QHat.lean`

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57 declarations, 0 contain sorry, 0 declared as axioms

The profinite completion of `ℤ` and the finite adeles of `ℚ`

A self-contained construction of ZHat, the profinite completion of ℤ, defined as the inverse limit of the rings ℤ/Nℤ together with the natural reduction maps. We also define the finite adele ring QHat as ZHat ⊗_ℤ ℚ.

Main definitions

ZHat: the profinite completion of ℤ, defined explicitly as compatible systems of elements of ℤ/Nℤ.
QHat: the finite adele ring QHat = ZHat ⊗_ℤ ℚ.

Imports (6)

Imported by 3 files

def ZHat We define the profinite completion of ℤ explicitly as compatible elements of ℤ/Nℤ for all positive integers N line 41 source
Source documentation
We define the profinite completion of ℤ explicitly as compatible elements of ℤ/Nℤ for all positive integers N. We declare it as a subring of ∏_{N ≥ 1} (ℤ/Nℤ), and then promote it to a type.
```
def ZHat : Type := {
…
```
instance ZHat.(instance at line 67) line 67 source
def ZHat.toZMod The canonical "reduce mod M" map from ZHat to ℤ/Mℤ line 72 source
Source documentation
The canonical "reduce mod M" map from ZHat to ℤ/Mℤ.
```
def toZMod (M : ℕ+) : ZHat →+* ZMod M where
…
```
Mentions: ZHat
lemma ZHat.toZMod_def line 79 source
lemma ZHat.prop line 81 source
lemma ZHat.ext line 84 source
lemma ZHat.zero_val line 87 source
lemma ZHat.one_val line 88 source
lemma ZHat.ofNat_val line 89 source
lemma ZHat.natCast_val line 91 source
lemma ZHat.intCast_val line 92 source
instance ZHat.commRing line 94 source
lemma ZHat.zeroNeOne line 97 source
instance ZHat.nontrivial line 103 source
instance ZHat.charZero line 106 source
def ZHat.e A nonarchimedean analogue `0! line 118 source
Source documentation
A nonarchimedean analogue 0! + 1! + 2! + ⋯ of e = 1/0! + 1/1! + 1/2! + ⋯. It is defined as the function whose value at ZMod n is the sum of i! for 0 ≤ i < n.
```
def e : ZHat := ⟨fun (n : ℕ+) ↦ ∑ i ∈ range (n : ℕ), i !, by
…
```
Mentions: ZHat
lemma ZHat.e_def line 130 source
lemma ZHat._root_.Nat.sum_factorial_lt_factorial_succ line 132 source
lemma ZHat._root_.Nat.sum_factorial_lt_two_mul_factorial line 144 source
lemma ZHat.e_factorial_succ line 156 source
lemma ZHat.e_not_in_Int Nonarchimedean $e$ is not an integer line 167 source
Source documentation
Nonarchimedean $e$ is not an integer.
```
lemma e_not_in_Int : ∀ a : ℤ, e ≠ a := by
…
```
Mentions: ZHat.e
lemma ZHat.torsionfree_aux line 220 source
lemma ZHat.nat_mul_apply line 228 source
lemma ZHat.pnat_mul_apply line 231 source
theorem ZHat.eq_zero_of_mul_eq_zero line 234 source
lemma ZHat.torsionfree ZHat is torsion-free line 248 source
Source comment
ZHat is torsion-free. LaTeX proof in the notes.
```
lemma torsionfree (N : ℕ+) : Function.Injective (fun z : ZHat ↦ N * z) := by
…
```
Mentions: ZHat
instance ZHat.ZHat_flat line 254 source
lemma ZHat.y_mul_N_eq_z line 272 source
lemma ZHat.multiples LaTeX proof in the notes line 283 source
Source comment
LaTeX proof in the notes.
```
lemma multiples (N : ℕ+) (z : ZHat) : (∃ (y : ZHat), N * y = z) ↔ z N = 0 := by
…
```
Mentions: ZHat
lemma ZHat.nat_dense ZHat has division by positive naturals, with remainder a smaller natural line 312 source
Source comment
`ZHat` has division by positive naturals, with remainder a smaller natural. In other words, the naturals are dense in `ZHat`.
```
lemma nat_dense (N : ℕ+) (z : ZHat) : ∃ (q : ZHat) (r : ℕ), z = N * q + r ∧ r < N := by
…
```
Mentions: ZHat
abbrev QHat The "profinite completion" of ℚ is defined to be ℚ ⊗ ZHat, with ZHat the profinite completion of ℤ line 324 source
Source documentation
The "profinite completion" of ℚ is defined to be ℚ ⊗ ZHat, with ZHat the profinite completion of ℤ.
```
abbrev QHat := ℚ ⊗[ℤ] ZHat
```
Mentions: ZHat
example (example at line 326) line 326 source
lemma QHat.canonicalForm line 330 source
def QHat.IsCoprime An element z of ℤ̂ is coprime to N if its image in ℤ/Nℤ is a unit line 352 source
Source documentation
An element z of ℤ̂ is coprime to N if its image in ℤ/Nℤ is a unit.
```
def IsCoprime (N : ℕ+) (z : ZHat) : Prop := IsUnit (z N)
…
```
Mentions: ZHat
lemma QHat.isUnit_iff_coprime line 355 source
lemma QHat.isCoprime_iff_coprime line 367 source
abbrev QHat.i₂ The canonical inclusion ℤ̂ → ℚ̂ := ℚ ⊗_ℤ ℤ̂ sending z to 1 ⊗ z line 372 source
Source documentation
The canonical inclusion ℤ̂ → ℚ̂ := ℚ ⊗_ℤ ℤ̂ sending z to 1 ⊗ z.
```
noncomputable abbrev i₂ : ZHat →ₐ[ℤ] QHat := Algebra.TensorProduct.includeRight
```
Mentions: ZHat, QHat
lemma QHat.injective_zHat line 373 source
instance QHat.nontrivial_QHat line 386 source
abbrev QHat.i₁ The canonical inclusion ℚ → ℚ̂ = ℚ ⊗_ℤ ℤ̂ sending q to q ⊗ 1 line 390 source
Source documentation
The canonical inclusion ℚ → ℚ̂ = ℚ ⊗_ℤ ℤ̂ sending q to q ⊗ 1.
```
noncomputable abbrev i₁ : ℚ →ₐ[ℤ] QHat := Algebra.TensorProduct.includeLeft
```
Mentions: QHat
lemma QHat.injective_rat line 391 source
theorem QHat.PNat.lcm_comm line 394 source
lemma QHat.lowestTerms line 398 source
abbrev QHat.ratsub The image of ℚ inside ℚ̂ as an additive subgroup line 543 source
Source documentation
The image of ℚ inside ℚ̂ as an additive subgroup.
```
noncomputable abbrev ratsub : AddSubgroup QHat :=
…
```
Mentions: QHat
abbrev QHat.zHatsub The image of ℤ̂ inside ℚ̂ as an additive subgroup line 547 source
Source documentation
The image of ℤ̂ inside ℚ̂ as an additive subgroup.
```
noncomputable abbrev zHatsub : AddSubgroup QHat :=
…
```
Mentions: QHat
abbrev QHat.zsub The image of ℤ inside ℚ̂ as an additive subgroup line 551 source
Source documentation
The image of ℤ inside ℚ̂ as an additive subgroup.
```
noncomputable abbrev zsub : AddSubgroup QHat :=
…
```
Mentions: QHat
lemma QHat.ZMod.isUnit_natAbs line 554 source
lemma QHat._root_.Algebra.TensorProduct.one_tmul_intCast line 559 source
lemma QHat.rat_meet_zHat line 565 source
lemma QHat.rat_join_zHat line 591 source
abbrev QHat.unitsratsub The image of ℚˣ inside ℚ̂ˣ as a subgroup line 617 source
Source documentation
The image of ℚˣ inside ℚ̂ˣ as a subgroup.
```
noncomputable abbrev unitsratsub : Subgroup QHatˣ :=
…
```
abbrev QHat.unitszHatsub The image of ℤ̂ˣ inside ℚ̂ˣ as a subgroup line 621 source
Source documentation
The image of ℤ̂ˣ inside ℚ̂ˣ as a subgroup.
```
noncomputable abbrev unitszHatsub : Subgroup QHatˣ :=
…
```
abbrev QHat.unitszsub The image of ℤˣ inside ℚ̂ˣ as a subgroup line 625 source
Source documentation
The image of ℤˣ inside ℚ̂ˣ as a subgroup.
```
noncomputable abbrev unitszsub : Subgroup QHatˣ :=
…
```
lemma QHat.unitsrat_meet_unitszHat line 629 source
lemma QHat._root_.Algebra.TensorProduct.intCast_tmul_one line 693 source
lemma QHat._root_.Algebra.TensorProduct.one_tmul_natCast line 700 source
lemma QHat.unitsrat_join_unitszHat this needs that ℤ is a PID line 707 source
Source comment
this needs that ℤ is a PID.

depends on sorry (Lean reported 2 error(s) in this file) what this means
Statement uses: Eq, Subgroup, Units, QHat, Int, Rat, ZHat, Max.max, QHat.unitsratsub, QHat.unitszHatsub, Top.top
Proof uses: Eq.mpr, LE.le, id, congrArg, propext, eq_top_iff, Membership.mem, PNat, Exists, TensorProduct.tmul, HDiv.hDiv, OfNat.ofNat, QHat.canonicalForm, Inv.inv, TensorProduct, HMul.hMul, rfl, AlgHom, QHat.i₂, Algebra.TensorProduct.includeRight and 112 more
206 instances, coercions and auxiliary constants left out.

Tactic trace: 95 steps
Open to load the checked before-and-after goal states.
```
lemma unitsrat_join_unitszHat : unitsratsub ⊔ unitszHatsub = ⊤ := by
…
```
Mentions: QHat.unitsratsub, QHat.unitszHatsub

FLT/Data/QHat.lean

The profinite completion of ℤ and the finite adeles of ℚ

Main definitions

`FLT/Data/QHat.lean`

The profinite completion of `ℤ` and the finite adeles of `ℚ`