LiftFunctor.lean`

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14 declarations, 1 contain sorry, 0 declared as axioms

The functor of continuous representations

For a profinite group G and a proartinian local ring 𝓞, the functor repnFunctor n G 𝓞 sends a proartinian 𝓞-algebra R to the set of continuous representations G → GLₙ(R).

Imports (4)

Imported by 2 files

def Deformation.repnFunctor repnFunctor n G 𝓞 is the functor taking R to continuous reps G → GLₙ(R) line 40 source
Source documentation
repnFunctor n G 𝓞 is the functor taking R to continuous reps G → GLₙ(R).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Group, TopologicalSpace, CommRing, CategoryTheory.Functor, Deformation.ProartinianCat
Proof uses: CategoryTheory.Functor.mk, ContinuousMonoidHom, Matrix.GeneralLinearGroup, Matrix, Quiver.Hom, TypeCat.ofHom, ContinuousMonoidHom.comp, Units, Units.mapₜ, ContinuousAlgHom.toContinuousMonoidHom, ContinuousAlgHom.mapMatrix, Deformation.ProartinianCat.Hom.hom
22 instances, coercions and auxiliary constants left out.
```
def repnFunctor : ProartinianCat 𝓞 ⥤ Type u where
…
```
Mentions: Deformation.ProartinianCat
lemma Deformation.repnFunctor_map line 46 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Group, TopologicalSpace, CommRing, Deformation.ProartinianCat, Quiver.Hom, ContinuousMonoidHom, Matrix.GeneralLinearGroup, Matrix, Eq, TypeCat.Fun, CategoryTheory.Functor.obj, Deformation.repnFunctor, CategoryTheory.ConcreteCategory.hom, CategoryTheory.Functor.map, MonoidHom, Matrix.GeneralLinearGroup.map, AlgHom.toRingHom, Deformation.ProartinianCat.Hom.hom
Proof uses: rfl
45 instances, coercions and auxiliary constants left out.
def Deformation.toRepresentation Turn an element in repnFunctor into an actual Representation line 52 source
Source documentation
Turn an element in repnFunctor into an actual Representation.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Group, TopologicalSpace, CommRing, Deformation.ProartinianCat, CategoryTheory.Functor.obj, Deformation.repnFunctor, Representation
Proof uses: MonoidHom.comp, Units, LinearMap, RingHom.id, Units.coeHom, Matrix.GeneralLinearGroup, LinearMap.GeneralLinearGroup, Matrix, MulEquiv.toMonoidHom, Matrix.GeneralLinearGroup.toLin, ContinuousMonoidHom.toMonoidHom
35 instances, coercions and auxiliary constants left out.
```
def toRepresentation {R} (ρ : (repnFunctor n G 𝓞).obj R) :
    Representation R G (n → R) :=
…
```
Mentions: Deformation.repnFunctor
def Deformation.toFramedGaloisRep line 59 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, CommRing, Field, Deformation.ProartinianCat, CategoryTheory.Functor.obj, Deformation.repnFunctor, Field.absoluteGaloisGroup, AlgebraicClosure, FramedGaloisRep
Proof uses: Equiv, ContinuousMonoidHom, Matrix.GeneralLinearGroup, Matrix, Equiv.symm, FramedGaloisRep.GL
38 instances, coercions and auxiliary constants left out.
lemma Deformation.toFramedGaloisRep_map line 65 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, CommRing, Field, Deformation.ProartinianCat, Quiver.Hom, CategoryTheory.Functor.obj, Deformation.repnFunctor, Field.absoluteGaloisGroup, AlgebraicClosure, Eq, FramedGaloisRep, Deformation.toFramedGaloisRep, TypeCat.Fun, CategoryTheory.ConcreteCategory.hom, CategoryTheory.Functor.map, FramedGaloisRep.baseChange, ContinuousAlgHom, Deformation.ProartinianCat.Hom.hom, ContinuousAlgHom.cont
Proof uses: Equiv.injective, ContinuousMonoidHom, Matrix.GeneralLinearGroup, Matrix, FramedGaloisRep.GL, ContinuousMonoidHom.ext, Equiv, Units.ext, Matrix.ext, of_eq_true, Eq.trans, True, congr, congrArg, MonoidHom, Matrix.GeneralLinearGroup.map, AlgHom, RingHom, congrFun', Equiv.apply_symm_apply and 4 more
73 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
def Deformation.repnQuotFunctor line 79 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Group, TopologicalSpace, CommRing, IsLocalRing, CategoryTheory.Functor, Deformation.ProartinianCat
Proof uses: CategoryTheory.Functor.mk, MulAction.orbitRel.Quotient, Subtype, ConjAct, Matrix.GeneralLinearGroup, Membership.mem, Subgroup, Matrix, Subgroup.comap, MulEquiv.toMonoidHom, ConjAct.ofConjAct, MonoidHom.ker, Deformation.ProartinianCat.residueField, Matrix.GeneralLinearGroup.map, AlgHom.toRingHom, Deformation.ProartinianCat.Hom.hom, Deformation.ProartinianCat.toResidueField, ContinuousMonoidHom, Quiver.Hom, TypeCat.ofHom and 7 more
40 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
def Deformation.toRepnQuot line 98 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Group, TopologicalSpace, CommRing, IsLocalRing, Quiver.Hom, CategoryTheory.Functor, Deformation.ProartinianCat, Deformation.repnFunctor, Deformation.repnQuotFunctor
Proof uses: CategoryTheory.NatTrans.mk, TypeCat.ofHom, CategoryTheory.Functor.obj, Quotient.mk'', MulAction.orbitRel, Subtype, ConjAct, Matrix.GeneralLinearGroup, Membership.mem, Subgroup, Matrix, Subgroup.comap, MulEquiv.toMonoidHom, ConjAct.ofConjAct, MonoidHom.ker, Deformation.ProartinianCat.residueField, Matrix.GeneralLinearGroup.map, AlgHom.toRingHom, Deformation.ProartinianCat.Hom.hom, Deformation.ProartinianCat.toResidueField and 1 more
36 instances, coercions and auxiliary constants left out.
def Deformation.liftFunctor line 104 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Group, TopologicalSpace, CommRing, IsLocalRing, CategoryTheory.Functor.obj, Deformation.ProartinianCat, Deformation.repnFunctor, Deformation.ProartinianCat.residueField, CategoryTheory.Subfunctor
Proof uses: CategoryTheory.Subfunctor.ofIsTerminal, Deformation.ProartinianCat.isTerminalResidueField, Singleton.singleton, Set
9 instances, coercions and auxiliary constants left out.
def Deformation.deformationFunctor line 110 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Group, TopologicalSpace, CommRing, IsLocalRing, CategoryTheory.Functor.obj, Deformation.ProartinianCat, Deformation.repnFunctor, Deformation.ProartinianCat.residueField, CategoryTheory.Subfunctor, Deformation.repnQuotFunctor
Proof uses: CategoryTheory.Subfunctor.ofIsTerminal, Deformation.ProartinianCat.isTerminalResidueField, Singleton.singleton, Set, TypeCat.Fun, CategoryTheory.ConcreteCategory.hom, CategoryTheory.NatTrans.app, Deformation.toRepnQuot
12 instances, coercions and auxiliary constants left out.
def Deformation.flatFunctor The subfunctor of flat lifts sorry line 115 source
Source documentation
The subfunctor of flat lifts. This probably only makes sense when 𝓞 is v-adic.

depends on sorry what this means
Statement uses: Fintype, DecidableEq, CommRing, Field, NumberField, IsDedekindDomain.HeightOneSpectrum, NumberField.RingOfIntegers, CategoryTheory.Subfunctor, Deformation.ProartinianCat, Deformation.repnFunctor, Field.absoluteGaloisGroup, AlgebraicClosure
Proof uses: CategoryTheory.Subfunctor.mk, Set.ofPred, CategoryTheory.Functor.obj, GaloisRep.IsFlatAt, Deformation.toFramedGaloisRep, sorryAx, Lean.Name, Quiver.Hom, LE.le, Set, Set.preimage, TypeCat.Fun, CategoryTheory.ConcreteCategory.hom, CategoryTheory.Functor.map, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
43 instances, coercions and auxiliary constants left out.
```
def flatFunctor (v : Ω K) : Subfunctor (repnFunctor n (Γ K) 𝓞) where
…
```
Mentions: Deformation.repnFunctor
def Deformation.unramifiedFunctor The subfunctor of unramified (at v) representations line 121 source
Source documentation
The subfunctor of unramified (at v) representations.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, CommRing, Field, NumberField, IsDedekindDomain.HeightOneSpectrum, NumberField.RingOfIntegers, CategoryTheory.Subfunctor, Deformation.ProartinianCat, Deformation.repnFunctor, Field.absoluteGaloisGroup, AlgebraicClosure
Proof uses: CategoryTheory.Subfunctor.mk, Set.ofPred, CategoryTheory.Functor.obj, GaloisRep.IsUnramifiedAt, Deformation.toFramedGaloisRep, Quiver.Hom, Membership.mem, Set
36 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
def unramifiedFunctor (v : Ω K) : Subfunctor (repnFunctor n (Γ K) 𝓞) where
…
```
Mentions: Deformation.repnFunctor
def Deformation.traceConditionFunctor The subfunctor of representations whose trace is 2 on ker(Iᵥ → k(v)ˣ) line 131 source
Source documentation
The subfunctor of representations whose trace is 2 on ker(Iᵥ → k(v)ˣ).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, NumberField, IsDedekindDomain.HeightOneSpectrum, NumberField.RingOfIntegers, CategoryTheory.Subfunctor, Deformation.ProartinianCat, Deformation.repnFunctor, Fin, OfNat.ofNat, Nat, Field.absoluteGaloisGroup, AlgebraicClosure
Proof uses: CategoryTheory.Subfunctor.mk, Set.ofPred, CategoryTheory.Functor.obj, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Membership.mem, Subgroup, localTameAbelianInertiaGroup, Eq, LinearMap, RingHom.id, LinearMap.trace, GaloisRep, Module.End, GaloisRep.toLocal, Deformation.toFramedGaloisRep, Quiver.Hom, Set
61 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
def traceConditionFunctor (v : Ω K) : Subfunctor (repnFunctor (Fin 2) (Γ K) 𝓞) where
…
```
Mentions: Deformation.repnFunctor
def Deformation.narrowTraceConditionFunctor The subfunctor of representations whose trace is 2 on Iᵥ line 142 source
Source documentation
The subfunctor of representations whose trace is 2 on Iᵥ.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Field, NumberField, IsDedekindDomain.HeightOneSpectrum, NumberField.RingOfIntegers, CategoryTheory.Subfunctor, Deformation.ProartinianCat, Deformation.repnFunctor, Fin, OfNat.ofNat, Nat, Field.absoluteGaloisGroup, AlgebraicClosure
Proof uses: CategoryTheory.Subfunctor.mk, Set.ofPred, CategoryTheory.Functor.obj, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Membership.mem, Subgroup, localInertiaGroup, Eq, LinearMap, RingHom.id, LinearMap.trace, GaloisRep, Module.End, GaloisRep.toLocal, Deformation.toFramedGaloisRep, Quiver.Hom, Set
61 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
def narrowTraceConditionFunctor (v : Ω K) : Subfunctor (repnFunctor (Fin 2) (Γ K) 𝓞) where
…
```
Mentions: Deformation.repnFunctor
def Deformation.detConditionFunctor The subfunctor of representations with det = εₗ line 153 source
Source documentation
The subfunctor of representations with det = εₗ.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, CommRing, Field, Nat, Fact, Nat.Prime, Algebra, PadicInt, CategoryTheory.Subfunctor, Deformation.ProartinianCat, Deformation.repnFunctor, Field.absoluteGaloisGroup, AlgebraicClosure
Proof uses: CategoryTheory.Subfunctor.mk, Set.ofPred, CategoryTheory.Functor.obj, Eq, ContinuousMonoidHom, GaloisRep.det, Deformation.toFramedGaloisRep, RingHom, Algebra.algebraMap, MonoidHom, RingEquiv, Units, cyclotomicCharacter, AlgEquiv.toRingEquiv, Quiver.Hom, Membership.mem, Set
57 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
def detConditionFunctor (l : ℕ) [Fact l.Prime] [Algebra ℤ_[l] 𝓞] :
    Subfunctor (repnFunctor n (Γ K) 𝓞) where
…
```
Mentions: Deformation.repnFunctor