{"id":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom'","label":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom'","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom"],"nlp_text":"The product of the maps \\(K_v\\to L_w\\) for \\(w|v\\) is a natural ring map \\(K_v\\to \\prod _{w|v}L_w\\) lying over \\(K\\to L\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom'","latex_link":"Adele_miniproject.html#IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom'","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom'","lean_decl_name":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom'","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/DedekindDomain/Completion/BaseChange.lean","lean4_code":"/-- The canonical map `K_v → ∏_{w|v} L_w` extending K → L. -/\nnoncomputable def adicCompletionComapSemialgHom' (v : HeightOneSpectrum A) :\n v.adicCompletion K →ₛₐ[algebraMap K L] ∀ w : v.Extension B, w.1.adicCompletion L :=\n Pi.semialgHom _ _ fun i ↦ adicCompletionComapSemialgHom A K L B v i.1 i.2","line":205,"end_line":208,"proof_status":"formalized"} {"id":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv","label":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom'","IsDedekindDomain.HeightOneSpectrum.Extension.finite"],"nlp_text":"The induced \\(L\\)-algebra homomorphism \\(L\\otimes _KK_v\\to \\prod _{w|v}L_w\\) is an isomorphism of rings.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv","latex_link":"Adele_miniproject.html#IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv","lean_decl_name":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/DedekindDomain/Completion/BaseChange.lean","lean4_code":"/-- The L-algebra isomorphism `L ⊗[K] K_v ≅ ∏_{w|v} L_w`. -/\nnoncomputable def adicCompletionComapAlgEquiv (v : HeightOneSpectrum A) :\n L ⊗[K] v.adicCompletion K ≃ₐ[L] (∀ w : v.Extension B, w.1.adicCompletion L) :=\n AlgEquiv.ofBijective (tensorAdicCompletionComapAlgHom A K L B v) <|\n tensorAdicCompletionComapAlgHom_bijective A K L B v\n","line":889,"end_line":893,"proof_status":"statement_formalized"} {"id":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv_integral","label":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv_integral","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv"],"nlp_text":"The isomorphism \\(L\\otimes _KK_v\\to \\prod _{w|v}L_w\\) induces an isomorphism \\(B\\otimes _AA_v\\to \\prod _{w|v}B_w\\) for all \\(v\\) in the height one spectrum of \\(A\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv_integral","latex_link":"Adele_miniproject.html#IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv_integral","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv_integral","lean_decl_name":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv_integral","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/DedekindDomain/Completion/BaseChange.lean","lean4_code":"/-- The image of `B ⊗[A] 𝓞_v` in `∏_w L_w` is `∏_w 𝓞_w`. -/\ntheorem adicCompletionComapAlgEquiv_integral :\n AlgHom.range (((tensorAdicCompletionComapAlgHom A K L B v).restrictScalars B).comp\n (tensorAdicCompletionIntegersTo A K L B v)) =\n Subalgebra.pi Set.univ (fun _ ↦ adicCompletionIntegersSubalgebra _ _) := by\n have hlhs := tensorAdicCompletionIntegersToRange_eq_closureIntegers\n have hrhs := prodAdicCompletionsIntegers_eq_closureIntegers\n have hrange :\n SetLike.coe (algebraMap B ((w : Extension B v) → adicCompletion L w.1)).range =\n (tensorAdicCompletionComapAlgHom A K L B v) ''\n SetLike.coe (algebraMap B (L ⊗[K] adicCompletion K v)).range := by\n ext x\n simp [Algebra.algebraMap_eq_smul_one]\n have hrange' := adicCompletionComapAlgHom_map_closure_is_closed A K L B v\n rw [AlgHom.range_comp, ← SetLike.coe_set_eq, Subalgebra.coe_map, AlgHom.coe_restrictScalars',\n hlhs, hrhs, hrange, ← IsClosed.closure_eq hrange']\n exact closure_image_closure\n (tensorAdicCompletionComapLinearMap_isOpenQuotientMap A K L B v).continuous\n","line":594,"end_line":611,"proof_status":"formalized"} {"id":"IsDedekindDomain.FiniteAdeleRing.baseChangeAlgEquiv","label":"IsDedekindDomain.FiniteAdeleRing.baseChangeAlgEquiv","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv_integral","IsDedekindDomain.FiniteAdeleRing.baseChangeIntegralAlgEquiv","IsDedekindDomain.AKLB.tensorProduct_module_algEquiv"],"nlp_text":"This natural map \\(L\\otimes _K\\mathbb {A}_{A,K}^\\infty \\to \\mathbb {A}_{B,L}^\\infty \\) is an isomorphism.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.FiniteAdeleRing.baseChangeAlgEquiv","latex_link":"Adele_miniproject.html#IsDedekindDomain.FiniteAdeleRing.baseChangeAlgEquiv","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.FiniteAdeleRing.baseChangeAlgEquiv","lean_decl_name":"IsDedekindDomain.FiniteAdeleRing.baseChangeAlgEquiv","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/DedekindDomain/FiniteAdeleRing/BaseChange.lean","lean4_code":"/-- The `L`-algebra isomorphism `L ⊗_K 𝔸_K^∞ ≅ 𝔸_L^∞`. -/\ndef FiniteAdeleRing.baseChangeAlgEquiv :\n L ⊗[K] FiniteAdeleRing A K ≃ₐ[L] FiniteAdeleRing B L :=\n AlgEquiv.ofBijective\n (SemialgHom.baseChange_of_algebraMap <| FiniteAdeleRing.mapSemialgHom A K L B)\n (FiniteAdeleRing.baseChange_bijective A K L B)\n","line":282,"end_line":287,"proof_status":"formalized"} {"id":"IsDedekindDomain.FiniteAdeleRing.baseChangeContinuousAlgEquiv","label":"IsDedekindDomain.FiniteAdeleRing.baseChangeContinuousAlgEquiv","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.adicCompletionComapAlgEquiv_integral"],"nlp_text":"The induced \\(L\\)-algebra morphism \\(L\\otimes _K\\mathbb {A}_{A,K}^\\infty \\to \\mathbb {A}_{B,L}^\\infty \\) is a topological isomorphism.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.FiniteAdeleRing.baseChangeContinuousAlgEquiv","latex_link":"Adele_miniproject.html#IsDedekindDomain.FiniteAdeleRing.baseChangeContinuousAlgEquiv","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.FiniteAdeleRing.baseChangeContinuousAlgEquiv","lean_decl_name":"IsDedekindDomain.FiniteAdeleRing.baseChangeContinuousAlgEquiv","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/DedekindDomain/FiniteAdeleRing/BaseChange.lean","lean4_code":"/-- The continuous `L`-algebra isomorphism `L ⊗_K 𝔸_K^∞ ≅ 𝔸_L^∞` -/\nnoncomputable def FiniteAdeleRing.baseChangeContinuousAlgEquiv :\n L ⊗[K] FiniteAdeleRing A K ≃A[L] FiniteAdeleRing B L where\n __ := FiniteAdeleRing.baseChangeAlgEquiv A K L B\n __ := FiniteAdeleRing.baseChangeAdeleContinuousAlgEquiv A K L B\n","line":407,"end_line":411,"proof_status":"formalized"} {"id":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom.mapadicCompletionIntegers","label":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom.mapadicCompletionIntegers","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.valued_adicCompletionComap"],"nlp_text":"The map \\(i_v:K_v\\to L_w\\) sends the integer ring \\(A_v\\) into \\(B_w\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom.mapadicCompletionIntegers","latex_link":"Adele_miniproject.html#IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom.mapadicCompletionIntegers","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom.mapadicCompletionIntegers","lean_decl_name":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom.mapadicCompletionIntegers","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/DedekindDomain/Completion/BaseChange.lean","lean4_code":"/-- The canonical map K_v → L_w sends 𝓞_v to 𝓞_w. -/\nlemma adicCompletionComapSemialgHom.mapadicCompletionIntegers (v : HeightOneSpectrum A)\n (w : HeightOneSpectrum B) (hvw : w.comap A = v) :\n (adicCompletionComapSemialgHom A K L B v w hvw) '' (v.adicCompletionIntegers K) ≤\n w.adicCompletionIntegers L := by\n rintro y ⟨x, hx, rfl⟩\n rw [SetLike.mem_coe, mem_adicCompletionIntegers] at hx ⊢\n rw [valued_adicCompletionComap A K L B v w hvw]\n rwa [pow_le_one_iff]\n exact ramificationIdx_ne_zero A B (algebraMap_injective_of_field_isFractionRing A B K L) w","line":188,"end_line":197,"proof_status":"formalized"} {"id":"IsDedekindDomain.FiniteAdeleRing.mapSemialgHom","label":"IsDedekindDomain.FiniteAdeleRing.mapSemialgHom","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom.mapadicCompletionIntegers"],"nlp_text":"There’s a natural ring homomorphism \\(\\mathbb {A}_{A,K}^\\infty \\to \\mathbb {A}_{B,L}^\\infty \\) lying over \\(K\\to L\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.FiniteAdeleRing.mapSemialgHom","latex_link":"Adele_miniproject.html#IsDedekindDomain.FiniteAdeleRing.mapSemialgHom","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.FiniteAdeleRing.mapSemialgHom","lean_decl_name":"IsDedekindDomain.FiniteAdeleRing.mapSemialgHom","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/Mathlib/Topology/Algebra/UniformRing.lean","lean4_code":"noncomputable def mapSemialgHom {α : Type*} [CommRing α] [UniformSpace α]\n [IsTopologicalRing α] [IsUniformAddGroup α] {β : Type*} [UniformSpace β] [CommRing β]\n [IsUniformAddGroup β] [IsTopologicalRing β] (f : α →+* β) (hf : Continuous f) :\n Completion α →ₛₐ[f] Completion β where\n __ := UniformSpace.Completion.mapRingHom f hf\n map_smul' m x := by\n simp only [RingHom.toMonoidHom_eq_coe, OneHom.toFun_eq_coe, MonoidHom.toOneHom_coe,\n MonoidHom.coe_coe]\n rw [Algebra.smul_def, map_mul, Algebra.smul_def]\n congr\n exact extensionHom_coe _ _ m\n","line":16,"end_line":26,"proof_status":"formalized"} {"id":"NumberField.instCompactSpaceAdicCompletionIntegers","label":"NumberField.instCompactSpaceAdicCompletionIntegers","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"If \\(K\\) is a number field and \\(v\\) is a nonzero prime ideal of the integers of \\(K\\), then the integers of \\(K_v\\) is a compact open subgroup.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.instCompactSpaceAdicCompletionIntegers","latex_link":"Adele_miniproject.html#NumberField.instCompactSpaceAdicCompletionIntegers","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.instCompactSpaceAdicCompletionIntegers","lean_decl_name":"NumberField.instCompactSpaceAdicCompletionIntegers","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/NumberField/Completion/Finite.lean","lean4_code":"instance NumberField.instCompactSpaceAdicCompletionIntegers :\n CompactSpace (v.adicCompletionIntegers K) :=\n Valued.WithZeroMulInt.integer_compactSpace (v.adicCompletion K) inferInstance","line":44,"end_line":46,"proof_status":"statement_formalized"} {"id":"NumberField.AdeleRing.baseChangeEquiv","label":"NumberField.AdeleRing.baseChangeEquiv","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.FiniteAdeleRing.baseChangeAlgEquiv","NumberField.InfiniteAdeleRing.baseChangeEquiv"],"nlp_text":"If \\(K\\to L\\) is a ring homomorphism between two number fields then there is a natural isomorphism (both topological and algebraic) \\(L\\otimes _K\\mathbb {A}_K\\cong \\mathbb {A}_L\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.baseChangeEquiv","latex_link":"Adele_miniproject.html#NumberField.AdeleRing.baseChangeEquiv","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.baseChangeEquiv","lean_decl_name":"NumberField.AdeleRing.baseChangeEquiv","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/NumberField/InfiniteAdeleRing.lean","lean4_code":"/-- The canonical `L`-algebra homeomorphism from `L ⊗_K K_∞` to `L_∞` induced by the\n`K`-algebra base change map `K_∞ → L_∞`. -/\nnoncomputable\ndef baseChangeEquiv :\n L ⊗[K] InfiniteAdeleRing K ≃A[L] InfiniteAdeleRing L :=\n /- Because both `L ⊗[K] K_∞` and `L_∞` have the `K_∞` module topology, we obtain a _continuous_\n `L`-algebra equivalence, using the following set up of algebras\n ```\n K_∞ L\n \\ /\n \\ /\n K\n ```\n -/\n IsModuleTopology.continuousAlgEquivOfIsScalarTower K _ (baseChangeEquivAux K L)\n (by simp_rw [baseChangeEquivAux_apply]; exact SemialgHom.baseChange_of_algebraMap_tmul_right _)\n","line":111,"end_line":126,"proof_status":"formalized"} {"id":"NumberField.AdeleRing.cocompact","label":"NumberField.AdeleRing.cocompact","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.AdeleRing.baseChangeEquiv","Rat.AdeleRing.cocompact"],"nlp_text":"The quotient \\(\\mathbb {A}_K/K\\) is compact.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.cocompact","latex_link":"Adele_miniproject.html#NumberField.AdeleRing.cocompact","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.cocompact","lean_decl_name":"NumberField.AdeleRing.cocompact","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Topology/Defs/Filter.lean","lean4_code":"def Filter.cocompact : Filter X :=\n ⨅ (s : Set X) (_ : IsCompact s), 𝓟 sᶜ\n\nvariable (X) in\n/-- `Filter.coclosedCompact` is the filter generated by complements to closed compact sets.\nIn a Hausdorff space, this is the same as `Filter.cocompact`. -/\n","line":323,"end_line":328,"proof_status":"statement_formalized"} {"id":"NumberField.AdeleRing.baseChange_moduleTopology","label":"NumberField.AdeleRing.baseChange_moduleTopology","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.AdeleRing.baseChangeEquiv"],"nlp_text":"If \\(K\\to L\\) is a ring homomorphism between two number fields then the topology on \\(\\mathbb {A}_L\\) is the \\(\\mathbb {A}_K\\)-module topology, where the module structure comes from the natural map \\(\\mathbb {A}_K\\to \\mathbb {A}_L\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.instBaseChangeIsModuleTopology","latex_link":"Adele_miniproject.html#NumberField.AdeleRing.baseChange_moduleTopology","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.instBaseChangeIsModuleTopology","lean_decl_name":"NumberField.AdeleRing.instBaseChangeIsModuleTopology","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/NumberField/AdeleRing.lean","lean4_code":"instance instBaseChangeIsModuleTopology : IsModuleTopology (𝔸 K) (𝔸 L) := by\n exact IsModuleTopology.instProd' (A := InfiniteAdeleRing K)\n (B := FiniteAdeleRing (𝓞 K) K) (M := InfiniteAdeleRing L) (N := FiniteAdeleRing (𝓞 L) L)","line":67,"end_line":69,"proof_status":"formalized"} {"id":"NumberField.AdeleRing.zero_discrete","label":"NumberField.AdeleRing.zero_discrete","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.AdeleRing.baseChangeEquiv","Rat.AdeleRing.zero_discrete"],"nlp_text":"There’s an open subset of \\(\\mathbb {A}_{K}\\) whose intersection with \\(K\\) is \\(\\{ 0\\} \\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.zero_discrete","latex_link":"Adele_miniproject.html#NumberField.AdeleRing.zero_discrete","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.zero_discrete","lean_decl_name":"NumberField.AdeleRing.zero_discrete","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/NumberField/AdeleRing.lean","lean4_code":"theorem NumberField.AdeleRing.zero_discrete : ∃ U : Set (AdeleRing (𝓞 K) K),\n IsOpen U ∧ (algebraMap K (AdeleRing (𝓞 K) K)) ⁻¹' U = {0} := by\n obtain ⟨V, hV, hV0⟩ := Rat.AdeleRing.zero_discrete\n use (piEquiv ℚ K) '' {f | ∀i, f i ∈ V }\n constructor\n · rw [← (piEquiv ℚ K).coe_toHomeomorph, Homeomorph.isOpen_image, Set.setOf_forall]\n apply isOpen_iInter_of_finite\n intro i\n exact hV.preimage (continuous_apply i)\n rw [Set.eq_singleton_iff_unique_mem]\n constructor\n · rw [Set.eq_singleton_iff_unique_mem, Set.mem_preimage, map_zero] at hV0\n simp only [Set.mem_preimage, map_zero, Set.mem_image,\n EmbeddingLike.map_eq_zero_iff, exists_eq_right]\n exact fun _ => hV0.left\n intro x ⟨y, hy, hyx⟩\n apply (Module.Finite.equivPi ℚ K).injective\n set f := Module.Finite.equivPi ℚ K x\n let g := fun i => algebraMap ℚ (AdeleRing (𝓞 ℚ) ℚ) (f i)\n have hfg : ∀ i, (algebraMap _ _) (f i) = g i := fun i => rfl\n have hg := piEquiv_apply_of_algebraMap hfg\n simp only [LinearEquiv.symm_apply_apply, f, ← hyx, EquivLike.apply_eq_iff_eq] at hg\n subst hg\n ext i\n rw [map_zero, Pi.zero_apply, ← Set.mem_singleton_iff, ← hV0, Set.mem_preimage]\n exact hy i","line":411,"end_line":436,"proof_status":"statement_formalized"} {"id":"NumberField.AdeleRing.discrete","label":"NumberField.AdeleRing.discrete","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.AdeleRing.zero_discrete"],"nlp_text":"The additive subgroup \\(K\\) of \\(\\mathbb {A}_K\\) is discrete.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.discrete","latex_link":"Adele_miniproject.html#NumberField.AdeleRing.discrete","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.discrete","lean_decl_name":"NumberField.AdeleRing.discrete","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Topology/Category/TopCat/Basic.lean","lean4_code":"def discrete : Type u ⥤ TopCat.{u} where\n obj X := @of X ⊥\n map f := @ofHom _ _ ⊥ ⊥ <| @ContinuousMap.mk _ _ ⊥ ⊥ f continuous_bot\n\n","line":164,"end_line":167,"proof_status":"statement_formalized"} {"id":"a0000000185","label":"a0000000185","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"\\(\\mathbb {\\mathbb {A}_L^\\infty }\\) is homeomorphic to \\(\\prod _v(B\\otimes _AK_v,B\\otimes _AA_v)\\).","doc_url":null,"latex_link":"Adele_miniproject.html#a0000000185","lean_link":null,"lean_decl_name":"a0000000185","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} {"id":"NumberField.InfinitePlace.Completion.piExtension","label":"NumberField.InfinitePlace.Completion.piExtension","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"Let \\(v\\) be an infinite place of \\(K\\). There is a continuous \\(K\\)-algebra homomorphism \\(K_v \\to \\prod _{w\\mid v}L_w\\), whose restriction to \\(K\\) corresponds to the global embedding of \\(K\\) into \\((L_w)_w\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.piExtension","latex_link":"Adele_miniproject.html#NumberField.InfinitePlace.Completion.piExtension","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.piExtension","lean_decl_name":"NumberField.InfinitePlace.Completion.piExtension","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/NumberField/Completion/Infinite.lean","lean4_code":"/-- The map from `v.Completion` to the product of all completions of `L` lying above `v`. -/\ndef piExtension :\n v.Completion →ₛₐ[algebraMap K L] (wv : v.Extension L) → wv.1.Completion :=\n Pi.semialgHom _ _ fun wv => comapHom wv.2\n","line":49,"end_line":52,"proof_status":"formalized"} {"id":"NumberField.InfinitePlace.Completion.baseChange","label":"NumberField.InfinitePlace.Completion.baseChange","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.InfinitePlace.Completion.piExtension"],"nlp_text":"Let \\(v\\) be an infinite place of \\(K\\). There is a natural \\(L\\)-algebra homomorphism \\(L\\otimes _K K_v \\to \\prod _{w\\mid v}L_w\\), whose restriction to \\(1\\otimes _K K_v\\) corresponds to the map in 9.36.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.baseChange","latex_link":"Adele_miniproject.html#NumberField.InfinitePlace.Completion.baseChange","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.baseChange","lean_decl_name":"NumberField.InfinitePlace.Completion.baseChange","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/Deformations/RepresentationTheory/GaloisRep.lean","lean4_code":"/-- Make a `A`-linear galois rep on `M` into a `B`-linear rep on `B ⊗ M`. -/\nnoncomputable\ndef GaloisRep.baseChange [IsTopologicalRing B] [Algebra A B] [ContinuousSMul A B]\n [Module.Finite A M] [Module.Free A M]\n (ρ : GaloisRep K A M) : GaloisRep K B (B ⊗[A] M) :=\n letI := moduleTopology A (Module.End A M)\n letI := moduleTopology B (Module.End B (B ⊗[A] M))\n letI : ContinuousMul _ := ⟨IsModuleTopology.continuous_mul_of_finite B (Module.End B (B ⊗[A] M))⟩\n letI := IsModuleTopology.toContinuousAdd B (Module.End B (B ⊗[A] M))\n let F : Module.End A M →+* Module.End B (B ⊗[A] M) := Module.End.baseChangeHom A B M\n have : Continuous F := by\n have : IsTopologicalSemiring (Module.End B (B ⊗[A] M)) := ⟨⟩\n have : Continuous (algebraMap A (Module.End B (B ⊗[A] M))) := by\n rw [IsScalarTower.algebraMap_eq A B, RingHom.coe_comp]\n exact (continuous_algebraMap _ _).comp (continuous_algebraMap _ _)\n exact IsModuleTopology.continuous_of_ringHom (R := A) F (by simpa [F])\n .comp ⟨F, this⟩ ρ\n","line":189,"end_line":205,"proof_status":"formalized"} {"id":"NumberField.InfinitePlace.Completion.baseChangeEquiv","label":"NumberField.InfinitePlace.Completion.baseChangeEquiv","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.InfinitePlace.Completion.baseChange","NumberField.InfinitePlace.Completion.baseChange_surjective","NumberField.InfinitePlace.Completion.baseChange_injective","NumberField.InfinitePlace.Completion.instIsModuleTopologyValEqComapAlgebraMap_fLT"],"nlp_text":"Let \\(v\\) be an infinite place of \\(K\\). There is a natural \\(L\\)-algebra homeomorphism \\(L\\otimes _K K_v \\cong _L \\prod _{w\\mid v}L_w\\), whose restriction to \\(1\\otimes _K K_v\\) corresponds to the map in 9.36.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.baseChangeEquiv","latex_link":"Adele_miniproject.html#NumberField.InfinitePlace.Completion.baseChangeEquiv","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.baseChangeEquiv","lean_decl_name":"NumberField.InfinitePlace.Completion.baseChangeEquiv","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/NumberField/InfiniteAdeleRing.lean","lean4_code":"/-- The canonical `L`-algebra homeomorphism from `L ⊗_K K_∞` to `L_∞` induced by the\n`K`-algebra base change map `K_∞ → L_∞`. -/\nnoncomputable\ndef baseChangeEquiv :\n L ⊗[K] InfiniteAdeleRing K ≃A[L] InfiniteAdeleRing L :=\n /- Because both `L ⊗[K] K_∞` and `L_∞` have the `K_∞` module topology, we obtain a _continuous_\n `L`-algebra equivalence, using the following set up of algebras\n ```\n K_∞ L\n \\ /\n \\ /\n K\n ```\n -/\n IsModuleTopology.continuousAlgEquivOfIsScalarTower K _ (baseChangeEquivAux K L)\n (by simp_rw [baseChangeEquivAux_apply]; exact SemialgHom.baseChange_of_algebraMap_tmul_right _)\n","line":111,"end_line":126,"proof_status":"statement_formalized"} {"id":"NumberField.InfiniteAdeleRing.baseChangeEquivAux","label":"NumberField.InfiniteAdeleRing.baseChangeEquivAux","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.InfinitePlace.Completion.baseChangeEquiv"],"nlp_text":"There is a natural \\(L\\)-algebra isomorphism \\(L \\otimes _K K_{\\infty } \\cong _L L_{\\infty }\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfiniteAdeleRing.baseChangeEquivAux","latex_link":"Adele_miniproject.html#NumberField.InfiniteAdeleRing.baseChangeEquivAux","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfiniteAdeleRing.baseChangeEquivAux","lean_decl_name":"NumberField.InfiniteAdeleRing.baseChangeEquivAux","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/NumberField/InfiniteAdeleRing.lean","lean4_code":"/-- The $L$-algebra isomorphism $L\\otimes_K K_{\\infty} \\cong L_{\\infty}$. -/\nnoncomputable def baseChangeEquivAux :\n L ⊗[K] InfiniteAdeleRing K ≃ₐ[L] InfiniteAdeleRing L :=\n -- L ⊗ K_∞ ≃[K_∞] ∏ v, L ⊗ K_v\n Algebra.TensorProduct.piRight K L L Completion |>.trans\n -- lift the established equivalence L ⊗ K_v ≃[v.Completion] ∏ w ∣ v, L_w on fibers of comap\n (AlgEquiv.piCongrFiberwise\n (fun v : InfinitePlace K => (Completion.baseChangeEquiv L v).toAlgEquiv.symm)).symm\n","line":67,"end_line":74,"proof_status":"statement_formalized"} {"id":"NumberField.InfinitePlace.Completion.piEquiv","label":"NumberField.InfinitePlace.Completion.piEquiv","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.InfinitePlace.Completion.baseChangeEquiv"],"nlp_text":"Let \\(v\\) be an infinite place of \\(K\\). There is a natural \\(K_v\\)-linear homeomorphism \\(K_v^{[L:K]} \\cong _{K_v} \\prod _{w\\mid v}L_w\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.piEquiv","latex_link":"Adele_miniproject.html#NumberField.InfinitePlace.Completion.piEquiv","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.piEquiv","lean_decl_name":"NumberField.InfinitePlace.Completion.piEquiv","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/NumberField/InfiniteAdeleRing.lean","lean4_code":"/-- The $K_{\\infty}$-linear homeomorphism $K_{\\infty}^{[L:K]} \\cong L_{\\infty}$. -/\nnoncomputable\ndef piEquiv : let d := Module.finrank K L\n (Fin d → InfiniteAdeleRing K) ≃L[InfiniteAdeleRing K] InfiniteAdeleRing L := by\n -- I think we could remove convert if we make `InfiniteAdeleRing` an `abbrev`\n -- (K_∞)^d ≃[K_∞] ∏ v, K_v^d\n convert (ContinuousLinearEquiv.piScalarPiComm _ _).symm.trans\n -- lift the equivalence K_v^d ≃[v.Completion] ∏ w ∣ v, L_w on fibers of comap\n (ContinuousLinearEquiv.piScalarPiCongrFiberwise\n (fun v : InfinitePlace K => (Completion.piEquiv L v).symm)).symm\n","line":48,"end_line":57,"proof_status":"statement_formalized"} {"id":"NumberField.AdeleRing.locallyCompactSpace","label":"NumberField.AdeleRing.locallyCompactSpace","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"The adeles of a number field are locally compact.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.locallyCompactSpace","latex_link":"Adele_miniproject.html#NumberField.AdeleRing.locallyCompactSpace","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.AdeleRing.locallyCompactSpace","lean_decl_name":"NumberField.AdeleRing.locallyCompactSpace","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Analysis/Normed/Field/WithAbs.lean","lean4_code":"theorem locallyCompactSpace [LocallyCompactSpace L] (h : ∀ x, ‖f x‖ = v x) :\n LocallyCompactSpace (v.Completion) :=\n (isClosedEmbedding_extensionEmbedding_of_comp h).locallyCompactSpace\n\n","line":118,"end_line":121,"proof_status":"statement_formalized"} {"id":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComap_isModuleTopology","label":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComap_isModuleTopology","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom"],"nlp_text":"Giving \\(L_w\\) the \\(K_v\\)-algebra structure coming from the natural map \\(K_v\\to L_w\\), the \\(w\\)-adic topology on \\(L_w\\) is the \\(K_v\\)-module topology.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComap_isModuleTopology","latex_link":"Adele_miniproject.html#IsDedekindDomain.HeightOneSpectrum.adicCompletionComap_isModuleTopology","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComap_isModuleTopology","lean_decl_name":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComap_isModuleTopology","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/DedekindDomain/Completion/BaseChange.lean","lean4_code":"/-- L_w has the K_v-module topology. -/\nlemma adicCompletionComap_isModuleTopology\n (v : HeightOneSpectrum A) (w : HeightOneSpectrum B) (hvw : w.comap A = v) :\n -- temporarily make L_w a K_v-algebra\n letI := comap_algebra A K L B hvw\n -- then claim that L_w has the module topology.\n IsModuleTopology (v.adicCompletion K) (w.adicCompletion L) := by\n let Kv := adicCompletion K v\n let Lw := adicCompletion L w\n let _ : Algebra Kv Lw := comap_algebra A K L B hvw\n have : ContinuousSMul Kv Lw := comap_algebra_continuousSmul A K L B v w hvw\n have : Module.Finite Kv Lw := comap_algebra_finite A K L B v w hvw\n let iso : ((Fin (Module.finrank Kv Lw)) → Kv) ≃L[Kv] Lw :=\n ContinuousLinearEquiv.ofFinrankEq (Module.finrank_fin_fun Kv)\n apply IsModuleTopology.iso iso\n","line":334,"end_line":348,"proof_status":"statement_formalized"} {"id":"IsDedekindDomain.HeightOneSpectrum.prodAdicCompletionComap_isModuleTopology","label":"IsDedekindDomain.HeightOneSpectrum.prodAdicCompletionComap_isModuleTopology","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.adicCompletionComap_isModuleTopology"],"nlp_text":"For \\(v\\) fixed, the product topology on \\(\\prod _{w|v}L_w\\) is the \\(K_v\\)-module topology.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.prodAdicCompletionComap_isModuleTopology","latex_link":"Adele_miniproject.html#IsDedekindDomain.HeightOneSpectrum.prodAdicCompletionComap_isModuleTopology","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.prodAdicCompletionComap_isModuleTopology","lean_decl_name":"IsDedekindDomain.HeightOneSpectrum.prodAdicCompletionComap_isModuleTopology","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/DedekindDomain/Completion/BaseChange.lean","lean4_code":"/-- ∏_{w|v} L_w has the K_v-module topology. -/\nlemma prodAdicCompletionComap_isModuleTopology (v : HeightOneSpectrum A) :\n -- TODO: the `let _` in the statement below should not be required as it is an instance\n -- see mathlib PR #22488 for potential fix to this.\n -- Note that this one does not involve `adicCompletionIntegers` so the\n -- issue may not be to do with subtype vs. type implementation of\n -- `adicCompletionIntegers`.\n let _ := comap_pi_algebra A K L B v |>.toSMul\n -- the claim that L_w has the module topology.\n IsModuleTopology (v.adicCompletion K) (Π (w : v.Extension B), w.1.adicCompletion L) := by\n -- these are defs or lemmas so are required\n let _ (w : v.Extension B) := comap_algebra A K L B w.2 |>.toModule\n let _ (w : v.Extension B) := adicCompletionComap_isModuleTopology A K L B v w.1 w.2\n let _ := Extension.finite A K L B v\n exact IsModuleTopology.instPi\n","line":351,"end_line":365,"proof_status":"formalized"} {"id":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapContinuousAlgEquiv","label":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapContinuousAlgEquiv","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.prodAdicCompletionComap_isModuleTopology"],"nlp_text":"If we give \\(L\\otimes _KK_v\\) the \\(K_v\\)-module topology then the \\(L\\)-algebra isomorphism \\(L\\otimes _K K_v\\cong \\prod _{w|v}L_w\\) is also a homeomorphism.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapContinuousAlgEquiv","latex_link":"Adele_miniproject.html#IsDedekindDomain.HeightOneSpectrum.adicCompletionComapContinuousAlgEquiv","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapContinuousAlgEquiv","lean_decl_name":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapContinuousAlgEquiv","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/DedekindDomain/Completion/BaseChange.lean","lean4_code":"/-- The continuous L-algebra isomorphism `L ⊗[K] K_v ≅ ∏_{w|v} L_w`. -/\nnoncomputable def adicCompletionComapContinuousAlgEquiv (v : HeightOneSpectrum A) :\n L ⊗[K] v.adicCompletion K ≃A[L] ∀ w : v.Extension B, w.1.adicCompletion L :=\n {\n __ := adicCompletionComapAlgEquiv A K L B v\n __ := adicCompletionComapRightContinuousAlgEquiv A K L B v\n }\n","line":908,"end_line":914,"proof_status":"statement_formalized"} {"id":"NumberField.InfiniteAdeleRing.piEquiv","label":"NumberField.InfiniteAdeleRing.piEquiv","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.InfinitePlace.Completion.piEquiv"],"nlp_text":"There is a natural \\(K_{\\infty }\\)-linear homeomorphism \\(K_{\\infty }^{[L:K]} \\cong _{K_{\\infty }} L_{\\infty }\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfiniteAdeleRing.piEquiv","latex_link":"Adele_miniproject.html#NumberField.InfiniteAdeleRing.piEquiv","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfiniteAdeleRing.piEquiv","lean_decl_name":"NumberField.InfiniteAdeleRing.piEquiv","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/NumberField/InfiniteAdeleRing.lean","lean4_code":"/-- The $K_{\\infty}$-linear homeomorphism $K_{\\infty}^{[L:K]} \\cong L_{\\infty}$. -/\nnoncomputable\ndef piEquiv : let d := Module.finrank K L\n (Fin d → InfiniteAdeleRing K) ≃L[InfiniteAdeleRing K] InfiniteAdeleRing L := by\n -- I think we could remove convert if we make `InfiniteAdeleRing` an `abbrev`\n -- (K_∞)^d ≃[K_∞] ∏ v, K_v^d\n convert (ContinuousLinearEquiv.piScalarPiComm _ _).symm.trans\n -- lift the equivalence K_v^d ≃[v.Completion] ∏ w ∣ v, L_w on fibers of comap\n (ContinuousLinearEquiv.piScalarPiCongrFiberwise\n (fun v : InfinitePlace K => (Completion.piEquiv L v).symm)).symm\n","line":48,"end_line":57,"proof_status":"statement_formalized"} {"id":"NumberField.InfiniteAdeleRing.instIsModuleTopology_fLT","label":"NumberField.InfiniteAdeleRing.instIsModuleTopology_fLT","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.InfiniteAdeleRing.piEquiv"],"nlp_text":"\\(L_{\\infty }\\) has the \\(K_{\\infty }\\)-module topology.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfiniteAdeleRing.instIsModuleTopology_fLT","latex_link":"Adele_miniproject.html#NumberField.InfiniteAdeleRing.instIsModuleTopology_fLT","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfiniteAdeleRing.instIsModuleTopology_fLT","lean_decl_name":"NumberField.InfiniteAdeleRing.instIsModuleTopology_fLT","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/NumberField/InfiniteAdeleRing.lean","lean4_code":"instance : IsModuleTopology (InfiniteAdeleRing K) (InfiniteAdeleRing L) := by\n exact IsModuleTopology.iso (piEquiv K L)","line":59,"end_line":60,"proof_status":"statement_formalized"} {"id":"NumberField.InfiniteAdeleRing.baseChangeEquiv","label":"NumberField.InfiniteAdeleRing.baseChangeEquiv","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.InfiniteAdeleRing.baseChangeEquivAux"],"nlp_text":"If \\(K\\to L\\) is a ring homomorphism between two number fields then there is a natural isomorphism (both topological and algebraic) \\(L\\otimes _KK_\\infty \\cong L_\\infty \\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfiniteAdeleRing.baseChangeEquiv","latex_link":"Adele_miniproject.html#NumberField.InfiniteAdeleRing.baseChangeEquiv","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfiniteAdeleRing.baseChangeEquiv","lean_decl_name":"NumberField.InfiniteAdeleRing.baseChangeEquiv","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/NumberField/InfiniteAdeleRing.lean","lean4_code":"/-- The canonical `L`-algebra homeomorphism from `L ⊗_K K_∞` to `L_∞` induced by the\n`K`-algebra base change map `K_∞ → L_∞`. -/\nnoncomputable\ndef baseChangeEquiv :\n L ⊗[K] InfiniteAdeleRing K ≃A[L] InfiniteAdeleRing L :=\n /- Because both `L ⊗[K] K_∞` and `L_∞` have the `K_∞` module topology, we obtain a _continuous_\n `L`-algebra equivalence, using the following set up of algebras\n ```\n K_∞ L\n \\ /\n \\ /\n K\n ```\n -/\n IsModuleTopology.continuousAlgEquivOfIsScalarTower K _ (baseChangeEquivAux K L)\n (by simp_rw [baseChangeEquivAux_apply]; exact SemialgHom.baseChange_of_algebraMap_tmul_right _)\n","line":111,"end_line":126,"proof_status":"statement_formalized"} {"id":"IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux1","label":"IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux1","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.pi_tensorProduct"],"nlp_text":"The natural map \\(B\\otimes _A\\mathbb {A}_K^\\infty \\to R\\) is a \\(B\\)-algebra isomorphism.","doc_url":null,"latex_link":"Adele_miniproject.html#IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux1","lean_link":null,"lean_decl_name":"IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux1","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} {"id":"IsDedekindDomain.FiniteAdeleRing.baseChangeIntegralAlgEquiv","label":"IsDedekindDomain.FiniteAdeleRing.baseChangeIntegralAlgEquiv","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux1","IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux2"],"nlp_text":"The natural map \\(B\\otimes _A\\mathbb {A}_K^\\infty \\to \\mathbb {A}_L^\\infty \\) is a \\(B\\)-algebra isomorphism.","doc_url":null,"latex_link":"Adele_miniproject.html#IsDedekindDomain.FiniteAdeleRing.baseChangeIntegralAlgEquiv","lean_link":null,"lean_decl_name":"IsDedekindDomain.FiniteAdeleRing.baseChangeIntegralAlgEquiv","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} {"id":"IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux2","label":"IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux2","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["RestrictedProduct.relabelIso"],"nlp_text":"The ring \\(R\\) introduced above (the restricted product of the \\(B\\otimes _A K_v\\) with respect to the \\(B\\otimes _A A_v\\)) is isomorphic to \\(\\mathbb {A}_L\\).","doc_url":null,"latex_link":"Adele_miniproject.html#IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux2","lean_link":null,"lean_decl_name":"IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux2","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} {"id":"a0000000191","label":"a0000000191","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"If \\(X_v\\) and \\(Y_v\\) are topological spaces with open subspaces \\(C_v\\) and \\(D_v\\), then the obvious bijection \\(\\prod '_v(X_v \\times Y_v,C_v\\times D_v) \\cong \\left(\\prod '_v(X_v,C_v)\\right)\\times \\left(\\prod '_v(Y_v,D_v)\\right)\\) is a homeomorphism, where the restricted products have the restricted product topology and the binary product has the product topology.","doc_url":null,"latex_link":"Adele_miniproject.html#a0000000191","lean_link":null,"lean_decl_name":"a0000000191","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} {"id":"pi_tensorProduct_of_finitePresentation","label":"pi_tensorProduct_of_finitePresentation","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"If \\(R\\) is a commutative ring, if \\(M\\) is a finitely presented \\(R\\)-module and if \\(N_i\\) are a collection of \\(R\\)-modules, then the canonical map \\(M\\otimes _R\\prod _i N_i\\to \\prod _i(M\\otimes _R N_i)\\) is an isomorphism.","doc_url":null,"latex_link":"Adele_miniproject.html#pi_tensorProduct_of_finitePresentation","lean_link":null,"lean_decl_name":"pi_tensorProduct_of_finitePresentation","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} {"id":"IsDedekindDomain.pi_tensorProduct","label":"IsDedekindDomain.pi_tensorProduct","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["pi_tensorProduct_of_finitePresentation","IsDedekindDomain.AKLB.finitePresentation"],"nlp_text":"If \\(S\\) is a finite set of nonzero primes of \\(A\\) then the natural map \\(B\\otimes ((\\prod _{v\\in S}K_v)\\times (\\prod _{v\\notin S}A_v))\\to (\\prod _{v\\in S}(B\\otimes _AK_v))\\times (\\prod _{v\\notin S}(B\\otimes _AA_v))\\) is an isomorphism.","doc_url":null,"latex_link":"Adele_miniproject.html#IsDedekindDomain.pi_tensorProduct","lean_link":null,"lean_decl_name":"IsDedekindDomain.pi_tensorProduct","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} {"id":"NumberField.InfinitePlace.Completion.denseRange_algebraMap_subtype_pi","label":"NumberField.InfinitePlace.Completion.denseRange_algebraMap_subtype_pi","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"Let \\(S\\) be a set of infinite places of \\(K\\). The image of \\(K\\) under the embedding \\(K\\hookrightarrow (K_v)_{v\\in S}; k \\mapsto (k)_v\\) is dense in the product topology.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.denseRange_algebraMap_subtype_pi","latex_link":"Adele_miniproject.html#NumberField.InfinitePlace.Completion.denseRange_algebraMap_subtype_pi","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.denseRange_algebraMap_subtype_pi","lean_decl_name":"NumberField.InfinitePlace.Completion.denseRange_algebraMap_subtype_pi","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/NumberField/InfinitePlace/WeakApproximation.lean","lean4_code":"/--\n*Weak approximation for subcollections*: this is the result that `K` is dense in any subcollection\n`Π v ∈ S, Kᵥ` of completions at infinite places.\n-/\ntheorem denseRange_algebraMap_subtype_pi (p : InfinitePlace K → Prop) [NumberField K] :\n DenseRange <| algebraMap K ((v : Subtype p) → v.1.Completion) := by\n apply DenseRange.comp (g := Subtype.restrict p)\n (f := algebraMap K ((v : InfinitePlace K) → v.1.Completion))\n · exact Subtype.surjective_restrict (β := fun v => v.1.Completion) p |>.denseRange\n · exact InfiniteAdeleRing.denseRange_algebraMap K\n · exact continuous_pi (fun _ => continuous_apply _)\n","line":298,"end_line":308,"proof_status":"statement_formalized"} {"id":"NumberField.InfinitePlace.Completion.baseChange_surjective","label":"NumberField.InfinitePlace.Completion.baseChange_surjective","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.InfinitePlace.Completion.denseRange_algebraMap_subtype_pi"],"nlp_text":"For a fixed infinite place \\(v\\) of \\(K\\), the map \\(L\\otimes _K K_v \\to \\prod _{w\\mid v}L_w\\) is surjective.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.baseChange_surjective","latex_link":"Adele_miniproject.html#NumberField.InfinitePlace.Completion.baseChange_surjective","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.baseChange_surjective","lean_decl_name":"NumberField.InfinitePlace.Completion.baseChange_surjective","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/NumberField/Completion/Infinite.lean","lean4_code":"theorem baseChange_surjective : Function.Surjective (baseChange L v) := by\n -- Let `Bw` be a `K_v` basis of `Π v | w, L_w`\n let Bw := Module.finBasis v.Completion ((w : v.Extension L) → w.1.Completion)\n -- `L` is dense inside Π v | w, L_w\n have := denseRange_algebraMap_subtype_pi _ fun w : InfinitePlace L => w.comap (algebraMap K L) = v\n -- So there exists a vector `α ∈ L^d` whose matrix wrt `Bw` gets close to `1` (has non-zero det)\n classical\n let ⟨α, h⟩ := (DenseRange.piMap fun _ => this).exists_matrix_det_ne_zero\n (Basis.toMatrix_continuous Bw) Bw.toMatrix_self\n -- Therefore `α` is a basis under the image of `piExtension L v`, hence it's surjective\n rw [← isUnit_iff_ne_zero, ← Bw.det_apply, ← Module.Basis.is_basis_iff_det Bw] at h\n rw [← baseChangeRightOfAlgebraMap_coe, ← AlgHom.coe_toLinearMap, ← LinearMap.range_eq_top,\n ← top_le_iff, ← h.2, Submodule.span_le]\n rintro x ⟨i, rfl⟩\n exact ⟨α i ⊗ₜ 1, by simp⟩\n","line":106,"end_line":120,"proof_status":"statement_formalized"} {"id":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom","label":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.valuation_comap"],"nlp_text":"There’s a natural ring map \\(K_v\\to L_w\\) extending the map \\(K\\to L\\). It is defined by completing the inclusion \\(K\\to L\\) at the finite places \\(v\\) and \\(w\\) (which can be done because the previous lemma shows that the map is uniformly continuous for the \\(v\\)-adic and \\(w\\)-adic topologies).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom","latex_link":"Adele_miniproject.html#IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom","lean_decl_name":"IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/DedekindDomain/Completion/BaseChange.lean","lean4_code":"/-- If w of L divides v of K, `adicCompletionComapSemialgHom v w pf` is the canonical map\n`Kᵥ → L_w` lying above `K → L`. Here we actually use the type synonyms `WithVal K` and `WithVal L`.\n-/\nnoncomputable def adicCompletionComapSemialgHom (v : HeightOneSpectrum A) (w : HeightOneSpectrum B)\n (hvw : w.comap A = v) :\n v.adicCompletion K →ₛₐ[σ v w] w.adicCompletion L :=\n UniformSpace.Completion.mapSemialgHom _ <|\n adicValued.continuous_algebraMap A K L B v w hvw\n","line":148,"end_line":155,"proof_status":"formalized"} {"id":"IsDedekindDomain.HeightOneSpectrum.valued_adicCompletionComap","label":"IsDedekindDomain.HeightOneSpectrum.valued_adicCompletionComap","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.HeightOneSpectrum.adicCompletionComapSemialgHom"],"nlp_text":"If \\(i_v:K_v\\to L_w\\) denotes the map of the previous definition then for \\(x\\in K_v\\) we have \\(e\\times w(i(k))=v(k)\\), where \\(e\\) is the ramification index of \\(w/v\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.valued_adicCompletionComap","latex_link":"Adele_miniproject.html#IsDedekindDomain.HeightOneSpectrum.valued_adicCompletionComap","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.valued_adicCompletionComap","lean_decl_name":"IsDedekindDomain.HeightOneSpectrum.valued_adicCompletionComap","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/DedekindDomain/Completion/BaseChange.lean","lean4_code":"/--\nThe local ramification index for the extension L_w/K_v is equal to the global ramification\nindex for the extension w/v. In other words, if x in K_v and i:K_v->L_w then w(i(x))=v(x)^e\nwhere e is computed globally.\n-/\nlemma valued_adicCompletionComap\n (v : HeightOneSpectrum A) (w : HeightOneSpectrum B) (hvw : w.comap A = v) (x) :\n Valued.v (adicCompletionComapSemialgHom A K L B v w hvw x) = Valued.v x ^\n Ideal.ramificationIdx (algebraMap A B) (comap A w).asIdeal w.asIdeal := by\n revert x\n apply funext_iff.mp\n symm\n apply UniformSpace.Completion.ext\n · exact Valued.continuous_valuation.pow _\n · exact Valued.continuous_valuation.comp UniformSpace.Completion.continuous_extension\n intro a\n simp_rw [adicCompletionComap_coe, adicCompletion, Valued.valuedCompletion_apply,\n adicValued_apply']\n subst hvw\n rw [← valuation_comap A K L B w a]\n","line":166,"end_line":185,"proof_status":"formalized"} {"id":"a0000000181","label":"a0000000181","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"If \\(X_v\\) and \\(Y_v\\) are families of topological spaces indexed by \\(v\\in V\\), if \\(f_v:X_v\\to Y_v\\) is a continuous map sending the subset \\(C_v\\subseteq X_v\\) into \\(D_v\\subseteq Y_v\\) then there’s an induced continuous map \\(\\prod '_v(X_v,C_v)\\to \\prod '_v(Y_v,D_v)\\).","doc_url":null,"latex_link":"Adele_miniproject.html#a0000000181","lean_link":null,"lean_decl_name":"a0000000181","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} {"id":"NumberField.InfinitePlace.Completion.baseChange_injective","label":"NumberField.InfinitePlace.Completion.baseChange_injective","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["NumberField.InfinitePlace.Completion.finrank_pi_eq_finrank_tensorProduct"],"nlp_text":"For a fixed infinite place \\(v\\) of \\(K\\), the map \\(L\\otimes _K K_v \\to \\prod _{w\\mid v}L_w\\) is injective.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.baseChange_injective","latex_link":"Adele_miniproject.html#NumberField.InfinitePlace.Completion.baseChange_injective","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.baseChange_injective","lean_decl_name":"NumberField.InfinitePlace.Completion.baseChange_injective","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/NumberField/Completion/Infinite.lean","lean4_code":"theorem baseChange_injective :\n Function.Injective (baseChange L v) := by\n rw [← baseChangeRightOfAlgebraMap_coe, ← AlgHom.coe_toLinearMap,\n LinearMap.injective_iff_surjective_of_finrank_eq_finrank\n (finrank_pi_eq_finrank_tensorProduct L v).symm]\n simp [baseChange_surjective L v]\n","line":124,"end_line":129,"proof_status":"formalized"} {"id":"NumberField.InfinitePlace.Completion.instIsModuleTopologyValEqComapAlgebraMap_fLT","label":"NumberField.InfinitePlace.Completion.instIsModuleTopologyValEqComapAlgebraMap_fLT","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"If \\(w \\mid v\\) is an infinite place of \\(L\\) lying above the infinite place \\(v\\) of \\(K\\), then \\(L_w\\) has the \\(K_v\\)-module topology.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.instIsModuleTopologyValEqComapAlgebraMap_fLT","latex_link":"Adele_miniproject.html#NumberField.InfinitePlace.Completion.instIsModuleTopologyValEqComapAlgebraMap_fLT","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.instIsModuleTopologyValEqComapAlgebraMap_fLT","lean_decl_name":"NumberField.InfinitePlace.Completion.instIsModuleTopologyValEqComapAlgebraMap_fLT","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/NumberField/Completion/Infinite.lean","lean4_code":"instance : IsModuleTopology v.Completion wv.1.Completion :=\n IsModuleTopology.iso (FiniteDimensional.nonempty_continuousLinearEquiv_of_finrank_eq\n (Module.finrank_fin_fun v.Completion)).some","line":131,"end_line":133,"proof_status":"statement_formalized"} {"id":"RestrictedProduct.relabelIso","label":"RestrictedProduct.relabelIso","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"Let \\(V\\) and \\(W\\) be index sets, and let \\(f:W\\to V\\) be a map with finite fibres. Let \\(X_v\\) be sets, with subsets \\(C_v\\), let \\(Y_w\\) be sets with subsets \\(D_w\\), and say for all \\(v\\in V\\) we’re given a bijection \\(X_v\\to \\prod _{w|f(w)=v}Y_w\\), identifying \\(C_v\\) with \\(\\prod _{w:f(w)=v}D_w\\). Then there’s an induced bijection between the restricted products \\(\\prod '_v(X_v,C_v)\\) and \\(\\prod '_w(Y_w,D_w)\\).","doc_url":null,"latex_link":"Adele_miniproject.html#RestrictedProduct.relabelIso","lean_link":null,"lean_decl_name":"RestrictedProduct.relabelIso","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} {"id":"IsDedekindDomain.AKLB.finitePresentation","label":"IsDedekindDomain.AKLB.finitePresentation","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"\\(B\\) is a finitely-presented \\(A\\)-module.","doc_url":null,"latex_link":"Adele_miniproject.html#IsDedekindDomain.AKLB.finitePresentation","lean_link":null,"lean_decl_name":"IsDedekindDomain.AKLB.finitePresentation","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/AdjoinRoot.lean","lean4_code":"theorem finitePresentation : Algebra.FinitePresentation R (AdjoinRoot f) :=\n (Algebra.FinitePresentation.polynomial R).quotient (Submodule.fg_span_singleton f)\n\n/-- The adjoined root. -/\n","line":155,"end_line":158,"proof_status":"statement_formalized"} {"id":"Rat.AdeleRing.cocompact","label":"Rat.AdeleRing.cocompact","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"The quotient \\(\\mathbb {A}_{\\mathbb {Q}}/\\mathbb {Q}\\) is compact.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Rat.AdeleRing.cocompact","latex_link":"Adele_miniproject.html#Rat.AdeleRing.cocompact","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Rat.AdeleRing.cocompact","lean_decl_name":"Rat.AdeleRing.cocompact","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Topology/Defs/Filter.lean","lean4_code":"def Filter.cocompact : Filter X :=\n ⨅ (s : Set X) (_ : IsCompact s), 𝓟 sᶜ\n\nvariable (X) in\n/-- `Filter.coclosedCompact` is the filter generated by complements to closed compact sets.\nIn a Hausdorff space, this is the same as `Filter.cocompact`. -/\n","line":323,"end_line":328,"proof_status":"statement_formalized"} {"id":"IsDedekindDomain.AKLB.surjective_tensorProduct_map","label":"IsDedekindDomain.AKLB.surjective_tensorProduct_map","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.dvd_norm"],"nlp_text":"The \\(A\\)-bilinear map \\(B\\times K\\to L\\) sending \\((b,k)\\) to \\(bk\\) is surjective.","doc_url":null,"latex_link":"Adele_miniproject.html#IsDedekindDomain.AKLB.surjective_tensorProduct_map","lean_link":null,"lean_decl_name":"IsDedekindDomain.AKLB.surjective_tensorProduct_map","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} {"id":"IsDedekindDomain.FiniteAdeleRing.tensorProduct_algEquiv","label":"IsDedekindDomain.FiniteAdeleRing.tensorProduct_algEquiv","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":["IsDedekindDomain.dvd_norm"],"nlp_text":"The natural map \\(B\\otimes _AK\\to L\\) is a \\(B\\)-algebra isomorphism.","doc_url":null,"latex_link":"Adele_miniproject.html#IsDedekindDomain.FiniteAdeleRing.tensorProduct_algEquiv","lean_link":null,"lean_decl_name":"IsDedekindDomain.FiniteAdeleRing.tensorProduct_algEquiv","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} {"id":"a0000000182","label":"a0000000182","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"If all the \\(f_v\\) are homeomorphisms identifying \\(C_v\\) with \\(D_v\\) then the induced map on restricted products is also a homeomorphism (proof: apply the previous construction to \\(f_v\\) and their inverses)","doc_url":null,"latex_link":"Adele_miniproject.html#a0000000182","lean_link":null,"lean_decl_name":"a0000000182","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} {"id":"IsDedekindDomain.HeightOneSpectrum.Extension.finite","label":"IsDedekindDomain.HeightOneSpectrum.Extension.finite","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"There are only finitely many primes \\(w\\) of \\(B\\) lying above \\(v\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.Extension.finite","latex_link":"Adele_miniproject.html#IsDedekindDomain.HeightOneSpectrum.Extension.finite","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.Extension.finite","lean_decl_name":"IsDedekindDomain.HeightOneSpectrum.Extension.finite","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Order/Interval/Finset/Defs.lean","lean4_code":"Provide the `LocallyFiniteOrder` instance for `α →₀ β` where `β` is locally finite. Provide the\n`LocallyFiniteOrder` instance for `Π₀ i, β i` where all the `β i` are locally finite.\n\nFrom `LinearOrder α`, `NoMaxOrder α`, `LocallyFiniteOrder α`, we can also define an\norder isomorphism `α ≃ ℕ` or `α ≃ ℤ`, depending on whether we have `OrderBot α` or\n`NoMinOrder α` and `Nonempty α`. When `OrderBot α`, we can match `a : α` to `#(Iio a)`.\n\nWe can provide `SuccOrder α` from `LinearOrder α` and `LocallyFiniteOrder α` using\n\n```lean\n","line":69,"end_line":78,"proof_status":"formalized"} {"id":"IsDedekindDomain.dvd_norm","label":"IsDedekindDomain.dvd_norm","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"If \\(0\\not=b\\in B\\) then there exists \\(0\\not=a\\in A\\) such that \\(b\\) divides the image of \\(a\\) in \\(B\\).","doc_url":null,"latex_link":"Adele_miniproject.html#IsDedekindDomain.dvd_norm","lean_link":null,"lean_decl_name":"IsDedekindDomain.dvd_norm","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/NumberTheory/NumberField/Norm.lean","lean4_code":"theorem dvd_norm [IsGalois K L] (x : 𝓞 L) : x ∣ algebraMap (𝓞 K) (𝓞 L) (norm K x) := by\n classical\n have hint :\n IsIntegral ℤ (∏ σ ∈ univ.erase (AlgEquiv.refl : L ≃ₐ[K] L), σ x) :=\n IsIntegral.prod _ (fun σ _ =>\n ((RingOfIntegers.isIntegral_coe x).map σ))\n refine ⟨⟨_, hint⟩, ?_⟩\n ext\n rw [coe_algebraMap_norm K x, norm_eq_prod_automorphisms]\n simp [← Finset.mul_prod_erase _ _ (mem_univ AlgEquiv.refl)]\n\nvariable (F : Type*) [Field F] [Algebra K F] [Algebra.IsSeparable K F] [FiniteDimensional K F]\n\n","line":84,"end_line":96,"proof_status":"statement_formalized"} {"id":"Rat.AdeleRing.zero_discrete","label":"Rat.AdeleRing.zero_discrete","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"There’s an open subset of \\(\\mathbb {A}_{\\mathbb {Q}}\\) whose intersection with \\(\\mathbb {Q}\\) is \\(\\{ 0\\} \\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Rat.AdeleRing.zero_discrete","latex_link":"Adele_miniproject.html#Rat.AdeleRing.zero_discrete","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Rat.AdeleRing.zero_discrete","lean_decl_name":"Rat.AdeleRing.zero_discrete","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/NumberField/AdeleRing.lean","lean4_code":"theorem Rat.AdeleRing.zero_discrete : ∃ U : Set (AdeleRing (𝓞 ℚ) ℚ),\n IsOpen U ∧ (algebraMap ℚ (AdeleRing (𝓞 ℚ) ℚ)) ⁻¹' U = {0} := by\n let integralAdeles := {f : FiniteAdeleRing (𝓞 ℚ) ℚ |\n ∀ v , f v ∈ IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers ℚ v}\n use {f | ∀ v, f v ∈ (Metric.ball 0 1)} ×ˢ integralAdeles\n refine ⟨?_, ?_⟩\n · apply IsOpen.prod\n · rw [Set.setOf_forall]\n apply isOpen_iInter_of_finite\n intro v\n exact Metric.isOpen_ball.preimage (continuous_apply v)\n · exact RestrictedProduct.isOpen_forall_mem fun v ↦ Valued.isOpen_integer _\n · apply subset_antisymm\n · intro x hx\n rw [Set.mem_preimage] at hx\n simp only [Set.mem_singleton_iff]\n rw [show (algebraMap ℚ (AdeleRing (𝓞 ℚ) ℚ)) x =\n (algebraMap ℚ (InfiniteAdeleRing ℚ) x, algebraMap ℚ (FiniteAdeleRing (𝓞 ℚ) ℚ) x)\n from rfl] at hx\n rw [Set.mem_prod] at hx\n obtain ⟨h1, h2⟩ := hx\n dsimp only at h1 h2\n simp only [Metric.mem_ball, dist_zero_right, Set.mem_setOf_eq] at h1\n exact Rat.AdeleRing.integral_and_norm_lt_one x h2 h1\n · intro x\n simp only [Set.mem_singleton_iff, Set.mem_preimage]\n rintro rfl\n simp only [map_zero]\n change (0, 0) ∈ _\n simp only [Prod.mk_zero_zero]\n constructor\n · simp only [Metric.mem_ball, dist_zero_right, Set.mem_setOf_eq]\n intro v\n have : ‖(0:InfiniteAdeleRing ℚ) v‖ = 0 := by\n simp only [norm_eq_zero]\n rfl\n simp [this, zero_lt_one]\n · simp only [integralAdeles, Set.mem_setOf_eq]\n intro v\n apply zero_mem","line":368,"end_line":407,"proof_status":"statement_formalized"} {"id":"IsDedekindDomain.HeightOneSpectrum.valuation_comap","label":"IsDedekindDomain.HeightOneSpectrum.valuation_comap","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"If \\(i:K\\to L\\) denotes the inclusion then for \\(k\\in K\\) we have \\(e\\times w(i(k))=v(k)\\), where \\(e\\) is the ramification index of \\(w/v\\) (recall that valuations here are written additively, unlike in mathlib).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.valuation_comap","latex_link":"Adele_miniproject.html#IsDedekindDomain.HeightOneSpectrum.valuation_comap","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.HeightOneSpectrum.valuation_comap","lean_decl_name":"IsDedekindDomain.HeightOneSpectrum.valuation_comap","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/DedekindDomain/IntegralClosure.lean","lean4_code":"/-- If w | v then for x ∈ K we have w(x)=v(x)^e where e is the ramification index. -/\nlemma valuation_comap (w : HeightOneSpectrum B) (x : K) :\n (comap A w).valuation K x ^\n (Ideal.ramificationIdx (algebraMap A B) (comap A w).asIdeal w.asIdeal) =\n w.valuation L (algebraMap K L x) := by\n obtain ⟨x, y, hy, rfl⟩ := IsFractionRing.div_surjective (A := A) x\n simp [valuation, ← IsScalarTower.algebraMap_apply A K L, IsScalarTower.algebraMap_apply A B L,\n ← intValuation_comap A B (algebraMap_injective_of_field_isFractionRing A B K L), div_pow]\n","line":116,"end_line":123,"proof_status":"formalized"} {"id":"a0000000183","label":"a0000000183","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"In the same setup as definition 9.26 (\\(V,W\\) index sets, \\(f:W\\to V\\), \\(C_v\\subseteq X_v\\) and \\(D_w\\subseteq Y_w\\), bijections \\(b_v:X_v\\to \\prod _{w:f(w)=v}Y_w\\) identifying \\(C_v\\) with \\(\\prod _{w:f(w)=v}D_w\\)), if all the \\(X_v\\) and \\(Y_w\\) are furthermore topological spaces, all the \\(C_v\\) and \\(D_w\\) are open, and all the \\(b_v\\) are homeomorphisms, then the induced map \\(\\prod '_v(X_v,C_v)\\to \\prod '_w(Y_w,D_w)\\) is also a homeomorphism.","doc_url":null,"latex_link":"Adele_miniproject.html#a0000000183","lean_link":null,"lean_decl_name":"a0000000183","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} {"id":"NumberField.InfinitePlace.Completion.finrank_pi_eq_finrank_tensorProduct","label":"NumberField.InfinitePlace.Completion.finrank_pi_eq_finrank_tensorProduct","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"For a fixed infinite place \\(v\\) of \\(K\\), we have \\(\\text{dim}_{K_v} \\prod _{w\\mid v} L_w = \\text{dim}_{K_v} L\\otimes _K K_v\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.finrank_pi_eq_finrank_tensorProduct","latex_link":"Adele_miniproject.html#NumberField.InfinitePlace.Completion.finrank_pi_eq_finrank_tensorProduct","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/NumberField.InfinitePlace.Completion.finrank_pi_eq_finrank_tensorProduct","lean_decl_name":"NumberField.InfinitePlace.Completion.finrank_pi_eq_finrank_tensorProduct","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/NumberField/Completion/Infinite.lean","lean4_code":"theorem finrank_pi_eq_finrank_tensorProduct [NumberField K] :\n Module.finrank v.Completion ((w : v.Extension L) → w.1.Completion) =\n Module.finrank v.Completion (L ⊗[K] v.Completion) := by\n rw [(TensorProduct.RightActions.Algebra.TensorProduct.comm\n K v.Completion L).symm.toLinearEquiv.finrank_eq,\n Module.finrank_tensorProduct, Module.finrank_self, one_mul,\n finrank_prod_eq_finrank]\n","line":97,"end_line":103,"proof_status":"statement_formalized"} {"id":"IsDedekindDomain.AKLB.tensorProduct_module_algEquiv","label":"IsDedekindDomain.AKLB.tensorProduct_module_algEquiv","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_9.html","dependencies":[],"nlp_text":"If \\(M\\) is any \\(K\\)-module then the canonical map \\(B\\otimes _A M\\to L\\otimes _K M\\) is an isomorphism.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.linearEquivTensorProductModule","latex_link":"Adele_miniproject.html#IsDedekindDomain.AKLB.tensorProduct_module_algEquiv","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsDedekindDomain.linearEquivTensorProductModule","lean_decl_name":"IsDedekindDomain.linearEquivTensorProductModule","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch9","source_path":"FLT/DedekindDomain/IntegralClosure.lean","lean4_code":"/-- The canonical `A`-linear isomorphism `L ⊗ M ≅ B ⊗ M` for any `K`-module `M`. -/\nnoncomputable def linearEquivTensorProductModule : L ⊗[K] M ≃ₗ[A] B ⊗[A] M :=\n let f₁ : L ⊗[K] M ≃ₗ[A] L ⊗[A] M := IsLocalization.moduleTensorEquiv (nonZeroDivisors A) K L M\n |>.restrictScalars A\n let f₂ : L ≃ₗ[A] B ⊗[A] K := LinearEquivTensorProduct A K L B\n |>.restrictScalars A\n |>.trans (TensorProduct.comm A K B)\n let f₃ : L ⊗[A] M ≃ₗ[A] (B ⊗[A] K) ⊗[A] M := TensorProduct.congr f₂ (LinearEquiv.refl A M)\n let f₄ : (B ⊗[A] K) ⊗[A] M ≃ₗ[A] B ⊗[A] (K ⊗[A] M) :=\n TensorProduct.assoc A B K M\n let f₅ : B ⊗[A] (K ⊗[A] M) ≃ₗ[A] B ⊗[A] M := TensorProduct.congr (LinearEquiv.refl A B)\n (IsLocalization.moduleLid (nonZeroDivisors A) K M |>.restrictScalars A)\n f₁.trans f₃ |>.trans f₄ |>.trans f₅\n","line":245,"end_line":257,"proof_status":"formalized"}