Representable.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Pinned FLT blueprint introduction

6 declarations, 2 contain sorry, 0 declared as axioms

Representability of deformation functors

Statements (mostly placeholders) towards the representability of the deformation functor of an absolutely irreducible mod-l Galois representation, following the framework of de Smit and Lenstra.

Imports (3)

Imported by 1 file

FLT.lean

lemma Deformation.isCorepresentable_deformationFunctor sorry line 36 source
depends on sorry what this means
Statement uses: Fintype, DecidableEq, Group, TopologicalSpace, T0Space, TotallyDisconnectedSpace, CompactSpace, CommRing, IsLocalRing, IsNoetherianRing, Finite, IsLocalRing.ResidueField, IsAdicComplete, IsLocalRing.maximalIdeal, CategoryTheory.Functor.obj, Deformation.ProartinianCat, Deformation.repnFunctor, Deformation.ProartinianCat.residueField, Representation.IsAbsolutelyIrreducible, Deformation.toRepresentation and 4 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
def Deformation.SLiftFunctor The functor of S-lifts of ρ, consisting of representations of GL₂ satisfying: 1 line 79 source
Source documentation
The functor of S-lifts of ρ, consisting of representations of GL₂ satisfying:
1. It is a lift of ρ.
2. Its determinant is equal to εₗ, the l-adic cyclotomic character.
3. It is unramified outside l and S.
4. Its trace is 2 on ker(Iᵥ → k(v)ˣ).
5. It is flat at all v | l.
depends on sorry what this means
Statement uses: CommRing, IsLocalRing, Field, NumberField, Nat, Fact, Nat.Prime, Algebra, PadicInt, Finset, IsDedekindDomain.HeightOneSpectrum, NumberField.RingOfIntegers, CategoryTheory.Functor.obj, Deformation.ProartinianCat, Deformation.repnFunctor, Fin, OfNat.ofNat, Field.absoluteGaloisGroup, AlgebraicClosure, Deformation.ProartinianCat.residueField and 1 more
Proof uses: Min.min, Deformation.liftFunctor, Deformation.detConditionFunctor, iInf, Not, Membership.mem, Ideal, IsDedekindDomain.HeightOneSpectrum.asIdeal, Deformation.unramifiedFunctor, Deformation.traceConditionFunctor, Deformation.flatFunctor
50 instances, coercions and auxiliary constants left out.
```
def SLiftFunctor : Subfunctor (repnFunctor (Fin 2) (Γ K) 𝓞) :=
…
```
Mentions: Deformation.repnFunctor
def Deformation.narrowSLiftFunctor The functor of narrow S-lifts of ρ, consisting of representations of GL₂ satisfying: 1 line 94 source
Source documentation
The functor of narrow S-lifts of ρ, consisting of representations of GL₂ satisfying:
1. It is a lift of ρ.
2. Its determinant is equal to εₗ, the l-adic cyclotomic character.
3. It is unramified outside l and S.
4. Its trace is 2 on Iᵥ.
5. It is flat at all v | l.
depends on sorry what this means
Statement uses: CommRing, IsLocalRing, Field, NumberField, Nat, Fact, Nat.Prime, Algebra, PadicInt, Finset, IsDedekindDomain.HeightOneSpectrum, NumberField.RingOfIntegers, CategoryTheory.Functor.obj, Deformation.ProartinianCat, Deformation.repnFunctor, Fin, OfNat.ofNat, Field.absoluteGaloisGroup, AlgebraicClosure, Deformation.ProartinianCat.residueField and 1 more
Proof uses: Min.min, Deformation.liftFunctor, Deformation.detConditionFunctor, iInf, Not, Membership.mem, Ideal, IsDedekindDomain.HeightOneSpectrum.asIdeal, Deformation.unramifiedFunctor, Deformation.narrowTraceConditionFunctor, Deformation.flatFunctor
50 instances, coercions and auxiliary constants left out.
```
def narrowSLiftFunctor : Subfunctor (repnFunctor (Fin 2) (Γ K) 𝓞) :=
…
```
Mentions: Deformation.repnFunctor
lemma Deformation.isCorepresentable_narrowSLiftFunctor sorry line 104 source
depends on sorry what this means
Statement uses: CommRing, IsLocalRing, IsNoetherianRing, Finite, IsLocalRing.ResidueField, IsAdicComplete, IsLocalRing.maximalIdeal, Field, NumberField, NumberField.IsTotallyReal, Even, Nat, Module.finrank, Rat, Fact, Nat.Prime, LT.lt, OfNat.ofNat, Algebra, PadicInt and 21 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
82 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
def Deformation.narrowSLiftUniversalRing The universal lifting ring Rᵘⁿⁱᵛ representing the functor of narrow S-lifts line 109 source
Source documentation
The universal lifting ring Rᵘⁿⁱᵛ representing the functor of narrow S-lifts.

depends on sorry what this means
Statement uses: CommRing, IsLocalRing, IsNoetherianRing, Finite, IsLocalRing.ResidueField, IsAdicComplete, IsLocalRing.maximalIdeal, Field, NumberField, NumberField.IsTotallyReal, Even, Nat, Module.finrank, Rat, Fact, Nat.Prime, LT.lt, OfNat.ofNat, Algebra, PadicInt and 19 more
Proof uses: Exists.choose, Nonempty, CategoryTheory.Functor.CorepresentableBy, CategoryTheory.Subfunctor.toFunctor
83 instances, coercions and auxiliary constants left out.
```
def narrowSLiftUniversalRing : ProartinianCat 𝓞 :=
…
```
Mentions: Deformation.ProartinianCat
def Deformation.narrowSLiftUniversalRingCorepresentableBy The universal lifting ring Rᵘⁿⁱᵛ represents the functor of narrow S-lifts line 113 source
Source documentation
The universal lifting ring Rᵘⁿⁱᵛ represents the functor of narrow S-lifts.

depends on sorry what this means
Statement uses: CommRing, IsLocalRing, IsNoetherianRing, Finite, IsLocalRing.ResidueField, IsAdicComplete, IsLocalRing.maximalIdeal, Field, NumberField, NumberField.IsTotallyReal, Even, Nat, Module.finrank, Rat, Fact, Nat.Prime, LT.lt, OfNat.ofNat, Algebra, PadicInt and 22 more
Proof uses: Nonempty.some, Exists.choose, Nonempty
84 instances, coercions and auxiliary constants left out.
```
def narrowSLiftUniversalRingCorepresentableBy :
    (narrowSLiftFunctor 𝓞 l S ρ).toFunctor.CorepresentableBy
      (narrowSLiftUniversalRing 𝓞 hK l hl S hS ρ hρ) :=
…
```
Mentions: Deformation.narrowSLiftFunctor, Deformation.narrowSLiftUniversalRing