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TwoSidedIdeal.mem_leAddSubgroup' {α} [NonUnitalNonAssocRing α] {G : AddSubgroup α} {x : α} :
x ∈ leAddSubgroup G ↔ (span {x} : Set α) ⊆ G | by
conv_rhs => rw [← sub_zero x]
rfl | lemma | TwoSidedIdeal.mem_leAddSubgroup' | Patching.Utils | FLT/Patching/Utils/Lemmas.lean | [
"Mathlib.Data.Set.Card",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.RingTheory.Filtration",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Separation.Profinite"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
TwoSidedIdeal.mem_leAddSubgroup {α} [Ring α] {G : AddSubgroup α} {x : α} :
x ∈ leAddSubgroup G ↔ ∀ a b, a * x * b ∈ G | by
constructor
· intro hx a b
exact hx (mul_mem_right _ _ _ (mul_mem_left _ _ _ ((sub_zero x).symm ▸ mem_span_singleton)))
· intro H a ha
simpa using mem_span_iff.mp ha (.mk' { x | ∀ a b, a * x * b ∈ G }
(by simp [G.zero_mem]) (by simp +contextual [mul_add, add_mul, G.add_mem])
(by simp) (fun ... | lemma | TwoSidedIdeal.mem_leAddSubgroup | Patching.Utils | FLT/Patching/Utils/Lemmas.lean | [
"Mathlib.Data.Set.Card",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.RingTheory.Filtration",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Separation.Profinite"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
exists_twoSidedIdeal_isOpen_and_subset {α} [TopologicalSpace α]
[CompactSpace α] [T2Space α] [TotallyDisconnectedSpace α]
[Ring α] [IsTopologicalRing α] {U : Set α} (hU : U ∈ 𝓝 0) :
∃ I : TwoSidedIdeal α, IsOpen (X := α) I ∧ (I : Set α) ⊆ U | by
obtain ⟨G, hG, hGU⟩ := exists_addSubgroup_isOpen_and_subset hU
refine ⟨_, isOpen_iff_mem_nhds.mpr ?_, (TwoSidedIdeal.leAddSubgroup_subset G).trans hGU⟩
intro x hx
replace hx := TwoSidedIdeal.mem_leAddSubgroup.mp hx
suffices
∀ s t, IsCompact s → IsCompact t →
∃ V ∈ 𝓝 x, ∀ a ∈ s, ∀ b ∈ V, ∀ c ∈ t,... | theorem | exists_twoSidedIdeal_isOpen_and_subset | Patching.Utils | FLT/Patching/Utils/Lemmas.lean | [
"Mathlib.Data.Set.Card",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.RingTheory.Filtration",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Separation.Profinite"
] | [
"TwoSidedIdeal.leAddSubgroup_subset",
"TwoSidedIdeal.mem_leAddSubgroup"
] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
exists_ideal_isOpen_and_subset {α} [TopologicalSpace α]
[CompactSpace α] [T2Space α] [TotallyDisconnectedSpace α]
[Ring α] [IsTopologicalRing α] {U : Set α} (hU : U ∈ 𝓝 0) :
∃ I : Ideal α, IsOpen (X := α) I ∧ (I : Set α) ⊆ U | by
obtain ⟨I, hI, hIU⟩ := exists_twoSidedIdeal_isOpen_and_subset hU
exact ⟨I.asIdeal, hI, hIU⟩ | theorem | exists_ideal_isOpen_and_subset | Patching.Utils | FLT/Patching/Utils/Lemmas.lean | [
"Mathlib.Data.Set.Card",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.RingTheory.Filtration",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Separation.Profinite"
] | [
"exists_twoSidedIdeal_isOpen_and_subset"
] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
WellFoundedGT.exists_eq_sup {α} [CompleteLattice α] [WellFoundedGT α]
(f : ℕ →o α) : ∃ i, f i = ⨆ i, f i | by
obtain ⟨n, hn⟩ := wellFoundedGT_iff_monotone_chain_condition.mp ‹WellFoundedGT α› f
exact ⟨n, le_antisymm (le_iSup _ _) (iSup_le fun i ↦
(le_total i n).elim (f.2 ·) (fun h ↦ (hn _ h).ge))⟩ | lemma | WellFoundedGT.exists_eq_sup | Patching.Utils | FLT/Patching/Utils/Lemmas.lean | [
"Mathlib.Data.Set.Card",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.RingTheory.Filtration",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Separation.Profinite"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
WellFoundedLT.exists_eq_inf {α} [CompleteLattice α] [WellFoundedLT α]
(f : ℕ →o αᵒᵈ) : ∃ i, f i = (⨅ i, f i : α) | WellFoundedGT.exists_eq_sup (α := αᵒᵈ) f | lemma | WellFoundedLT.exists_eq_inf | Patching.Utils | FLT/Patching/Utils/Lemmas.lean | [
"Mathlib.Data.Set.Card",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.RingTheory.Filtration",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Separation.Profinite"
] | [
"WellFoundedGT.exists_eq_sup"
] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
IsLocalRing.maximalIdeal_pow_card_smul_top_le {R M}
[CommRing R] [IsLocalRing R] [IsNoetherianRing R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [Finite (M ⧸ N)] : maximalIdeal R ^ Nat.card (M ⧸ N) • ⊤ ≤ N | by
let f (n) : Submodule R (M ⧸ N) := maximalIdeal R ^ n • ⊤
have hf : ∀ i j, i ≤ j → f j ≤ f i :=
fun i j h ↦ Submodule.smul_mono (Ideal.pow_le_pow_right h) le_rfl
have H : ∃ i, f i = ⊥ := by
obtain ⟨i, hi⟩ := WellFoundedLT.exists_eq_inf ⟨f, hf⟩
have := Ideal.iInf_pow_smul_eq_bot_of_isLocalRing (R :=... | lemma | IsLocalRing.maximalIdeal_pow_card_smul_top_le | Patching.Utils | FLT/Patching/Utils/Lemmas.lean | [
"Mathlib.Data.Set.Card",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.RingTheory.Filtration",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Separation.Profinite"
] | [
"WellFoundedLT.exists_eq_inf"
] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
Submodule.comap_smul_of_le_range {R M M'} [CommRing R] [AddCommGroup M]
[AddCommGroup M'] [Module R M] [Module R M']
(f : M →ₗ[R] M') (S : Submodule R M') (hS : S ≤ LinearMap.range f) (I : Ideal R) :
(I • S).comap f = (I • S.comap f) ⊔ LinearMap.ker f | by
rw [← comap_map_eq, map_smul'', Submodule.map_comap_eq, inf_eq_right.mpr hS] | theorem | Submodule.comap_smul_of_le_range | Patching.Utils | FLT/Patching/Utils/Lemmas.lean | [
"Mathlib.Data.Set.Card",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.RingTheory.Filtration",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Separation.Profinite"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
Submodule.comap_smul_of_surjective {R M M'} [CommRing R] [AddCommGroup M]
[AddCommGroup M'] [Module R M] [Module R M']
(f : M →ₗ[R] M') (S : Submodule R M') (hS : Function.Surjective f) (I : Ideal R) :
(I • S).comap f = (I • S.comap f) ⊔ LinearMap.ker f | comap_smul_of_le_range f S (le_top.trans_eq (LinearMap.range_eq_top_of_surjective f hS).symm) I | theorem | Submodule.comap_smul_of_surjective | Patching.Utils | FLT/Patching/Utils/Lemmas.lean | [
"Mathlib.Data.Set.Card",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.RingTheory.Filtration",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Separation.Profinite"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
Pi.liftQuotientₗ {ι R M : Type*} [CommRing R] [AddCommGroup M] [Module R M] [Finite ι]
(f : (ι → R) →ₗ[R] M) (I : Ideal R) : (ι → R ⧸ I) →ₗ[R] M ⧸ (I • ⊤ : Submodule R M) | by
refine Submodule.liftQ _ (Submodule.mkQ _ ∘ₗ f) ?_ ∘ₗ
(((Algebra.linearMap R (R ⧸ I)).compLeft ι).quotKerEquivOfSurjective ?_).symm.toLinearMap
· intro x hx
replace hx : ∀ i, x i ∈ I := by
simpa [funext_iff, Ideal.Quotient.eq_zero_iff_mem] using hx
cases nonempty_fintype ι
classical
hav... | def | Pi.liftQuotientₗ | Patching.Utils | FLT/Patching/Utils/Lemmas.lean | [
"Mathlib.Data.Set.Card",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.RingTheory.Filtration",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Separation.Profinite"
] | [] | The canonical descent of a linear map `f : (ι → R) →ₗ[R] M` from the standard finite free
module to a linear map between the corresponding quotients modulo the ideal `I`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
Pi.liftQuotientₗ_surjective {ι R M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
[Finite ι] (f : (ι → R) →ₗ[R] M) (I : Ideal R) (hf : Function.Surjective f) :
Function.Surjective (Pi.liftQuotientₗ f I) | by
simp only [liftQuotientₗ, LinearMap.coe_comp, LinearEquiv.coe_coe, EquivLike.surjective_comp]
rw [← LinearMap.range_eq_top, Submodule.range_liftQ, LinearMap.range_eq_top]
exact (Submodule.mkQ_surjective _).comp hf | lemma | Pi.liftQuotientₗ_surjective | Patching.Utils | FLT/Patching/Utils/Lemmas.lean | [
"Mathlib.Data.Set.Card",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.RingTheory.Filtration",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Separation.Profinite"
] | [
"Pi.liftQuotientₗ"
] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
Pi.liftQuotientₗ_bijective {ι R M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
[Finite ι] (f : (ι → R) →ₗ[R] M) (I : Ideal R) (hf : Function.Surjective f)
(hf' : LinearMap.ker f ≤ LinearMap.ker ((Algebra.linearMap R (R ⧸ I)).compLeft ι)) :
Function.Bijective (Pi.liftQuotientₗ f I) | by
refine ⟨?_, liftQuotientₗ_surjective f I hf⟩
simp only [liftQuotientₗ, LinearMap.coe_comp, LinearEquiv.coe_coe, EquivLike.injective_comp]
rw [← LinearMap.ker_eq_bot, Submodule.ker_liftQ, ← le_bot_iff, Submodule.map_le_iff_le_comap,
Submodule.comap_bot, Submodule.ker_mkQ, LinearMap.ker_comp, Submodule.ker... | lemma | Pi.liftQuotientₗ_bijective | Patching.Utils | FLT/Patching/Utils/Lemmas.lean | [
"Mathlib.Data.Set.Card",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.RingTheory.Filtration",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Separation.Profinite"
] | [
"Pi.liftQuotientₗ",
"Submodule.comap_smul_of_surjective"
] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
IsModuleTopology.compactSpace
(R M : Type*) [CommRing R] [TopologicalSpace R] [AddCommGroup M]
[Module R M] [TopologicalSpace M] [IsModuleTopology R M]
[CompactSpace R] [Module.Finite R M] : CompactSpace M | letI : ContinuousAdd M := toContinuousAdd R M
⟨Submodule.isCompact_of_fg (Module.Finite.fg_top (R := R))⟩ | lemma | IsModuleTopology.compactSpace | Patching.Utils | FLT/Patching/Utils/Lemmas.lean | [
"Mathlib.Data.Set.Card",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.RingTheory.Filtration",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Separation.Profinite"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
disjoint_nonZeroDivisors_of_mem_minimalPrimes
{R : Type*} [CommRing R] (p : Ideal R) (hp : p ∈ minimalPrimes R) :
Disjoint (p : Set R) (nonZeroDivisors R) | by
classical
rw [← Set.subset_compl_iff_disjoint_right, Set.subset_def]
simp only [SetLike.mem_coe, Set.mem_compl_iff, mem_nonZeroDivisors_iff_right, not_forall]
intro x hxp
have := hp.1.1
have : p.map (algebraMap R (Localization.AtPrime p)) ≤ nilradical _ := by
rw [nilradical, Ideal.radical_eq_sInf, le... | lemma | disjoint_nonZeroDivisors_of_mem_minimalPrimes | Patching.Utils | FLT/Patching/Utils/Lemmas.lean | [
"Mathlib.Data.Set.Card",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.RingTheory.Filtration",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Separation.Profinite"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
ModuleTypeCardLT (N : ℕ) : Type _ | Σ (n : Fin N) (_ : AddCommGroup (Fin n)), Module R (Fin n) | def | ModuleTypeCardLT | Patching.Utils | FLT/Patching/Utils/StructureFiniteness.lean | [
"Mathlib.Algebra.Ring.Ext"
] | [] | The type of all finite `R`-modules of cardinality less than `N`, presented as a sigma
type over `Fin N`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
ModuleTypeCardLT.ofModule (N : ℕ) (M : Type*) [AddCommGroup M] [Module R M]
[Finite M] (hM : Nat.card M < N) : ModuleTypeCardLT R N | ⟨⟨Nat.card M, hM⟩, (Finite.equivFin M).symm.addCommGroup, (Finite.equivFin M).symm.module R⟩ | def | ModuleTypeCardLT.ofModule | Patching.Utils | FLT/Patching/Utils/StructureFiniteness.lean | [
"Mathlib.Algebra.Ring.Ext"
] | [
"ModuleTypeCardLT"
] | Pick a representative in `ModuleTypeCardLT R N` for a given finite `R`-module `M`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
ModuleTypeCardLT.equivOfModule (N : ℕ) {M : Type*} [AddCommGroup M] [Module R M]
[Finite M] (hM : Nat.card M < N) : M ≃ₗ[R] Fin ((ModuleTypeCardLT.ofModule R N M hM).1) | ((show M ≃ Fin ((ModuleTypeCardLT.ofModule R N M hM).1)
from Finite.equivFin M).symm.linearEquiv R).symm | def | ModuleTypeCardLT.equivOfModule | Patching.Utils | FLT/Patching/Utils/StructureFiniteness.lean | [
"Mathlib.Algebra.Ring.Ext"
] | [
"ModuleTypeCardLT.ofModule"
] | The canonical linear equivalence between a finite `R`-module `M` and its representative
in `ModuleTypeCardLT R N`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
AlgebraTypeCardLT (N : ℕ) : Type _ | Σ (n : Fin N) (_ : Ring (Fin n)), Algebra R (Fin n) | def | AlgebraTypeCardLT | Patching.Utils | FLT/Patching/Utils/StructureFiniteness.lean | [
"Mathlib.Algebra.Ring.Ext"
] | [] | The type of all finite `R`-algebras of cardinality less than `N`, presented as a sigma
type over `Fin N`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
AlgebraTypeCardLT.ofAlgebra (N : ℕ) (M : Type*) [Ring M] [Algebra R M]
[Finite M] (hM : Nat.card M < N) : AlgebraTypeCardLT R N | ⟨⟨Nat.card M, hM⟩, (Finite.equivFin M).symm.ring, (Finite.equivFin M).symm.algebra R⟩ | def | AlgebraTypeCardLT.ofAlgebra | Patching.Utils | FLT/Patching/Utils/StructureFiniteness.lean | [
"Mathlib.Algebra.Ring.Ext"
] | [
"AlgebraTypeCardLT"
] | Pick a representative in `AlgebraTypeCardLT R N` for a given finite `R`-algebra `M`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
AlgebraTypeCardLT.equivOfAlgebra (N : ℕ) {M : Type*} [Ring M] [Algebra R M]
[Finite M] (hM : Nat.card M < N) : M ≃ₐ[R] Fin ((AlgebraTypeCardLT.ofAlgebra R N M hM).1) | ((show M ≃ Fin ((AlgebraTypeCardLT.ofAlgebra R N M hM).1)
from Finite.equivFin M).symm.algEquiv R).symm | def | AlgebraTypeCardLT.equivOfAlgebra | Patching.Utils | FLT/Patching/Utils/StructureFiniteness.lean | [
"Mathlib.Algebra.Ring.Ext"
] | [
"AlgebraTypeCardLT.ofAlgebra"
] | The canonical algebra equivalence between a finite `R`-algebra `M` and its
representative in `AlgebraTypeCardLT R N`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
TopologicalModuleTypeCardLT (N : ℕ) : Type _ | Σ' (n : Fin N) (_ : AddCommGroup (Fin n)) (_ : TopologicalSpace (Fin n)) (_ : T2Space (Fin n))
(_ : Module R (Fin n)), ContinuousSMul R (Fin n) | def | TopologicalModuleTypeCardLT | Patching.Utils | FLT/Patching/Utils/StructureFiniteness.lean | [
"Mathlib.Algebra.Ring.Ext"
] | [] | The type of all finite Hausdorff topological `R`-modules of cardinality less than `N`,
with continuous scalar multiplication. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
TopologicalModuleTypeCardLT.ofModule (N : ℕ) (M : Type*) [AddCommGroup M]
[Module R M] [TopologicalSpace M] [T2Space M] [ContinuousSMul R M]
[Finite M] (hM : Nat.card M < N) : TopologicalModuleTypeCardLT R N | ⟨⟨Nat.card M, hM⟩, (Finite.equivFin M).symm.addCommGroup, .coinduced (Finite.equivFin M)
inferInstance,
letI := TopologicalSpace.coinduced (Finite.equivFin M) inferInstance
Topology.IsEmbedding.t2Space (f := (Finite.equivFin M).symm)
⟨⟨by rw [(Finite.equivFin M).induced_symm.symm]⟩, (Finite.equivFin M).... | def | TopologicalModuleTypeCardLT.ofModule | Patching.Utils | FLT/Patching/Utils/StructureFiniteness.lean | [
"Mathlib.Algebra.Ring.Ext"
] | [
"TopologicalModuleTypeCardLT"
] | Pick a representative in `TopologicalModuleTypeCardLT R N` for a given finite
topological `R`-module `M`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
TopologicalModuleTypeCardLT.equivOfModule (N : ℕ) (M : Type*) [AddCommGroup M] [Module R M]
[TopologicalSpace M] [T2Space M] [ContinuousSMul R M]
[Finite M] (hM : Nat.card M < N) :
M ≃L[R] Fin (TopologicalModuleTypeCardLT.ofModule R N M hM).1 | where
__ := ((show M ≃ Fin ((ModuleTypeCardLT.ofModule R N M hM).1) from
Finite.equivFin M).symm.linearEquiv R).symm
__ := (Finite.equivFin M).toHomeomorph (Y := Fin (ofModule R N M hM).1) (fun _ ↦ Iff.rfl) | def | TopologicalModuleTypeCardLT.equivOfModule | Patching.Utils | FLT/Patching/Utils/StructureFiniteness.lean | [
"Mathlib.Algebra.Ring.Ext"
] | [
"ModuleTypeCardLT.ofModule",
"TopologicalModuleTypeCardLT.ofModule"
] | The canonical continuous linear equivalence between a finite topological `R`-module `M`
and its representative in `TopologicalModuleTypeCardLT R N`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
TopologicalAlgebraTypeCardLT (N : ℕ) :
Type _ | Σ' (n : Fin N) (_ : Ring (Fin n)) (_ : TopologicalSpace (Fin n)) (_ : T2Space (Fin n))
(_ : Algebra R (Fin n)), ContinuousSMul R (Fin n) | def | TopologicalAlgebraTypeCardLT | Patching.Utils | FLT/Patching/Utils/StructureFiniteness.lean | [
"Mathlib.Algebra.Ring.Ext"
] | [] | The type of all finite Hausdorff topological `R`-algebras of cardinality less than `N`,
with continuous scalar multiplication. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
TopologicalAlgebraTypeCardLT.ofAlgebra (N : ℕ) (M : Type*) [Ring M]
[Algebra R M] [TopologicalSpace M] [T2Space M] [ContinuousSMul R M]
[Finite M] (hM : Nat.card M < N) : TopologicalAlgebraTypeCardLT R N | ⟨⟨Nat.card M, hM⟩, (Finite.equivFin M).symm.ring, .coinduced (Finite.equivFin M) inferInstance,
letI := TopologicalSpace.coinduced (Finite.equivFin M) inferInstance
Topology.IsEmbedding.t2Space (f := (Finite.equivFin M).symm)
⟨⟨congr_fun (Finite.equivFin M).induced_symm.symm inferInstance⟩,
(Finite.equi... | def | TopologicalAlgebraTypeCardLT.ofAlgebra | Patching.Utils | FLT/Patching/Utils/StructureFiniteness.lean | [
"Mathlib.Algebra.Ring.Ext"
] | [
"TopologicalAlgebraTypeCardLT",
"TopologicalModuleTypeCardLT.ofModule"
] | Pick a representative in `TopologicalAlgebraTypeCardLT R N` for a given finite
topological `R`-algebra `M`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
TopologicalAlgebraTypeCardLT.equivOfAlgebra (N : ℕ) (M : Type*) [Ring M]
[Algebra R M] [TopologicalSpace M] [T2Space M] [ContinuousSMul R M]
[Finite M] (hM : Nat.card M < N) :
M ≃ₐ[R] Fin (TopologicalAlgebraTypeCardLT.ofAlgebra R N M hM).1 | ((show M ≃ Fin ((AlgebraTypeCardLT.ofAlgebra R N M hM).1)
from Finite.equivFin M).symm.algEquiv R).symm | def | TopologicalAlgebraTypeCardLT.equivOfAlgebra | Patching.Utils | FLT/Patching/Utils/StructureFiniteness.lean | [
"Mathlib.Algebra.Ring.Ext"
] | [
"Algebra.TopologicallyFG",
"AlgebraTypeCardLT.ofAlgebra",
"TopologicalAlgebraTypeCardLT.ofAlgebra"
] | The canonical algebra equivalence between a finite topological `R`-algebra `M` and its
representative in `TopologicalAlgebraTypeCardLT R N`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
TopologicalAlgebraTypeCardLT.isHomeomorph_equivOfAlgebra (N : ℕ) (M : Type*) [Ring M]
[Algebra R M] [TopologicalSpace M] [T2Space M] [ContinuousSMul R M]
[Finite M] (hM : Nat.card M < N) : IsHomeomorph (equivOfAlgebra (R := R) N M hM) | ((Finite.equivFin M).toHomeomorph (Y := Fin (ofAlgebra R N M hM).1)
(fun _ ↦ Iff.rfl)).isHomeomorph | lemma | TopologicalAlgebraTypeCardLT.isHomeomorph_equivOfAlgebra | Patching.Utils | FLT/Patching/Utils/StructureFiniteness.lean | [
"Mathlib.Algebra.Ring.Ext"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
Algebra.TopologicallyFG [IsTopologicalRing S] : Prop where
out : ∃ s : Finset S, Dense (Algebra.adjoin R (s : Set S) : Set S) | class | Algebra.TopologicallyFG | Patching.Utils | FLT/Patching/Utils/TopologicallyFG.lean | [] | [] | An `R`-algebra `S` is topologically finitely generated if there is a finite subset
`s ⊆ S` such that the `R`-subalgebra generated by `s` is dense in `S`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
Algebra.TopologicallyFG.module_ext (s : Set S)
(hs' : Dense (Algebra.adjoin R (s : Set S) : Set S)) {m₁ m₂ : Module S M}
(hm₁ : letI := m₁; IsScalarTower R S M) (hm₂ : letI := m₂; IsScalarTower R S M)
(hm₁' : letI := m₁; ContinuousSMul S M) (hm₂' : letI := m₂; ContinuousSMul S M)
(H : ∀ x ∈ s, ∀ m : M, ... | by
ext r m
induction r using hs'.induction with
| mem x hx =>
induction hx using Algebra.adjoin_induction generalizing m with
| mem x hx => exact H x hx m
| algebraMap r =>
exact .trans (letI := m₁; algebraMap_smul ..) (.symm (letI := m₂; algebraMap_smul ..))
| add x y hx hy hx' hy' =>
... | lemma | Algebra.TopologicallyFG.module_ext | Patching.Utils | FLT/Patching/Utils/TopologicallyFG.lean | [] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
Rigidification | (D ⊗[F] (FiniteAdeleRing (𝓞 F) F) ≃ₐ[FiniteAdeleRing (𝓞 F) F]
Matrix (Fin 2) (Fin 2) (FiniteAdeleRing (𝓞 F) F)) | abbrev | IsQuaternionAlgebra.NumberField.Rigidification | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [] | A rigidification of a quaternion algebra D over a number field F
is a fixed choice of `𝔸_F^∞`-algebra isomorphism `D ⊗[F] 𝔸_F^∞ = M₂(𝔸_F^∞)`. In other
words, it is a choice of splitting of `D ⊗[F] Fᵥ` (i.e. an isomorphism to `M₂(Fᵥ)`)
for all finite places `v` together with a guarantee that the isomorphism works
on ... | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
IsUnramified : Prop | Nonempty (Rigidification F D) | def | IsQuaternionAlgebra.NumberField.IsUnramified | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [] | A quaternion algebra over a number field is unramified if it is split
at all finite places. This is implemented as the existence of a rigidification
of `D`, that is, an isomorphism `D ⊗[F] 𝔸_F^∞ = M₂(𝔸_F^∞)`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
M2.localFullLevel (v : HeightOneSpectrum (𝓞 F)) :
Subring (Matrix (Fin 2) (Fin 2) (v.adicCompletion F)) | (v.adicCompletionIntegers F).matrix | def | IsDedekindDomain.M2.localFullLevel | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [] | `M_2(O_v)` as a subring of `M_2(F_v)`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
GL2.localFullLevel (v : HeightOneSpectrum (𝓞 F)) :
Subgroup (GL (Fin 2) (v.adicCompletion F)) | MonoidHom.range (Units.map
(RingHom.mapMatrix (v.adicCompletionIntegers F).subtype).toMonoidHom) | def | IsDedekindDomain.GL2.localFullLevel | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [] | `GL₂(𝒪ᵥ)` as a subgroup of `GL₂(Fᵥ)`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
M2.localFullLevel.isOpen (v : HeightOneSpectrum (𝓞 F)) :
IsOpen (M2.localFullLevel v).carrier | (NumberField.isOpenAdicCompletionIntegers F v).matrix | theorem | IsDedekindDomain.M2.localFullLevel.isOpen | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [
"NumberField.isOpenAdicCompletionIntegers"
] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
M2.localFullLevel.isCompact (v : HeightOneSpectrum (𝓞 F)) :
IsCompact (M2.localFullLevel v).carrier | (isCompact_iff_compactSpace.mpr (NumberField.instCompactSpaceAdicCompletionIntegers F v)).matrix | theorem | IsDedekindDomain.M2.localFullLevel.isCompact | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [
"NumberField.instCompactSpaceAdicCompletionIntegers"
] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
GL2.localFullLevel.isOpen (v : HeightOneSpectrum (𝓞 F)) :
IsOpen (GL2.localFullLevel v).carrier | sorry | theorem | IsDedekindDomain.GL2.localFullLevel.isOpen | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
GL2.localFullLevel.isCompact (v : HeightOneSpectrum (𝓞 F)) :
IsCompact (GL2.localFullLevel v).carrier | sorry | theorem | IsDedekindDomain.GL2.localFullLevel.isCompact | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
GL2.mem_localFullLevel {v : HeightOneSpectrum (𝓞 F)} {x : GL (Fin 2) (v.adicCompletion F)}
(hx : x ∈ localFullLevel v) :
∃ x' : GL (Fin 2) (v.adicCompletionIntegers F),
Units.map ((v.adicCompletionIntegers F).subtype.mapMatrix.toMonoidHom) x' = x | hx | lemma | IsDedekindDomain.GL2.mem_localFullLevel | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
GL2.mem_localFullLevel' {v : HeightOneSpectrum (𝓞 F)} {x : GL (Fin 2) (v.adicCompletion F)}
(hx : x ∈ localFullLevel v) :
∃ x' : GL (Fin 2) (v.adicCompletionIntegers F), ∀ i j, x' i j = x i j | by
refine (mem_localFullLevel hx).imp ?_
simp only [RingHom.toMonoidHom_eq_coe, Units.ext_iff, Units.coe_map, MonoidHom.coe_coe,
RingHom.mapMatrix_apply]
rintro y hy
simp [← hy] | lemma | IsDedekindDomain.GL2.mem_localFullLevel' | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
GL2.v_det_val_mem_localFullLevel_eq_one {v : HeightOneSpectrum (𝓞 F)}
{x : GL (Fin 2) (v.adicCompletion F)} (hx : x ∈ localFullLevel v) :
Valued.v x.val.det = 1 | by
obtain ⟨y, hy⟩ := mem_localFullLevel hx
have hd : IsUnit y.det.val := by simp
rw [Valued.isUnit_valuationSubring_iff] at hd
simpa [← hy, Matrix.det_fin_two] using hd | lemma | IsDedekindDomain.GL2.v_det_val_mem_localFullLevel_eq_one | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [
"Valued.isUnit_valuationSubring_iff"
] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
GL2.v_le_one_of_mem_localFullLevel (v : HeightOneSpectrum (𝓞 F)) {x}
(hx : x ∈ localFullLevel v) (i j) : Valued.v (x i j) ≤ 1 | by
simp only [localFullLevel, Units.map, RingHom.mapMatrix, Matrix.map, ValuationSubring.subtype,
Subring.subtype, RingHom.coe_mk, MonoidHom.coe_mk, OneHom.coe_mk, RingHom.toMonoidHom_eq_coe,
RingHom.coe_monoidHom_mk, Units.inv_eq_val_inv, Matrix.coe_units_inv, MonoidHom.mem_range,
MonoidHom.mk'_apply, Ma... | lemma | IsDedekindDomain.GL2.v_le_one_of_mem_localFullLevel | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
GL2.mem_localFullLevel_iff_v_le_one_and_v_det_eq_one {v : HeightOneSpectrum (𝓞 F)}
{x : GL (Fin 2) (v.adicCompletion F)} :
x ∈ localFullLevel v ↔ (∀ (i j), Valued.v (x i j) ≤ 1) ∧ Valued.v x.val.det = 1 | ⟨fun h ↦ ⟨GL2.v_le_one_of_mem_localFullLevel _ h, GL2.v_det_val_mem_localFullLevel_eq_one h⟩, by
intro ⟨h₁, h₂⟩
let M : Matrix (Fin 2) (Fin 2) (v.adicCompletionIntegers F) :=
Matrix.of fun i j => ⟨x i j, h₁ i j⟩
have det_eq : M.det = x.val.det := by
rw [Matrix.det_fin_two, Matrix.det_fin_two]; s... | lemma | IsDedekindDomain.GL2.mem_localFullLevel_iff_v_le_one_and_v_det_eq_one | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
GL2.localTameLevel (v : HeightOneSpectrum (𝓞 F)) :
Subgroup (GL (Fin 2) (v.adicCompletion F)) | where
carrier := {x ∈ localFullLevel v |
Valued.v (x.val 0 0 - x.val 1 1) < 1 ∧ Valued.v (x.val 1 0) < 1}
mul_mem' {a b} ha hb := by
simp_all only [Set.mem_setOf_eq, Units.val_mul]
refine ⟨Subgroup.mul_mem _ ha.1 hb.1, ?_, ?_⟩
· simp only [Matrix.mul_apply, Fin.isValue, Fin.sum_univ_two]
conve... | def | IsDedekindDomain.GL2.localTameLevel | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [] | local U_1(v), defined as a subgroup of GL₂(Fᵥ) given by
matrices in GL₂(𝒪ᵥ) congruent to (a *;0 a) mod v. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
GL2.localTameLevel.isOpen (v : HeightOneSpectrum (𝓞 F)) :
IsOpen (GL2.localTameLevel v).carrier | sorry | theorem | IsDedekindDomain.GL2.localTameLevel.isOpen | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
GL2.localTameLevel.isCompact (v : HeightOneSpectrum (𝓞 F)) :
IsCompact (GL2.localTameLevel v).carrier | sorry | theorem | IsDedekindDomain.GL2.localTameLevel.isCompact | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
IsDedekindDomain.FiniteAdeleRing.toAdicCompletion (v : HeightOneSpectrum (𝓞 F)) :
FiniteAdeleRing (𝓞 F) F →ₐ[F] HeightOneSpectrum.adicCompletion F v | where
__ := RestrictedProduct.evalRingHom _ v
commutes' _ := rfl | def | IsDedekindDomain.FiniteAdeleRing.toAdicCompletion | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [] | The canonical F-algebra morphism from `𝔸_F^∞` (the finite adeles of a number field F) to
the local component `F_v` for `v` a finite place of `𝓞 F`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
GL2.toAdicCompletion
(v : HeightOneSpectrum (𝓞 F)) :
GL (Fin 2) (FiniteAdeleRing (𝓞 F) F) →*
GL (Fin 2) (v.adicCompletion F) | Units.map (RingHom.mapMatrix (FiniteAdeleRing.toAdicCompletion v)).toMonoidHom | def | IsDedekindDomain.FiniteAdeleRing.GL2.toAdicCompletion | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [
"FiniteAdeleRing.toAdicCompletion"
] | The canonical group homomorphism from `GL_2(𝔸_F^∞)` to the local component `GL_2(F_v)` for `v`
a finite place. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
GL2.restrictedProduct :
GL (Fin 2) (FiniteAdeleRing (𝓞 F) F) ≃ₜ*
Πʳ (v : HeightOneSpectrum (𝓞 F)),
[(GL (Fin 2) (v.adicCompletion F)), (M2.localFullLevel v).units] | ContinuousMulEquiv.restrictedProductMatrixUnits (NumberField.isOpenAdicCompletionIntegers F) | def | IsDedekindDomain.FiniteAdeleRing.GL2.restrictedProduct | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [
"ContinuousMulEquiv.restrictedProductMatrixUnits",
"NumberField.isOpenAdicCompletionIntegers"
] | `GL_2(𝔸_F^∞)` is isomorphic and homeomorphic to the
restricted product of the local components `GL_2(F_v)`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
GL2.TameLevel (S : Finset (HeightOneSpectrum (𝓞 F))) :
Subgroup (GL (Fin 2) (FiniteAdeleRing (𝓞 F) F)) | where
carrier := {x | (∀ v, GL2.toAdicCompletion v x ∈ GL2.localFullLevel v) ∧
(∀ v ∈ S, GL2.toAdicCompletion v x ∈ GL2.localTameLevel v)}
mul_mem' {a b} ha hb := by simp_all [mul_mem]
one_mem' := by simp_all [one_mem]
inv_mem' {x} hx := by simp_all | def | IsDedekindDomain.HeightOneSpectrum.GL2.TameLevel | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [] | If `F` is a number field and `S` is a finite set of finite places of `𝓞 F` then
`GL2.TameLevel S` is the subgroup of `GL₂(𝔸_F^∞)` consisting of things in `GL₂(𝓞ᵥ)` for
all places, and furthermore in the local "`U₁(v)`" subgroup `(a *;0 a) mod v` for all `v ∈ S`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
GL2.TameLevel.isOpen : IsOpen (GL2.TameLevel S).carrier | sorry | theorem | IsDedekindDomain.HeightOneSpectrum.GL2.TameLevel.isOpen | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
GL2.TameLevel.isCompact : IsCompact (GL2.TameLevel S).carrier | sorry | theorem | IsDedekindDomain.HeightOneSpectrum.GL2.TameLevel.isCompact | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
QuaternionAlgebra.TameLevel (r : Rigidification F D) :
Subgroup (D ⊗[F] (FiniteAdeleRing (𝓞 F) F))ˣ | Subgroup.comap (Units.map r.toMonoidHom) (GL2.TameLevel S) | def | IsDedekindDomain.HeightOneSpectrum.QuaternionAlgebra.TameLevel | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [] | The subgroup of `(D ⊗ 𝔸_F^∞)ˣ` corresponding to the subgroup `U₁(S)` of `GL₂(𝔸_F^∞)`
(that is, matrices congruent to `(a *; 0 a) mod v` for all `v ∈ S`) via the rigidification `r`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
Rigidification.continuous_toFun (r : Rigidification F D) :
Continuous r | letI : ∀ (i : HeightOneSpectrum (𝓞 F)),
Algebra (FiniteAdeleRing (𝓞 F) F) ((i.adicCompletion F)) :=
fun i ↦ (RestrictedProduct.evalRingHom _ i).toAlgebra
IsModuleTopology.continuous_of_linearMap r.toLinearMap | theorem | IsDedekindDomain.HeightOneSpectrum.Rigidification.continuous_toFun | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
Rigidification.continuous_invFun (r : Rigidification F D) :
Continuous r.symm | by
haveI : ContinuousAdd (D ⊗[F] FiniteAdeleRing (𝓞 F) F) :=
IsModuleTopology.toContinuousAdd (FiniteAdeleRing (𝓞 F) F) (D ⊗[F] (FiniteAdeleRing (𝓞 F) F))
exact IsModuleTopology.continuous_of_linearMap r.symm.toLinearMap | theorem | IsDedekindDomain.HeightOneSpectrum.Rigidification.continuous_invFun | QuaternionAlgebra | FLT/QuaternionAlgebra/NumberField.lean | [
"FLT.Mathlib.Topology.Algebra.Valued.ValuationTopology",
"FLT.Mathlib.Topology.Instances.Matrix"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
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