v : Valuation K NNReal) (O : Type u₂) [CommRing O] [Algebra O K] (hv : v.Integers O) (p : ℕ) [Fact (Nat.Prime p)] [Fact (v ↑p ≠ 1)] (f : PreTilt K v O hv p) : NNReal

The valuation Perfection(O/(p)) → ℝ≥0 as a function. Given f ∈ Perfection(O/(p)), if f = 0 then output 0; otherwise output preVal(f(n))^(p^n) for any n such that f(n) ≠ 0.

Equations

• One or more equations did not get rendered due to their size.

theorem PreTilt.coeff_nat_find_add_ne_zero {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} [Fact (Nat.Prime p)] [Fact (v ↑p ≠ 1)] {f : PreTilt K v O hv p} {h : ∃ (n : ℕ), (Perfection.coeff (ModP K v O hv p) p n) f ≠ 0} (k : ℕ) : (Perfection.coeff (ModP K v O hv p) p (Nat.find h + k)) f ≠ 0

theorem PreTilt.valAux_eq {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} [Fact (Nat.Prime p)] [Fact (v ↑p ≠ 1)] {f : PreTilt K v O hv p} {n : ℕ} (hfn : (Perfection.coeff (ModP K v O hv p) p n) f ≠ 0) : PreTilt.valAux K v O hv p f = ModP.preVal K v O hv p ((Perfection.coeff (ModP K v O hv p) p n) f) ^ p ^ n

theorem PreTilt.valAux_zero {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} [Fact (Nat.Prime p)] [Fact (v ↑p ≠ 1)] : PreTilt.valAux K v O hv p 0 = 0

theorem PreTilt.valAux_one {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} [Fact (Nat.Prime p)] [Fact (v ↑p ≠ 1)] : PreTilt.valAux K v O hv p 1 = 1

theorem PreTilt.valAux_mul {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} [Fact (Nat.Prime p)] [Fact (v ↑p ≠ 1)] (f g : PreTilt K v O hv p) : PreTilt.valAux K v O hv p (f * g) = PreTilt.valAux K v O hv p f * PreTilt.valAux K v O hv p g

theorem PreTilt.valAux_add {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} [Fact (Nat.Prime p)] [Fact (v ↑p ≠ 1)] (f g : PreTilt K v O hv p) : PreTilt.valAux K v O hv p (f + g) ≤ PreTilt.valAux K v O hv p f ⊔ PreTilt.valAux K v O hv p g

noncomputable def PreTilt.val (K : Type u₁) [Field K] (v : Valuation K NNReal) (O : Type u₂) [CommRing O] [Algebra O K] (hv : v.Integers O) (p : ℕ) [Fact (Nat.Prime p)] [Fact (v ↑p ≠ 1)] : Valuation (PreTilt K v O hv p) NNReal

The valuation Perfection(O/(p)) → ℝ≥0. Given f ∈ Perfection(O/(p)), if f = 0 then output 0; otherwise output preVal(f(n))^(p^n) for any n such that f(n) ≠ 0.

Equations

• PreTilt.val K v O hv p = { toFun := PreTilt.valAux K v O hv p, map_zero' := ⋯, map_one' := ⋯, map_mul' := ⋯, map_add_le_max' := ⋯ }

theorem PreTilt.map_eq_zero {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] {hv : v.Integers O} {p : ℕ} [Fact (Nat.Prime p)] [Fact (v ↑p ≠ 1)] {f : PreTilt K v O hv p} : (PreTilt.val K v O hv p) f = 0 ↔ f = 0

instance PreTilt.instIsDomain (K : Type u₁) [Field K] (v : Valuation K NNReal) (O : Type u₂) [CommRing O] [Algebra O K] (hv : v.Integers O) (p : ℕ) [Fact (v ↑p ≠ 1)] [hp : Fact (Nat.Prime p)] : IsDomain (PreTilt K v O hv p)

def Tilt (K : Type u₁) [Field K] (v : Valuation K NNReal) (O : Type u₂) [CommRing O] [Algebra O K] (hv : v.Integers O) (p : ℕ) [Fact (Nat.Prime p)] [Fact (v ↑p ≠ 1)] : Type u₂

The tilt of a field, as defined in Perfectoid Spaces by Peter Scholze, as in [Sch11]. Given a field K with valuation K → ℝ≥0 and ring of integers O, this is implemented as the fraction field of the perfection of O/(p).

Equations

• Tilt K v O hv p = FractionRing (PreTilt K v O hv p)

noncomputable instance Tilt.instField (K : Type u₁) [Field K] (v : Valuation K NNReal) (O : Type u₂) [CommRing O] [Algebra O K] (hv : v.Integers O) (p : ℕ) [Fact (Nat.Prime p)] [Fact (v ↑p ≠ 1)] : Field (Tilt K v O hv p)

Equations

• Tilt.instField K v O hv p = FractionRing.field (PreTilt K v O hv p)