full_collection/ClassFieldTheory.jsonl

{"id":"def:down","label":"down","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:ind'"],"nlp_text":"For any representation \\(M\\), there is a surjective morphism \\(\\mathrm{ind}_1'(M) \\to M\\), which takes a finitely supported function \\(f : G \\to _0 M\\) to the sum \\(\\sum _{x \\in G} f (x)\\). We define \\(\\mathrm{down}(M)\\) to be the kernel of this map. There is therefore a short exact sequence \\[ 0 \\to \\mathrm{down}(M) \\to \\mathrm{ind}_1'(M) \\to M \\to 0. \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Representation.ind₁'_π","latex_link":"sect0002.html#def:down","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Representation.ind₁'_π","lean_decl_name":"Representation.ind₁'_π","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/IndCoind/Finite.lean","lean4_code":"@[simps] def ind₁'_π : (G →₀ V) →ₗ[R] V where\n  toFun f := f.sum (fun _ ↦ (1 : V →ₗ[R] V))\n  map_add' _ _ :=  sum_add_index' (congrFun rfl) fun _ _ ↦ congrFun rfl\n  map_smul' _ _ := by simp","line":221,"end_line":224,"proof_status":"formalized"}
{"id":"cor:Tate up down isos","label":"Tate up down isos","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:down","lem:induced' trivial Tate","def:Tate long exact sequence","def:up"],"nlp_text":"If the group \\(G\\) is finite then for every subgroup \\(S\\) of \\(G\\) and every \\(n \\in {\\mathbb Z}\\) we have isomorphisms \\[ H^n_{\\mathrm{Tate}}(S, \\mathrm{up}(M)) \\cong H^{n+1}_{\\mathrm{Tate}}(S,M), \\qquad H^{n+1}_{\\mathrm{Tate}}(S, \\mathrm{down}(M)) \\cong H^{n}_{\\mathrm{Tate}}(S,M). \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.instIsIso_shortExact_upSES","latex_link":"sect0002.html#cor:Tate up down isos","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.instIsIso_shortExact_upSES","lean_decl_name":"groupCohomology.instIsIso_shortExact_upSES","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/Functors/UpDown.lean","lean4_code":"instance instIsIso_shortExact_upSES (M : Rep R G) (n : ℤ) :\n    IsIso (TateCohomology.δ (shortExact_upSES M) n) := by\n  have _ : TrivialTateCohomology (coind₁'.obj M) := inferInstance\n  exact ShortComplex.ShortExact.isIso_δ\n    (TateCohomology.map_tateComplexFunctor_shortExact (shortExact_upSES M))\n    n (n + 1) (by rfl) (by simp; exact isZero_of_trivialTateCohomology' (coind₁'.obj M) n)\n    (by simp; exact isZero_of_trivialTateCohomology' (coind₁'.obj M) (n + 1))","line":260,"end_line":266,"proof_status":"formalized"}
{"id":"def:up iso down","label":"up iso down","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:down","def:up","lem:map2 image"],"nlp_text":"We have a commutative square with vertical isomorphisms: \\[ \\begin{matrix} \\mathrm{ind}_1’(M) & \\stackrel{\\mathrm{map}_2}\\to & \\mathrm{ind}_1’(M) \\\\ \\downarrow & & \\downarrow \\\\ \\mathrm{coind}_1’(M) & \\stackrel{\\mathrm{map}_1}\\to & \\mathrm{coind}_1’(M) \\\\ \\end{matrix}. \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Representation.map₁","latex_link":"sect0002.html#def:up iso down","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Representation.map₁","lean_decl_name":"Representation.map₁","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"ClassFieldTheory/Cohomology/FiniteCyclic/UpDown.lean","lean4_code":"@[simps] def map₁ : (G → A) →ₗ[R] (G → A) where\n  toFun f x := f x - f ((gen G)⁻¹ * x)\n  map_add' _ _ := by\n    ext; simp [add_sub_add_comm]\n  map_smul' _ _ := by\n    ext; simp [← smul_sub]\n","line":67,"end_line":72,"proof_status":"formalized"}
{"id":"cor:cohomology sub Sylow","label":"cohomology sub Sylow","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["cor:Tate up down isos","lem:cor comp rest"],"nlp_text":"Let \\(M\\) be a representation of a finite group \\(G\\) and let \\(S_p\\) be a Sylow \\(p\\)-subgroup of \\(G\\) for some prime number \\(p\\). Then for any \\(n \\in {\\mathbb Z}\\), \\(H^n_{\\mathrm{Tate}}(G,M)[p^\\infty ]\\) is isomorphic to an \\(R\\)-submodule of \\(H^n_{\\mathrm{Tate}}(S_p,M)\\).","doc_url":null,"latex_link":"sect0002.html#cor:cohomology sub Sylow","lean_link":null,"lean_decl_name":"cohomology sub Sylow","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":"cor:cohomology G-torsion","label":"cohomology G-torsion","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["cor:Tate up down isos","lem:cor comp rest"],"nlp_text":"If \\(M\\) is a representation of a finite group \\(G\\) then for all \\(n \\in {\\mathbb Z}\\) and all \\(\\sigma \\in H^n_{\\mathrm{Tate}}(G,M)\\) we have \\(|G| \\cdot \\sigma = 0\\).","doc_url":null,"latex_link":"sect0002.html#cor:cohomology G-torsion","lean_link":null,"lean_decl_name":"cohomology G-torsion","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":"cor:periodic cohomology","label":"periodic cohomology","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["cor:Tate up down isos","def:up iso down"],"nlp_text":"Let \\(G\\) be a finite cyclic group. For all \\(n {\\gt} 0\\) and all representations \\(M\\) we have an isomorphism \\(H^{n}(G,M) \\cong H^{n+2}(G,M)\\). Similarly for all integers \\(n\\) we have isomorphisms \\(H^{n}_{\\mathrm{Tate}}(G,M) \\cong H^{n+2}_{\\mathrm{Tate}}(G,M)\\).","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.periodicCohomology","latex_link":"sect0002.html#cor:periodic cohomology","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.periodicCohomology","lean_decl_name":"Rep.periodicCohomology","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"ClassFieldTheory/Cohomology/FiniteCyclic/UpDown.lean","lean4_code":"/-- The non-zero cohomology of a finite cyclic group is 2-periodic. -/\ndef periodicCohomology (m n : ℕ) [NeZero m] [NeZero n] (hmn : m ≡ n [MOD 2]) :\n    functor R G m ≅ functor R G n := calc\n      functor R G m\n    ≅ functor R G (m + 2 * (n / 2)) := periodicCohomologyAux ..\n  _ ≅ functor R G (n + 2 * (m / 2)) := by\n    refine eqToIso ?_\n    congr 1\n    obtain ⟨k, rfl⟩ | ⟨k, rfl⟩ := m.even_or_odd <;> obtain ⟨l, rfl⟩ | ⟨l, rfl⟩ := n.even_or_odd\n    · omega\n    · simp [Nat.ModEq, ← two_mul] at hmn\n    · simp [Nat.ModEq, ← two_mul] at hmn\n    · omega\n  _ ≅ functor R G n := (periodicCohomologyAux ..).symm\n","line":386,"end_line":399,"proof_status":"formalized"}
{"id":"thm:global cohomology bound","label":"global cohomology bound","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["cor:cohomology sub Sylow","cor:cohomology G-torsion","cor:H1 H2 cyclic idele class","thm:inflation restriction sequence"],"nlp_text":"If \\(l/k\\) is any finite Galois extension then \\(H^1(l/k, \\mathrm{Cl}_{l}) \\cong 0\\) and \\(|H^2(l/k, \\mathrm{Cl}_{l})| \\le [l:k]\\).","doc_url":null,"latex_link":"sect0004.html#thm:global cohomology bound","lean_link":null,"lean_decl_name":"global cohomology bound","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":"lem:local H2 upper bound","label":"local H2 upper bound","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["cor:cohomology sub Sylow","cor:cohomology G-torsion","lem:local H2 l*","thm:inflation restriction sequence"],"nlp_text":"Let \\(l/k\\) be a Galois extension of local fields. Then \\(|H^2(l/k,l^\\times )| \\le [l:k]\\).","doc_url":null,"latex_link":"sect0003.html#lem:local H2 upper bound","lean_link":null,"lean_decl_name":"local H2 upper bound","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":"thm:triviality criterion","label":"triviality criterion","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["cor:cohomology sub Sylow","cor:cohomology G-torsion","thm:triviality criterion solvable"],"nlp_text":"Let \\(M\\) be a representation of a finite group \\(G\\) (no longer assumed to be solvable). Suppose we have positive natural numbers \\(e\\) and \\(o\\) with \\(e\\) even and \\(o\\) odd, such that for all subgroups \\(S\\) of \\(G\\) we have \\[ H^e(S,M) =0, \\qquad H^o(S,M) = 0. \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.trivialCohomology_of_even_of_odd","latex_link":"sect0002.html#thm:triviality criterion","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.trivialCohomology_of_even_of_odd","lean_decl_name":"groupCohomology.trivialCohomology_of_even_of_odd","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/TrivialityCriterion.lean","lean4_code":"theorem groupCohomology.trivialCohomology_of_even_of_odd [Finite G]\n    (M : Rep R G) (n m : ℕ)\n    -- todo: don't quantify over all types\n    (h_even : ∀ (H : Type) [Group H] {φ : H →* G} (_ : Function.Injective φ),\n      IsZero (groupCohomology (M ↓ φ) (2 * n + 2)))\n    (h_odd : ∀ (H : Type) [Group H] {φ : H →* G} (_ : Function.Injective φ),\n      IsZero (groupCohomology (M ↓ φ) (2 * m + 1))) :\n    M.TrivialCohomology := by\n  constructor\n  -- let `S` be a subgroup of `G`\n  intro S u\n  refine @ModuleCat.isZero_of_subsingleton R _ _\n    (@Unique.instSubsingleton _ ⟨⟨0⟩, fun x => (?_ : x = 0)⟩)\n  -- `Hᵘ⁺¹(S, M)` is torsion\n  have hx : Nat.card S • x = 0 := by\n    classical\n    have : Fintype S := Fintype.ofFinite S\n    apply torsion_of_finite_of_neZero\n  -- it suffices to show that for every prime `p`, it has no `p^∞` torsion\n  have hk : 0 < Nat.card S := Nat.card_pos\n  generalize Nat.card S = k at hx hk\n  induction k using Nat.recOnPrimePow with\n  | zero => simp at hk\n  | one => simpa using hx\n  | prime_pow_mul a p c hp ha hc ih =>\n    refine ih ?_ (Nat.pos_of_mul_pos_left hk)\n    -- let `v` be an arbitrary Sylow-`p` subgroup of `S`\n    obtain ⟨v⟩ : Nonempty (Sylow p S) := inferInstance\n    rw [mul_smul] at hx\n    -- the `p^∞` torsion injects into `Hᵘ⁺¹(v,M)`, so it suffices that `Hᵘ⁺¹(v,M)` is trivial\n    haveI : Fact p.Prime := ⟨hp⟩\n    apply (groupCohomology_Sylow (Nat.add_one_pos u) (M ↓ S.subtype) (a • x) p v ⟨c, hx⟩).mtr\n    -- `Hᵘ⁺¹(v,M)` is trivial if `M` has trivial cohomology\n    refine @Subsingleton.eq_zero _ _ (ModuleCat.subsingleton_of_isZero\n      (@isZero_of_trivialCohomology R v _ _ _ ?_ (u + 1) _)) _\n    -- `v` is a `p`-group, so it is solvable\n    have : Fact p.Prime := ⟨hp⟩\n    have : Group.IsNilpotent v := v.isPGroup'.isNilpotent -- todo: make this an instance?\n    have : Fintype v := Fintype.ofFinite v\n    classical\n    -- therefore `M` has trivial cohomology if it has an even and an odd trivial cohomology\n    apply trivialCohomology_of_even_of_odd_of_solvable (M ↓ S.subtype ↓ v.toSubgroup.subtype) n m\n    · -- the even trivial cohomology for `G` lifts to `v`\n      intro H  _ φ hφ\n      refine .of_iso (h_even H (φ := (S.subtype.comp v.toSubgroup.subtype).comp φ)\n        ((S.subtype_injective.comp v.toSubgroup.subtype_injective).comp hφ)) ?_\n      apply (functor R H (2 * n + 2)).mapIso\n      refine Iso.trans ?_ ((Action.resComp (ModuleCat R) φ _).app M)\n      apply (res φ).mapIso\n      exact (Action.resComp (ModuleCat R) _ S.subtype).app M\n    · -- the odd trivial cohomology for `G` lifts to `v`\n      intro H  _ φ hφ\n      refine .of_iso (h_odd H (φ := (S.subtype.comp v.toSubgroup.subtype).comp φ)\n        ((S.subtype_injective.comp v.toSubgroup.subtype_injective).comp hφ)) ?_\n      apply (functor R H (2 * m + 1)).mapIso\n      refine Iso.trans ?_ ((Action.resComp (ModuleCat R) φ _).app M)\n      apply (res φ).mapIso\n      exact (Action.resComp (ModuleCat R) _ S.subtype).app M","line":96,"end_line":153,"proof_status":"formalized"}
{"id":"lem:finite field trivial","label":"finite field trivial","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["cor:periodic cohomology","thm:additive field trivial","thm:hilbert 90","lem:herbrand finite"],"nlp_text":"The Galois modules \\({\\mathbb F}_l\\) and \\({\\mathbb F}_l^\\times \\) have trivial cohomology.","doc_url":null,"latex_link":"sect0003.html#lem:finite field trivial","lean_link":null,"lean_decl_name":"finite field trivial","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":"lem:herbrand ses","label":"herbrand ses","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["cor:periodic cohomology","def:herbrand quotient"],"nlp_text":"Suppose we have a short exact sequence of representations of a finite cyclic group \\(G\\): \\[ 0 \\to A \\to B \\to C \\to 0. \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.herbrandQuotient_ne_zero_of_shortExact₁","latex_link":"sect0002.html#lem:herbrand ses","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.herbrandQuotient_ne_zero_of_shortExact₁","lean_decl_name":"Rep.herbrandQuotient_ne_zero_of_shortExact₁","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"ClassFieldTheory/Cohomology/FiniteCyclic/HerbrandQuotient/SES.lean","lean4_code":"lemma herbrandQuotient_ne_zero_of_shortExact₁\n    (h₂ : S.X₂.herbrandQuotient ≠ 0) (h₃ : S.X₃.herbrandQuotient ≠ 0) :\n    S.X₁.herbrandQuotient ≠ 0 := by\n  rw [herbrandQuotient_ne_zero] at *\n  obtain ⟨_, _⟩ :\n      Finite ((herbrandSixTermSequence hS).sc 3).X₃ ∧\n      Finite ((herbrandSixTermSequence hS).sc 0).X₃ := by simpa using h₂\n  obtain ⟨_, _⟩ :\n      Finite ((herbrandSixTermSequence hS).sc 0).X₁ ∧\n      Finite ((herbrandSixTermSequence hS).sc 3).X₁ := by simpa using h₃\n  exact ⟨(herbrandSixTermSequence_exactAt hS 3).moduleCat_finite,\n    (herbrandSixTermSequence_exactAt hS 0).moduleCat_finite⟩\n","line":109,"end_line":120,"proof_status":"formalized"}
{"id":"lem:H2 cyclic Z","label":"H2 cyclic Z","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["cor:periodic cohomology"],"nlp_text":"Let \\(G\\) be a finite cyclic group of order \\(n\\). Then \\(H^1(G,{\\mathbb Z}) \\cong 0\\) and \\(H^2(G,{\\mathbb Z}) \\cong {\\mathbb Z}/n{\\mathbb Z}\\).","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.evenTrivialInt","latex_link":"sect0002.html#lem:H2 cyclic Z","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.evenTrivialInt","lean_decl_name":"groupCohomology.evenTrivialInt","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"ClassFieldTheory/Cohomology/FiniteCyclic/UpDown.lean","lean4_code":"/-- The even Tate cohomology of a trivial representation of a finite cyclic group of order `N` is\n`ℤ/Nℤ`. -/\ndef evenTrivialInt [IsCyclic G] (hG : Nat.card G = N) (hn : Even n) :\n    (tateCohomology n).obj (trivial ℤ G ℤ) ≅ .of ℤ (ZMod N) := calc\n  (tateCohomology n).obj (trivial ℤ G ℤ)\n    ≅ (tateCohomology 0).obj (trivial ℤ G ℤ) :=\n    (periodicTateCohomology _ _ <| by simp [Int.modEq_iff_dvd, hn.two_dvd]).app _\n  _ ≅ .of ℤ (ZMod N) := zeroTrivialInt hG\n","line":453,"end_line":460,"proof_status":"formalized"}
{"id":"a0000000035","label":"a0000000035","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(M\\) be a representation of a finite group \\(G\\) over the ring \\({\\mathbb Z}\\), and suppose that for all subgroups \\(S\\) of \\(G\\) the following two conditions hold: \\(H^1(S,M) \\cong 0\\), \\(H^2(S,M) \\cong {\\mathbb Z}/ |S| \\cdot {\\mathbb Z}\\). Then there is an isomorphism \\[ M^G / N_G M \\cong G^{\\mathrm{ab}}. \\]","doc_url":null,"latex_link":"sect0002.html#a0000000035","lean_link":null,"lean_decl_name":"a0000000035","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":"cor:coind₁' trivial cohomology","label":"coind₁' trivial cohomology","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:coind₁' iso coind₁","lem:coind₁ trivial cohomology"],"nlp_text":"The representation \\(\\mathrm{coind}_1'(M)\\) has trivial cohomology.","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.trivialCohomology_coind₁'","latex_link":"sect0002.html#cor:coind₁' trivial cohomology","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.trivialCohomology_coind₁'","lean_decl_name":"Rep.trivialCohomology_coind₁'","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/IndCoind/TrivialCohomology.lean","lean4_code":"instance trivialCohomology_coind₁' : (coind₁'.obj M).TrivialCohomology :=\n  .of_iso (coind₁'_obj_iso_coind₁ M)","line":111,"end_line":112,"proof_status":"formalized"}
{"id":"cor:up iso","label":"up iso","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["cor:coind₁' trivial cohomology","def:up"],"nlp_text":"Let \\(S\\) be any subgroup of \\(G\\) and let \\(n \\ge 1\\). Then the connecting map from the long exact sequence \\(H^{n}(S,\\mathrm{up}(M)) \\to H^{n+1}(S,M)\\) is an isomorphism. The corresponding map \\(H^{0}(S,\\mathrm{up}(M)) \\to H^{1}(S,M)\\) is surjective. The isomorphism \\(H^{n}(S,\\mathrm{up}(-)) \\cong H^{n+1}(S,-)\\) is an isomorphism of functors. This means that for every morphism \\(f : M \\to N\\) of representations, the following square commutes: \\[ \\begin{matrix} H^{n}(S,\\mathrm{up}(M)) & \\cong & H^{n+1}(S,M) \\\\ \\downarrow & & \\downarrow \\\\ H^{n}(S,\\mathrm{up}(N)) & \\cong & H^{n+1}(S,N) \\end{matrix}. \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.dimensionShift.δ_up_zero_epi","latex_link":"sect0002.html#cor:up iso","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.dimensionShift.δ_up_zero_epi","lean_decl_name":"Rep.dimensionShift.δ_up_zero_epi","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/Functors/UpDown.lean","lean4_code":"/--\nThe connecting homomorphism from `H⁰(G,up M)` to `H¹(G,M)` is\nan epimorphism (i.e. surjective).\n-/\ninstance δ_up_zero_epi : Epi (δ (shortExact_upSES M) 0 1 rfl) := by\n  refine epi_δ_of_isZero (shortExact_upSES M) 0 ?_\n  simpa only [upSES_X₂, zero_add] using isZero_of_trivialCohomology","line":99,"end_line":105,"proof_status":"formalized"}
{"id":"def:corestriction","label":"corestriction","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["cor:up iso"],"nlp_text":"Let \\(S\\) be a subgroup of finite index in \\(G\\) and let \\(\\{ r_i\\} \\) be a set of representatives for the left cosets \\(r_i S\\). For any representation \\(M\\) of \\(G\\) there is a linear map \\(N_{G/S} : M^S \\to M^G\\) defined by \\[ N_{G/S}(m) = \\sum _i r_i \\bullet m. \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.cores₀","latex_link":"sect0002.html#def:corestriction","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.cores₀","lean_decl_name":"groupCohomology.cores₀","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"ClassFieldTheory/Cohomology/Functors/Corestriction.lean","lean4_code":"/-- The corestriction functor on H^0 for S ⊆ G a finite index subgroup, as a\nfunctor `H^0(S,-) → H^0(G,-)`. -/\n@[simps]\ndef cores₀ : Rep.res S.subtype ⋙ functor R S 0 ⟶ functor R G 0 where\n  app M :=\n    (H0Iso (M ↓ S.subtype)).hom ≫ (ModuleCat.ofHom (Rep.cores₀_obj M)) ≫ (H0Iso M).inv\n  naturality := by\n    intro X Y f\n    simp_rw [← Category.assoc]\n    rw [(H0Iso Y).comp_inv_eq]\n    simp_rw [Category.assoc]\n    rw [functor_map, map_id_comp_H0Iso_hom, (H0Iso X).inv_hom_id_assoc, Functor.comp_map,\n      functor_map, map_id_comp_H0Iso_hom_assoc, (H0Iso (X ↓ S.subtype)).cancel_iso_hom_left]\n    ext x\n    have comm := congr(∑ i : G ⧸ S, ModuleCat.Hom.hom $(f.comm i.out) x.val)\n    simpa [Rep.cores₀_obj] using comm.symm\n","line":90,"end_line":105,"proof_status":"statement_formalized"}
{"id":"lem:cor comp rest","label":"cor comp rest","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:corestriction","lem:inflation restriction naturality"],"nlp_text":"For all \\(\\sigma \\in H^n(G,M)\\) we have \\(\\mathrm{cor}(\\mathrm{rest}(\\sigma )) = [G:S] \\cdot \\sigma \\).","doc_url":null,"latex_link":"sect0002.html#lem:cor comp rest","lean_link":null,"lean_decl_name":"cor comp rest","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":"cor:ind₁' trivial homology","label":"ind₁' trivial homology","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:ind₁ trivial homology","lem:ind₁' iso ind₁"],"nlp_text":"The representation \\(\\mathrm{ind}_1'(M)\\) has trivial homology.","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.trivialHomology_ind₁'","latex_link":"sect0002.html#cor:ind₁' trivial homology","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.trivialHomology_ind₁'","lean_decl_name":"Rep.trivialHomology_ind₁'","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/IndCoind/TrivialCohomology.lean","lean4_code":"instance trivialHomology_ind₁' : (ind₁'.obj M).TrivialHomology :=\n  .of_iso (ind₁'_obj_iso_ind₁ M)","line":108,"end_line":109,"proof_status":"formalized"}
{"id":"lem:local inv fundamental class","label":"local inv fundamental class","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:unramified fundamental class","def:local inv"],"nlp_text":"The fundamental class \\(\\sigma _{l/k}\\) has local invariant \\(\\frac{1}{[l:k]}\\).","doc_url":null,"latex_link":"sect0003.html#lem:local inv fundamental class","lean_link":null,"lean_decl_name":"local inv fundamental class","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":"lem:local unram inv infl","label":"local unram inv infl","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:local inv fundamental class"],"nlp_text":"Let \\(m / l / k\\) be a tower of unramified extensions of local fields and let \\(\\mathrm{infl}: H^2(l/k,l^\\times ) \\to H^2(m/k,m^\\times )\\) be the inflation map. Then \\[ \\mathrm{inv}_{m/k} \\circ \\mathrm{infl}= \\mathrm{inv}_{l/k}. \\]","doc_url":null,"latex_link":"sect0003.html#lem:local unram inv infl","lean_link":null,"lean_decl_name":"local unram inv infl","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":"thm:trivial cohomology implies trivial Tate","label":"trivial cohomology implies trivial Tate","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["cor:up and down trivial cohomology"],"nlp_text":"Let \\(M\\) be a representation of a finite group \\(G\\), and assume that \\(M\\) has trivial cohomology. Then \\(M\\) has trivial Tate cohomology.","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.tateCohomology_of_trivialCohomology","latex_link":"sect0002.html#thm:trivial cohomology implies trivial Tate","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.tateCohomology_of_trivialCohomology","lean_decl_name":"Rep.tateCohomology_of_trivialCohomology","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/TrivialityCriterion.lean","lean4_code":"instance Rep.tateCohomology_of_trivialCohomology [Fintype G] (M : Rep R G) [M.TrivialCohomology] :\n    M.TrivialTateCohomology := by\n  constructor\n  intro H n\n  let : Fintype H := .ofFinite H\n  obtain hn | hn := Int.lt_or_le 0 n\n  · lift n to ℕ using hn.le\n    rw [Nat.cast_pos, ← Nat.exists_eq_add_one] at hn\n    obtain ⟨n, rfl⟩ := hn\n    rw [Nat.cast_add, Nat.cast_one]\n    exact .of_iso (TrivialCohomology.isZero H) ((TateCohomology.isoGroupCohomology (n + 1)).app _)\n  · classical\n    induction n, hn using Int.le_induction_down generalizing M with\n    | base =>\n      refine .of_iso ?_ (δDownResIsoTate M H.subtype_injective _)\n      refine .of_iso ?_ ((TateCohomology.isoGroupCohomology 1).app _)\n      apply isZero_of_trivialCohomology\n    | pred n hmn ih =>\n      refine .of_iso ?_ (δDownResIsoTate M H.subtype_injective _)\n      rw [sub_add_cancel]\n      apply ih","line":168,"end_line":188,"proof_status":"statement_formalized"}
{"id":"def:reciprocity iso","label":"reciprocity iso","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["thm:trivial cohomology implies trivial Tate","lem:Tate -1 aug","thm:splitting module trivial"],"nlp_text":"The theorem implies that we have isomorphisms for all \\(n\\in {\\mathbb Z}\\) (which depend on \\(\\sigma \\)): \\[ H^{n}_{\\mathrm{Tate}}(G,\\mathrm{aug}(R,G)) \\cong H^{n+1}_{\\mathrm{Tate}}(G,M) . \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.split.reciprocityIso","latex_link":"sect0002.html#def:reciprocity iso","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.split.reciprocityIso","lean_decl_name":"Rep.split.reciprocityIso","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"ClassFieldTheory/Cohomology/SplittingModule.lean","lean4_code":"def reciprocityIso (N : Rep ℤ G) (τ : H2 N) [FiniteClassFormation τ] :\n    (tateCohomology 0).obj N ≅ .of ℤ (Additive (Abelianization G)) := calc\n  (tateCohomology 0).obj N\n    ≅ (tateCohomology (-2 + 2)).obj N := .refl _\n  _ ≅ (tateCohomology (-2)).obj (trivial ℤ G ℤ) := (tateCohomologyIso τ (-2)).symm\n  _ ≅ groupHomology (trivial ℤ G ℤ) 1 := (TateCohomology.isoGroupHomology _ 1 rfl).app _\n  _ ≅ .of ℤ (Additive (Abelianization G)) :=\n    groupHomology.H1TrivialAddEquiv.toIntLinearEquiv.toModuleIso\n","line":381,"end_line":388,"proof_status":"statement_formalized"}
{"id":"lem:reciprocity formula","label":"reciprocity formula","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:reciprocity iso"],"nlp_text":"The reciprocity isomorphism for a fundamental class \\(\\sigma \\in H^2(G,M)\\) is given by \\[ \\mathrm{reciprocity}(gG' \\otimes 1) \\equiv \\sum _{x \\in G} \\sigma '(x,g) \\bmod N_G M. \\]","doc_url":null,"latex_link":"sect0002.html#lem:reciprocity formula","lean_link":null,"lean_decl_name":"reciprocity formula","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":"lem:subgroup compatibility","label":"subgroup compatibility","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:reciprocity formula"],"nlp_text":"Let \\((R,G,M)\\) be a finite class formation with a fundamental class \\(\\sigma _G\\) and let \\(S\\) be a subgroup of \\(G\\). Then \\((R, S, M \\downarrow S)\\) is a finite class formation. The restriction \\(\\sigma _S\\) of \\(\\sigma _G\\) to \\(S\\) is a fundamental class in \\(H^2(S,M)\\). Furthermore there is a commuting square \\[ \\begin{matrix} S^{\\mathrm{ab}} \\otimes R & \\cong & M^S / N_SM \\\\ \\downarrow & & \\downarrow \\\\ G^{\\mathrm{ab}} \\otimes R & \\cong & M^G / N_GM \\end{matrix} \\]","doc_url":null,"latex_link":"sect0002.html#lem:subgroup compatibility","lean_link":null,"lean_decl_name":"subgroup compatibility","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":"lem:local unramified reciprocity","label":"local unramified reciprocity","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:reciprocity formula","lem:unramified fundamental class"],"nlp_text":"Let \\(l/k\\) be a finite unramified extension of local fields and let \\(F_k\\) be the Frobenius element in \\(\\mathrm{Gal}(l/k)\\). Let \\(\\pi _k\\) be a uniformizer of \\(k\\). Then we have \\[ \\mathrm{reciprocity}(F_k)= \\pi _k, \\]","doc_url":null,"latex_link":"sect0003.html#lem:local unramified reciprocity","lean_link":null,"lean_decl_name":"local unramified reciprocity","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":"def:norm submodule","label":"norm submodule","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:subgroup compatibility"],"nlp_text":"For a finite subgroup \\(S\\) of \\(G\\), we shall call \\(N_{G/S}M^S\\) the norm submodule corresponding to \\(S\\). This is a submodule of \\(M^G\\) containing \\(N_G M\\). By the commutative diagram in 73, the image of \\(SG'/G' \\otimes R\\) under the reciprocity map is equal to \\(N_{G/S}(M^S) / N_G M\\).","doc_url":null,"latex_link":"sect0002.html#def:norm submodule","lean_link":null,"lean_decl_name":"norm submodule","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"cor:norm limitiation","label":"norm limitiation","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:norm submodule"],"nlp_text":"Let \\((R,G,M)\\) be a finite class formation. Then \\[ N_G M = N_{G/G'} (M^{G'}). \\]","doc_url":null,"latex_link":"sect0002.html#cor:norm limitiation","lean_link":null,"lean_decl_name":"norm limitiation","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":"cor:norm submodule mono","label":"norm submodule mono","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:norm submodule"],"nlp_text":"Let \\(({\\mathbb Z},G,M)\\) be a finite class formation and let \\(S_1\\) and \\(S_2\\) be two subgroups of \\(G\\). Then \\(S_1 G' \\subseteq S_2 G'\\) if and only if \\(N_{G/S_1}(M^{S_1}) \\subseteq N_{G/S_2}(M^{S_2})\\).","doc_url":null,"latex_link":"sect0002.html#cor:norm submodule mono","lean_link":null,"lean_decl_name":"norm submodule mono","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":"cor:abelian norm submodule mono","label":"abelian norm submodule mono","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:norm submodule"],"nlp_text":"Suppose \\(({\\mathbb Z},G,M)\\) is a finite class formation with \\(G\\) abelian. Let \\(S_1\\) and \\(S_2\\) be subgroups of \\(G\\). Then \\(S_1 \\subseteq S_2\\) if and only if \\(N_{G/S_1}(M^{S_1}) \\subseteq N_{G/S_2}(M^{S_2})\\)","doc_url":null,"latex_link":"sect0002.html#cor:abelian norm submodule mono","lean_link":null,"lean_decl_name":"abelian norm submodule mono","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":"thm:local norm limitation","label":"local norm limitation","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["cor:norm limitiation","thm:local fund class"],"nlp_text":"Let \\(l/k\\) be a finite Galois extension of local fields and let \\(l^{\\mathrm{ab}}\\) be the maximal subfield of \\(l\\) which is an abelian extension of \\(k\\), i.e. the fixed field of the commutator subgroup of \\(\\mathrm{Gal}(l/k)\\). Then \\(N_{l/k}(l^\\times ) = N_{l^\\mathrm{ab}/k}(l^{\\mathrm{ab}\\times })\\).","doc_url":null,"latex_link":"sect0003.html#thm:local norm limitation","lean_link":null,"lean_decl_name":"local norm limitation","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":"lem:local abelian classification","label":"local abelian classification","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["cor:norm submodule mono","thm:local fund class"],"nlp_text":"Let \\(l_1\\) and \\(l_2\\) be two abelian extensions of \\(k\\) contained in a field \\(m\\). Then \\(l_1 \\subseteq l_2\\) if and only if \\(N_{l_1/k}(l_1^\\times ) \\supseteq N_{l_2/k}(l_2^\\times )\\) and \\(l_1 = l_2\\) if and only if \\(N_{l_1/k}(l_1^\\times ) = N_{l_2/k}(l_2^\\times )\\).","doc_url":null,"latex_link":"sect0003.html#lem:local abelian classification","lean_link":null,"lean_decl_name":"local abelian classification","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":"lem:norm composite intersection","label":"norm composite intersection","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:subgroup compatibility","thm:local fund class"],"nlp_text":"Let \\(l/k\\) be a finite abelian extension and let \\(m_1\\) and \\(m_2\\) be two intermediate fields between \\(k\\) and \\(l\\) such that \\(l = m_1 m_2\\). Then we have \\[ N_{l/k}(l^\\times ) = N_{m_1/k} (m_1^\\times ) \\cap N_{m_2/k} (m_2^\\times ). \\]","doc_url":null,"latex_link":"sect0003.html#lem:norm composite intersection","lean_link":null,"lean_decl_name":"norm composite intersection","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":"lem:local image inertia","label":"local image inertia","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:subgroup compatibility","lem:local unramified norms"],"nlp_text":"Let \\(l/k\\) be a finite abelian extension of local fields and let \\(I \\subseteq \\mathrm{Gal}(l/k)\\) be the inertia subgroup. Then the image of \\(I\\) in \\(k^\\times / N(l^\\times )\\) is the subgroup \\({\\mathcal O}_k^\\times / N({\\mathcal O}_l^\\times )\\).","doc_url":null,"latex_link":"sect0003.html#lem:local image inertia","lean_link":null,"lean_decl_name":"local image inertia","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":"thm:Kronecker Weber","label":"Kronecker Weber","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:local image inertia","lem:local cyclotomic norms"],"nlp_text":"Let \\(l/{\\mathbb Q}\\) be a finite abelian extension. Then there exists a natural number \\(n\\) such that \\(l\\) is isomorphic to a subfield of \\({\\mathbb Q}(\\zeta _n)\\).","doc_url":null,"latex_link":"sect0003.html#thm:Kronecker Weber","lean_link":null,"lean_decl_name":"Kronecker Weber","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":"lem:homology 1 trivial","label":"homology 1 trivial","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:group homology"],"nlp_text":"If \\(M\\) is a trivial representation of \\(G\\) then \\(H_1(G,M) \\cong G^{\\mathrm{ab}} \\otimes {\\mathbb Z}M\\). The isomorphism takes a 1-cycle \\(\\mathrm{single}(x,m)\\) to \\(xG' \\otimes m\\) for \\(x \\in G\\) and \\(m \\in M\\). (This result is already in Mathlib).","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupHomology.H1AddEquivOfIsTrivial","latex_link":"sect0002.html#lem:homology 1 trivial","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupHomology.H1AddEquivOfIsTrivial","lean_decl_name":"groupHomology.H1AddEquivOfIsTrivial","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/RepresentationTheory/Homological/GroupHomology/LowDegree.lean","lean4_code":"def H1AddEquivOfIsTrivial :\n    H1 A ≃+ (Additive <| Abelianization G) ⊗[ℤ] A :=\n  LinearEquiv.toAddEquiv <| LinearEquiv.ofLinear\n    (H1ToTensorOfIsTrivial A) (lift <| mkH1OfIsTrivial A)\n    (ext <| LinearMap.toAddMonoidHom_injective <| by\n      ext g a\n      simp [TensorProduct.mk_apply, TensorProduct.lift.tmul, mkH1OfIsTrivial_apply,\n        H1ToTensorOfIsTrivial_H1π_single g a])\n    (LinearMap.toAddMonoidHom_injective <|\n      (H1Iso A).symm.toLinearEquiv.toAddEquiv.comp_left_injective <|\n      QuotientAddGroup.addMonoidHom_ext _ <|\n      (cycles₁IsoOfIsTrivial A).symm.toLinearEquiv.toAddEquiv.comp_left_injective <| by\n        ext\n        simp only [H1ToTensorOfIsTrivial, Iso.toLinearEquiv, AddMonoidHom.coe_comp,\n          LinearMap.toAddMonoidHom_coe, LinearMap.coe_comp, AddMonoidHom.coe_toIntLinearMap]\n        change TensorProduct.lift _ (QuotientAddGroup.lift _ _ _ ((H1Iso A).hom _)) = _\n        simpa [AddSubgroup.subtype, cycles₁IsoOfIsTrivial_inv_apply (A := A),\n          -π_comp_H1Iso_inv_apply] using (π_comp_H1Iso_inv_apply A _).symm)\n\n","line":976,"end_line":994,"proof_status":"formalized"}
{"id":"lem:Tate -1 aug","label":"Tate -1 aug","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:homology 1 trivial","lem:Tate cohomology is cohomology or homology","def:augmentation module"],"nlp_text":"Let \\(S\\) be a subgroup of a finite group \\(G\\). Then there is an isomorphism of \\(R\\)-modules \\[ S^{\\mathrm{ab}} \\otimes _{{\\mathbb Z}} R \\cong H^{-1}_{\\mathrm{Tate}}(S,\\mathrm{aug}(R,G)), \\]","doc_url":null,"latex_link":"sect0002.html#lem:Tate -1 aug","lean_link":null,"lean_decl_name":"Tate -1 aug","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":"lem:ind₁ trivial homology","label":"ind₁ trivial homology","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:trivial cohomology","lem:Shapiro","def:induced"],"nlp_text":"The representation \\(\\mathrm{ind}_1(G,A)\\) has trivial homology.","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.trivialHomology_ind₁","latex_link":"sect0002.html#lem:ind₁ trivial homology","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.trivialHomology_ind₁","lean_decl_name":"Rep.trivialHomology_ind₁","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/IndCoind/TrivialCohomology.lean","lean4_code":"instance trivialHomology_ind₁ (A : ModuleCat R) : TrivialHomology ((ind₁ G).obj A) :=\n  .of_iso (ind₁AsFinsuppIso _).symm","line":102,"end_line":103,"proof_status":"formalized"}
{"id":"lem:induced trivial Tate","label":"induced trivial Tate","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:ind₁ trivial homology","def:ind₁ to coind₁","lem:Tate cohomology is cohomology or homology","lem:coind₁ trivial cohomology"],"nlp_text":"If the group \\(G\\) is finite then \\(\\mathrm{ind}_1(G,A)\\) and \\(\\mathrm{coind}_1(G,A)\\) have trivial Tate cohomology.","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.trivialTateCohomology_coind₁","latex_link":"sect0002.html#lem:induced trivial Tate","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.trivialTateCohomology_coind₁","lean_decl_name":"Rep.trivialTateCohomology_coind₁","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/IndCoind/TrivialCohomology.lean","lean4_code":"instance trivialTateCohomology_coind₁ : TrivialTateCohomology ((coind₁ G).obj A) :=\n  .of_iso (coind₁AsPiIso A).symm","line":193,"end_line":194,"proof_status":"formalized"}
{"id":"lem:induced' trivial Tate","label":"induced' trivial Tate","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:induced trivial Tate","lem:ind₁' iso ind₁","lem:coind₁' iso coind₁"],"nlp_text":"If \\(M\\) is a representation of a finite group \\(G\\) then the representations \\(\\mathrm{ind}_1'(M)\\) and \\(\\mathrm{coind}_1'(M)\\) have trivial Tate cohomology.","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.trivialTateCohomology_coind₁","latex_link":"sect0002.html#lem:induced' trivial Tate","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.trivialTateCohomology_coind₁","lean_decl_name":"Rep.trivialTateCohomology_coind₁","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/IndCoind/TrivialCohomology.lean","lean4_code":"instance trivialTateCohomology_coind₁ : TrivialTateCohomology ((coind₁ G).obj A) :=\n  .of_iso (coind₁AsPiIso A).symm","line":193,"end_line":194,"proof_status":"formalized"}
{"id":"thm:additive field trivial","label":"additive field trivial","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:induced trivial Tate"],"nlp_text":"Let \\(l/k\\) be a finite Galois extension of fields. Then there is an isomorphism of \\(\\mathrm{Gal}(l/k)\\)-representations: \\[ l \\cong \\mathrm{ind}_1(k). \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.iso_ind₁","latex_link":"sect0003.html#thm:additive field trivial","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.iso_ind₁","lean_decl_name":"Rep.iso_ind₁","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/IndCoind/Finite.lean","lean4_code":"/-- For a finite Galois extension `L/K`, the isomorphism between `ind₁` of `K`\nand `L` in the category of `(L ≃ₐ[K] L)`-representations. -/\nnoncomputable def iso_ind₁ :\n    (Rep.ind₁ (L ≃ₐ[K] L)).obj (.of K K) ≅ .of (AlgEquiv.toLinearMapHom K L) := by\n  classical\n  refine (Rep.ind₁AsFinsuppIso (G := (L ≃ₐ[K] L)) (.of K K)).symm ≪≫\n    mkIso _ _ (LinearEquiv.toModuleIso\n      ((IsGalois.normalBasis K L).reindex (Equiv.inv (L ≃ₐ[K] L))).repr.symm) ?_\n  intro x f\n  simp only [LinearEquiv.toModuleIso_hom, Module.Basis.coe_repr_symm, Module.Basis.coe_reindex,\n    Equiv.inv_symm, Equiv.inv_apply, ModuleCat.hom_ofHom, AlgEquiv.toLinearMapHom]\n  rw [Finsupp.linearCombination_apply, Finsupp.linearCombination_apply,\n    Finsupp.sum_fintype _ _ fun i ↦ by exact zero_smul K _,\n    Finsupp.sum_fintype _ _ fun i ↦ by exact zero_smul K _]\n  simp only [Function.comp_apply, map_sum, map_smul]\n  apply Fintype.sum_equiv (.mulRight x)\n  simp [IsGalois.normalBasis_apply _⁻¹, IsGalois.normalBasis_apply (_ * _)]","line":600,"end_line":616,"proof_status":"formalized"}
{"id":"lem:unramified additive trivial","label":"unramified additive trivial","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["thm:additive field trivial"],"nlp_text":"If \\(l/k\\) is unramified then there is a normal basis for \\({\\mathcal O}_l\\) over \\({\\mathcal O}_k\\). Hence there is an isomorphism of Galois representations \\({\\mathcal O}_l \\cong \\mathrm{ind}_1 {\\mathcal O}_k\\). In particular \\({\\mathcal O}_l\\) has trivial cohomology.","doc_url":null,"latex_link":"sect0003.html#lem:unramified additive trivial","lean_link":null,"lean_decl_name":"unramified additive trivial","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":"cor:H1 H2 cyclic idele class","label":"H1 H2 cyclic idele class","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:herbrand idele class group","thm:first inequality"],"nlp_text":"If \\(l/k\\) is cyclic then \\(|H^2(l/k, \\mathrm{Cl}_{l})| = [l:k]\\) and \\(H^1(l/k, \\mathrm{Cl}_{l}) = 0\\).","doc_url":null,"latex_link":"sect0004.html#cor:H1 H2 cyclic idele class","lean_link":null,"lean_decl_name":"H1 H2 cyclic idele class","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":"def:ind₁ to coind₁","label":"ind₁ to coind₁","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:induced"],"nlp_text":"There is a morphism of representations \\(\\mathrm{ind}_1(G,A) \\to \\mathrm{coind}_1(G,A)\\), which takes a finitely supported function \\(f : G \\to _0 A\\) to the function \\(f\\). If the group \\(G\\) is finite then this map is an isomorphism. More precisely, this is an isomorphism of functors \\(\\mathrm{ind}_1(G,-) \\cong \\mathrm{coind}_1(G,-)\\).","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.ind₁_obj_iso_coind₁_obj","latex_link":"sect0002.html#def:ind₁ to coind₁","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.ind₁_obj_iso_coind₁_obj","lean_decl_name":"Rep.ind₁_obj_iso_coind₁_obj","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/IndCoind/Finite.lean","lean4_code":"abbrev ind₁_obj_iso_coind₁_obj [Finite G] : (ind₁ G).obj A ≅ (coind₁ G).obj A :=\n  indCoindIso _","line":566,"end_line":567,"proof_status":"formalized"}
{"id":"lem:ind₁' iso ind₁","label":"ind₁' iso ind₁","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:ind'","def:induced"],"nlp_text":"The representations \\(\\mathrm{ind}_1'(M)\\) and \\(\\mathrm{ind}_1(G,M)\\) are isomorphic; more precisely the functors \\(\\mathrm{ind}_1'\\) and \\(\\mathrm{ind}_1(G,-) \\circ \\mathbf{forget}\\) are isomorphic.","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Representation.ind₁'_lequiv","latex_link":"sect0002.html#lem:ind₁' iso ind₁","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Representation.ind₁'_lequiv","lean_decl_name":"Representation.ind₁'_lequiv","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/IndCoind/Finite.lean","lean4_code":"/--\nThe linear automorphism of `G →₀ V`, which gives an isomorphism\nbetween `ind₁' ρ` and `ind₁ R G V`.\n-/\n@[simps!] def ind₁'_lequiv : (G →₀ V) ≃ₗ[R] Ind₁V R G V where\n  toLinearMap := ind₁'_lmap ρ\n  invFun := ind₁'_invlmap ρ\n  left_inv f := by\n    rw [LinearMap.toFun_eq_coe]\n    induction f using Finsupp.induction_linear\n    · simp\n    · rename_i f g h1 h2\n      rw [map_add, map_add, h1, h2]\n    · rename_i g v\n      simp [ind₁'_invlmap, Submodule.quotEquivOfEqBot, Coinvariants.mk]\n      change (TensorProduct.lift ρ.ind₁'_invlmap_aux)\n        (LinearMap.id (R := R) ((fun₀ | g => (1 : R)) ⊗ₜ[R] (ρ g) v)) = fun₀ | g => v\n      simp [ind₁'_invlmap_aux]\n  right_inv f := by\n    rw [LinearMap.toFun_eq_coe]\n    induction f using Submodule.Quotient.induction_on with | H f\n    induction f using TensorProduct.induction_on with\n    | zero => simp\n    | add f g h1 h2 =>\n      rw [← Submodule.mkQ_apply, map_add, map_add, map_add,\n        Submodule.mkQ_apply, Submodule.mkQ_apply, h1, h2]\n    | tmul x y =>\n    simp only [ind₁'_invlmap, Submodule.quotEquivOfEqBot,\n      LinearEquiv.ofLinear_toLinearMap, LinearMap.coe_comp, Function.comp_apply]\n    change ρ.ind₁'_lmap (TensorProduct.lift ρ.ind₁'_invlmap_aux (LinearMap.id (R := R) (x ⊗ₜ[R] y)))\n      = Submodule.Quotient.mk (x ⊗ₜ[R] y)\n    simp only [LinearMap.id_coe, id_eq, TensorProduct.lift.tmul]\n    induction x using Finsupp.induction_linear with\n    | zero => simp\n    | add f g h1 h2 =>\n      rw [map_add, LinearMap.add_apply, map_add, h1, h2, TensorProduct.add_tmul]\n      rfl\n    | single g r =>\n      simp [ind₁'_invlmap_aux, Coinvariants.mk]\n      rw [← map_smul, ← Submodule.mkQ_apply]\n      congr\n      rw [TensorProduct.smul_tmul', Finsupp.smul_single, smul_eq_mul, mul_one]","line":270,"end_line":311,"proof_status":"formalized"}
{"id":"lem:d comp norm","label":"d comp norm","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:norm comm"],"nlp_text":"The composition \\(N_G \\circ d_0\\) is zero.","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.d_comp_tateNorm","latex_link":"sect0002.html#lem:d comp norm","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.d_comp_tateNorm","lean_decl_name":"groupCohomology.d_comp_tateNorm","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"ClassFieldTheory/Cohomology/TateCohomology.lean","lean4_code":"lemma d_comp_tateNorm (M : Rep R G) : (inhomogeneousChains M).d 1 0 ≫ tateNorm M = 0 := by\n  simp only [ChainComplex.of_x, CochainComplex.of_x, tateNorm, ← Category.assoc,\n    Preadditive.IsIso.comp_right_eq_zero]\n  simp [← comp_d₁₀_eq, -norm_hom]\n","line":65,"end_line":68,"proof_status":"formalized"}
{"id":"def:Tate cohomology","label":"Tate cohomology","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:d comp norm","lem:norm comp d","lem:norm naturality"],"nlp_text":"Recall that we have a cochain complex \\(C^n(G,M)\\), indexed by \\(n \\in {\\mathbb N}\\), whose zeroth term may be identified with \\(M\\). We also have a chain complex \\(C_n(G,M)\\) whose zeroth term may be identified with \\(M\\). By 17 and 18, we may glue these together with the map \\(N_G : M \\to M\\) to obtain a cochain complex indexed by \\({\\mathbb Z}\\): \\[ \\cdots \\to C_2(G,M) \\to C_1(G,M) \\stackrel{d_0}\\to C_0(G,M) \\stackrel{N_G}\\to C^0(G,M) \\stackrel{d^0}\\to C^1(G,M) \\to C^2(G,M) \\to \\cdots \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.tateComplex","latex_link":"sect0002.html#def:Tate cohomology","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.tateComplex","lean_decl_name":"groupCohomology.tateComplex","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"ClassFieldTheory/Cohomology/TateCohomology.lean","lean4_code":"/-- The Tate complex defined by connecting inhomogeneous chains and cochains with the Tate norm. -/\n@[simps! X]\ndef tateComplex (M : Rep R G) : CochainComplex (ModuleCat R) ℤ :=\n  CochainComplex.ConnectData.cochainComplex (tateComplexConnectData M)\n","line":78,"end_line":81,"proof_status":"formalized"}
{"id":"def:Tate long exact sequence","label":"Tate long exact sequence","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:Tate cohomology","lem:cochainsFunctor exact"],"nlp_text":"Since the functors \\(C^\\bullet (G,-)\\) and \\(C_\\bullet (G,-)\\) are both exact, it follows that \\(C^\\bullet _{\\mathrm{Tate}}(G,-)\\) is an exact functor. Hence, given any short exact sequence in \\(\\mathbf{Rep}(R,G)\\): \\[ 0 \\to A \\to B \\to C \\to 0, \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.TateCohomology.preservesFiniteLimits_tateComplexFunctor","latex_link":"sect0002.html#def:Tate long exact sequence","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.TateCohomology.preservesFiniteLimits_tateComplexFunctor","lean_decl_name":"groupCohomology.TateCohomology.preservesFiniteLimits_tateComplexFunctor","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/TateCohomology.lean","lean4_code":"instance preservesFiniteLimits_tateComplexFunctor :\n    PreservesFiniteLimits (tateComplexFunctor (R := R) (G := G)) :=\n  (((tateComplexFunctor (R := R) (G := G)).exact_tfae.out 0 3 rfl rfl).mp\n    fun _ ↦ map_tateComplexFunctor_shortExact).1","line":168,"end_line":171,"proof_status":"formalized"}
{"id":"lem:Tate cohomology is cohomology or homology","label":"Tate cohomology is cohomology or homology","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:Tate cohomology"],"nlp_text":"Let \\(G\\) be a finite group and \\(M\\) a representation of \\(G\\). The zeroth Tate cohomology \\(H^0_{\\mathrm{Tate}}(G,M)\\) is isomorphic to \\(M^G / N_G(M)\\). In particular if \\(M\\) is a trivial representation of \\(G\\) then \\(H^0_{\\mathrm{Tate}}(G,M) \\cong M / |G|M\\). For \\(n {\\gt}0 \\) we have (an isomorphism of functors in the variable \\(M\\)) \\[ H^{n}_{\\mathrm{Tate}}(G,M) \\cong H^{n}(G,M). \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.TateCohomology.isoGroupCohomology","latex_link":"sect0002.html#lem:Tate cohomology is cohomology or homology","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.TateCohomology.isoGroupCohomology","lean_decl_name":"groupCohomology.TateCohomology.isoGroupCohomology","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"ClassFieldTheory/Cohomology/TateCohomology.lean","lean4_code":"/-- The isomorphism between the `n`-th Tate cohomology and `n`-th group cohomology for `n : ℕ`\nnon-zero. -/\ndef isoGroupCohomology (n : ℕ) [NeZero n] :\n    tateCohomology.{u} n ≅ groupCohomology.functor.{u} R G n :=\n  NatIso.ofComponents (fun M ↦ (tateComplexConnectData M).homologyIsoPos _ _ rfl) fun {X Y} f ↦ by\n    simp [tateCohomology, CochainComplex.ConnectData.homologyMap_map_of_eq_succ (n := n)]\n","line":202,"end_line":207,"proof_status":"formalized"}
{"id":"def:herbrand quotient","label":"herbrand quotient","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:Tate cohomology"],"nlp_text":"Let \\(G\\) be a finite cyclic group and \\(M\\) a representation of \\(G\\). Recall that there are isomorphisms \\(H^n_{\\mathrm{Tate}}(G,M) \\cong H^{n+2}_{\\mathrm{Tate}}(G,M)\\). We define the Herbrand quotient of \\(M\\) to be \\[ h(G,M) = \\frac{|H^0_{\\mathrm{Tate}}(G,M)|}{|H^1_{\\mathrm{Tate}}(G,M)|}. \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Representation.herbrandQuotient","latex_link":"sect0002.html#def:herbrand quotient","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Representation.herbrandQuotient","lean_decl_name":"Representation.herbrandQuotient","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"./obsolete/HerbrandQuotient_old.lean","lean4_code":"def herbrandQuotient : ℚ :=\n  let _ := ρ.herbrandModule\n  herbrandModule.quotient R A\n\n","line":142,"end_line":145,"proof_status":"formalized"}
{"id":"lem:herbrand finite","label":"herbrand finite","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:Tate cohomology is cohomology or homology"],"nlp_text":"If \\(M\\) is finite then \\(h(G,M)=1\\).","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Representation.herbrandQuotient_of_finite","latex_link":"sect0002.html#lem:herbrand finite","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Representation.herbrandQuotient_of_finite","lean_decl_name":"Representation.herbrandQuotient_of_finite","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"ClassFieldTheory/Cohomology/FiniteCyclic/HerbrandQuotient/Finite.lean","lean4_code":"lemma herbrandQuotient_of_finite [Finite A] : ρ.herbrandQuotient = 1 := by\n  /-\n  Consider the linear maps `oneSubGen` and `norm : M → M` defined to be multiplication by `1 - gen G`\n  and norm respectively. The kernel of `oneSubGen` is the submodule of `G`-invariants, and the\n  cokernel of `oneSubGen` is the quotient module of coinvariants. We therefore have (for Tate\n  groups):\n\n    `H⁰(G, M) ≅ ker oneSubGen ⧸ range norm` and `H⁻¹(G, M) ≅ ker norm ⧸ range oneSubGen`.\n\n  The result follows because `|ker f| * |range f| = |M|` for any function `f : M → M`.\n  -/\n  simp only [herbrandQuotient, Submodule.natCard_quotient, tateB0_le_tateZ0,\n    Submodule.natCard_submoduleOf, tateBNeg1_le_tateZNeg1, div_div_div_eq]\n  norm_cast\n  rw [mul_comm]\n  simp [tateZ0, tateZNeg1, tateB0, tateBNeg1, -LinearMap.mem_range, -LinearMap.mem_ker]\n","line":16,"end_line":31,"proof_status":"formalized"}
{"id":"lem:cochainsFunctor exact","label":"cochainsFunctor exact","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The functor taking \\(M\\) to \\(C^\\bullet (G,M)\\) is exact. I.e. if \\(0 \\to A \\to B \\to C \\to 0\\) is a short exact sequence in \\(\\mathbf{Rep}(R,G)\\). Then the corresponding sequence of cochain complexes is exact: \\[ 0 \\to C^\\bullet (G,A) \\to C^\\bullet (G,B) \\to C^\\bullet (G,C) \\to 0. \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.map_cochainsFunctor_shortExact","latex_link":"sect0002.html#lem:cochainsFunctor exact","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.map_cochainsFunctor_shortExact","lean_decl_name":"groupCohomology.map_cochainsFunctor_shortExact","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/RepresentationTheory/Homological/GroupCohomology/LongExactSequence.lean","lean4_code":"lemma map_cochainsFunctor_shortExact :\n    ShortExact (X.map (cochainsFunctor k G)) :=\n  HomologicalComplex.shortExact_of_degreewise_shortExact _ fun i => {\n    exact := by\n      have : LinearMap.range X.f.hom.hom = LinearMap.ker X.g.hom.hom :=\n        (hX.exact.map (forget₂ (Rep k G) (ModuleCat k))).moduleCat_range_eq_ker\n      simp [moduleCat_exact_iff_range_eq_ker, LinearMap.range_compLeft,\n        LinearMap.ker_compLeft, this]\n    mono_f := letI := hX.mono_f; cochainsMap_id_f_map_mono X.f i\n    epi_g := letI := hX.epi_g; cochainsMap_id_f_map_epi X.g i }\n\nopen HomologicalComplex.HomologySequence\n\n/-- The short complex `Hⁱ(G, X₃) ⟶ Hʲ(G, X₁) ⟶ Hʲ(G, X₂)` associated to an exact\nsequence of representations `0 ⟶ X₁ ⟶ X₂ ⟶ X₃ ⟶ 0`. -/\nnoncomputable abbrev mapShortComplex₁ {i j : ℕ} (hij : i + 1 = j) :=\n  (snakeInput (map_cochainsFunctor_shortExact hX) _ _ hij).L₂'\n\nvariable (X) in\n/-- The short complex `Hⁱ(G, X₁) ⟶ Hⁱ(G, X₂) ⟶ Hⁱ(G, X₃)` associated to a short complex of\nrepresentations `X₁ ⟶ X₂ ⟶ X₃`. -/\nnoncomputable abbrev mapShortComplex₂ (i : ℕ) := X.map (functor k G i)\n\n/-- The short complex `Hⁱ(G, X₂) ⟶ Hⁱ(G, X₃) ⟶ Hʲ(G, X₁)`. -/\nnoncomputable abbrev mapShortComplex₃ {i j : ℕ} (hij : i + 1 = j) :=\n  (snakeInput (map_cochainsFunctor_shortExact hX) _ _ hij).L₁'\n\n/-- Exactness of `Hⁱ(G, X₃) ⟶ Hʲ(G, X₁) ⟶ Hʲ(G, X₂)`. -/\n","line":39,"end_line":66,"proof_status":"formalized"}
{"id":"def:cohomology long exact sequence","label":"cohomology long exact sequence","background_color":"#B0ECA3","border_color":"darkgreen","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:cochainsFunctor exact"],"nlp_text":"Given a short exact sequence \\(0 \\to A \\stackrel{f}\\to B \\stackrel{g}\\to C \\to 0\\) in \\(\\mathbf{Rep}(R,G)\\), the corresponding sequence of cochain complexes is exact: \\(0 \\to C^n(G,A) \\to C^n(G,B) \\to C^n(G,C) \\to 0\\). This implies that there exist “connecting homomorphisms” \\(\\delta : H^n(G,C) \\to H^{n+1}(G,A)\\), such that the following is a long exact sequence: \\[ 0 \\to H^0(G,A) \\stackrel{H^0(f)}\\to H^0(G,B) \\stackrel{H^0(g)}\\to H^0(G,C) \\stackrel{\\delta }\\to H^1(G,A) \\stackrel{H^1(f)}\\to H^1(G,B) \\stackrel{H^1(g)}\\to H^1(G,C) \\stackrel{\\delta }\\to \\cdots . \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.δ","latex_link":"sect0002.html#def:cohomology long exact sequence","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.δ","lean_decl_name":"groupCohomology.δ","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"./obsolete/HerbrandQuotient_old.lean","lean4_code":"@[simps] def δ₀ : A →ₗ[R] A where\n  toFun a := d₀ R • a\n  map_add' := sorry\n  map_smul' := sorry\n\n","line":53,"end_line":57,"proof_status":"formalized"}
{"id":"lem:inflation restriction naturality","label":"inflation restriction naturality","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:cohomology long exact sequence","def:inflation map","def:restriction map"],"nlp_text":"Let \\(S\\) be a subgroup of \\(G\\) and suppose we have a short exact sequence \\(0 \\to A \\to B \\to C \\to 0\\) in \\(\\mathbf{Rep}(R,G)\\). Then the sequence the sequence \\(0 \\to A \\downarrow S \\to B \\downarrow S \\to C \\downarrow S \\to 0\\) is exact in \\(\\mathbf{Rep}(R,S)\\). The following diagram commutes, where the rows are the long exact sequences for \\(0 \\to A \\to B \\to C \\to 0\\) and for its restriction to \\(S\\), and the vertical maps are restriction. \\[ \\begin{array}{ccccccc} H^n(G,A) & \\to & H^n(G,B) & \\to & H^n(G,C)& \\to & H^{n+1}(G,A)\\\\ \\downarrow & & \\downarrow & & \\downarrow & & \\downarrow \\\\ H^n(S,A) & \\to & H^n(S,B) & \\to & H^n(S,C)& \\to & H^{n+1}(S,A) \\end{array} \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.rest","latex_link":"sect0002.html#lem:inflation restriction naturality","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.rest","lean_decl_name":"groupCohomology.rest","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"ClassFieldTheory/Cohomology/Functors/Restriction.lean","lean4_code":"/--\nThe restriction map `Hⁿ(G,M) ⟶ Hⁿ(H,M)`, defined as a morphism of functors\n-/\ndef rest (n : ℕ) : functor R G n ⟶ Rep.res φ ⋙ functor R S n  where\n  app M               := map φ (𝟙 (M ↓ φ)) n\n  naturality M₁ M₂ f  := by\n    simp only [functor_obj, Functor.comp_obj, functor_map, Functor.comp_map]\n    rw [←map_comp, ←map_comp]\n    congr 1\n","line":169,"end_line":177,"proof_status":"formalized"}
{"id":"thm:inflation restriction sequence","label":"inflation restriction sequence","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:inflation restriction naturality","def:up","cor:coind₁' invariants trivial cohomology"],"nlp_text":"Let \\(S\\) be a normal subgroup of a group \\(G\\) and let \\(n\\) be a positive integer. Assume that for all natural numbers \\(0 {\\lt}i {\\lt} n\\) we have \\(H^{i}(S,M) \\cong 0\\). Then the following sequence is exact: \\[ 0 \\to H^{n}(G/S, M^S) \\to H^{n}(G,M) \\to H^{n}(S,M), \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.inflationRestriction","latex_link":"sect0002.html#thm:inflation restriction sequence","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.inflationRestriction","lean_decl_name":"groupCohomology.inflationRestriction","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":"ClassFieldTheory","source_path":"ClassFieldTheory/Cohomology/Functors/InflationRestriction.lean","lean4_code":"def inflationRestriction (n : ℕ) (M : Rep R G) : ShortComplex (ModuleCat R) where\n  X₁ := groupCohomology (M ↑ surj) (n + 1)\n  X₂ := groupCohomology M (n + 1)\n  X₃ := groupCohomology (M ↓ φ.ker.subtype) (n + 1)\n  f := (infl surj (n + 1)).app M\n  g := (rest φ.ker.subtype (n + 1)).app M\n  zero := sorry\n","line":35,"end_line":41,"proof_status":"statement_formalized"}
{"id":"lem:unramified units trivial","label":"unramified units trivial","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:finite field trivial","lem:serre_approx"],"nlp_text":"If \\(l/k\\) is unramified then \\({\\mathcal O}_l^\\times \\) has trivial group cohomology.","doc_url":null,"latex_link":"sect0003.html#lem:unramified units trivial","lean_link":null,"lean_decl_name":"unramified units trivial","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":"lem:herbrand local units","label":"herbrand local units","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:herbrand ses","lem:serre_cor","lem:herbrand finite"],"nlp_text":"If \\(l/k\\) is a cyclic extension then \\(h(l/k, {\\mathcal O}_l^\\times ) = 1\\).","doc_url":null,"latex_link":"sect0003.html#lem:herbrand local units","lean_link":null,"lean_decl_name":"herbrand local units","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":"lem:herbrand log lattice","label":"herbrand log lattice","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:herbrand ses","lem:herbrand L_S","lem:herbrand finite"],"nlp_text":"Let \\(l/k\\) be cyclic and let \\(M\\) be any Galois-invariant lattice in \\(V_S\\). Then \\(h(l/k,M) = \\prod _{v \\in S} |D_{\\hat v}|\\)","doc_url":null,"latex_link":"sect0004.html#lem:herbrand log lattice","lean_link":null,"lean_decl_name":"herbrand log lattice","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":"lem:herbrand local l*","label":"herbrand local l*","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:herbrand local units"],"nlp_text":"If \\(l/k\\) is a cyclic extension of local fields then \\(h(l/k, l^\\times )= [l:k]\\).","doc_url":null,"latex_link":"sect0003.html#lem:herbrand local l*","lean_link":null,"lean_decl_name":"herbrand local l*","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":"lem:herbrand S-units","label":"herbrand S-units","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:herbrand log lattice","thm:Dirichlet unit theorem"],"nlp_text":"Let \\(l/k\\) be a cyclic extension. Then \\[ h(l/k,{\\mathcal O}_{l,S}^\\times ) = \\frac{\\prod _{v\\in S} |D_{\\hat v}|}{[l:k]}. \\]","doc_url":null,"latex_link":"sect0004.html#lem:herbrand S-units","lean_link":null,"lean_decl_name":"herbrand S-units","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":"lem:local H2 l*","label":"local H2 l*","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:herbrand local l*","thm:hilbert 90"],"nlp_text":"If \\(l/k\\) is a cyclic extension of local fields then \\(|H^2(l/k,l^\\times )| = [l:k]\\).","doc_url":null,"latex_link":"sect0003.html#lem:local H2 l*","lean_link":null,"lean_decl_name":"local H2 l*","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":"lem:herbrand S-ideles","label":"herbrand S-ideles","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:herbrand local l*","lem:cohomology S-ideles decomp"],"nlp_text":"If \\(l/k\\) is a cyclic extension then we have \\[ h(l/k, {\\mathbb A}_{S,l} ^\\times ) = \\prod _{v \\in S} |D_{\\hat v}|. \\]","doc_url":null,"latex_link":"sect0004.html#lem:herbrand S-ideles","lean_link":null,"lean_decl_name":"herbrand S-ideles","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":"lem:herbrand idele class group","label":"herbrand idele class group","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:herbrand S-units","lem:herbrand S-ideles"],"nlp_text":"If \\(l/k\\) is cyclic then \\(h(l/k,\\mathrm{Cl}_l) = [l:k]\\).","doc_url":null,"latex_link":"sect0004.html#lem:herbrand idele class group","lean_link":null,"lean_decl_name":"herbrand idele class group","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":"lem:serre_cor","label":"serre_cor","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(l/k\\) be a finite Galois extension of local fields. There is a Galois-invariant open subgroup \\(M \\subseteq {\\mathcal O}_l^\\times \\) such that \\(H^{q+1}(l/k,M)=0\\) for all naturals \\(q\\).","doc_url":null,"latex_link":"sect0003.html#lem:serre_cor","lean_link":null,"lean_decl_name":"serre_cor","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":"thm:local fund class","label":"local fund class","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:local H2 upper bound","lem:unramified fundamental class","def:local inv"],"nlp_text":"For every finite Galois extension \\(l/k\\) of local fields, \\(({\\mathbb Z},\\mathrm{Gal}(l/k),l^\\times )\\) is a finite class formation. The element \\(\\sigma _{l/k}\\) defined above is a fundamental class. Corresponding the \\(\\sigma _{l/k}\\) there is a reciprocity isomorphism \\[ \\mathrm{Gal}(l/k) \\cong k^\\times / N(l^\\times ). \\]","doc_url":null,"latex_link":"sect0003.html#thm:local fund class","lean_link":null,"lean_decl_name":"local fund class","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":"lem:local cyclotomic norms","label":"local cyclotomic norms","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["thm:local fund class","lem:local isomorphism"],"nlp_text":"Let \\(l = {\\mathbb Q}_p(\\zeta )\\) where \\(\\zeta \\) is a primitive \\(p^n\\)-th root of unity for some \\(n {\\gt} 0\\). Then \\(N(l^\\times ) = p^{\\mathbb Z}\\times (1+p^n {\\mathbb Z}_p)\\).","doc_url":null,"latex_link":"sect0003.html#lem:local cyclotomic norms","lean_link":null,"lean_decl_name":"local cyclotomic norms","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":"lem:H0 idele class group finite","label":"H0 idele class group finite","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(H^0_{\\mathrm{Tate}}(l/k,\\mathrm{Cl}_l)\\) is finite.","doc_url":null,"latex_link":"sect0004.html#lem:H0 idele class group finite","lean_link":null,"lean_decl_name":"H0 idele class group finite","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":"def:L-function","label":"L-function","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:H0 idele class group finite"],"nlp_text":"Let \\(\\chi : H^0_{\\mathrm{Tate}}(l/k,\\mathrm{Cl}_l) \\to {\\mathbb C}^\\times \\) be a character. For a non-zero ideal \\(I\\) of \\({\\mathcal O}_{k,S}\\), we shall write \\(\\chi (I)\\) in place of \\(\\chi (\\iota (I))\\). We define the \\(L\\)-function of \\(\\chi \\) by \\[ L(s,\\chi ) = \\sum _{I} \\chi (I) \\cdot N(I)^{-s} = \\prod _{P} \\frac{1}{1-\\chi (P) \\cdot N(P)^{-s}}. \\]","doc_url":null,"latex_link":"sect0004.html#def:L-function","lean_link":null,"lean_decl_name":"L-function","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem:L-function bound","label":"L-function bound","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:L-function"],"nlp_text":"If \\(\\chi \\) is a non-trivial character then \\(L(s,\\chi )\\) is bounded on the interval \\((1,2)\\).","doc_url":null,"latex_link":"sect0004.html#lem:L-function bound","lean_link":null,"lean_decl_name":"L-function bound","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":"lem:density bound","label":"density bound","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:L-function bound"],"nlp_text":"Let \\(M\\) be a set of primes of \\({\\mathcal O}_S\\) whose image in \\(H^0_{\\mathrm{Tate}}(l/k,\\mathrm{Cl}_l)\\) is zero. There exists a real number \\(c\\) depending only on the fields \\(k\\) and \\(l\\), such that for all \\(s {\\gt} 1\\) we have: \\[ \\sum _{p \\in M} N(P)^{-s} \\le \\frac{1}{|H^0_{\\mathrm{Tate}}(l/k,\\mathrm{Cl}_l)|} \\log \\left(\\frac{1}{s-1}\\right) + c. \\]","doc_url":null,"latex_link":"sect0004.html#lem:density bound","lean_link":null,"lean_decl_name":"density bound","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":"thm:first inequality","label":"first inequality","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:Dirichlet density split","lem:density bound"],"nlp_text":"For any finite Galois extension \\(l/k\\) be have \\[ |H^0_{\\mathrm{Tate}}(l/k, \\mathrm{Cl}_{l}) | \\le [l : k]. \\]","doc_url":null,"latex_link":"sect0004.html#thm:first inequality","lean_link":null,"lean_decl_name":"first inequality","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":"def:inflation map","label":"inflation map","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:inflation restriction functors"],"nlp_text":"If \\(S\\) is a normal subgroup of \\(G\\), then we write \\(H^n(G/S,M^S)\\) for the cohomology groups of the representation \\(M^S\\) of \\(G/S\\). If \\(f : (G/S)^n \\to M^S\\) is an element of \\(C^n(G/S,M^S)\\), then we may “inflate” \\(f\\) to a function \\(G^n \\to M\\). This inflation process defines a map of cochain complexes \\(C^\\bullet (G/S,M^S) \\to C^\\bullet (G,M)\\), and hence a map of cohomology groups, called the inflation map: \\[ \\mathrm{infl}: H^n(G/S,M^H) \\to H^n(G,M). \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.cochain_infl","latex_link":"sect0002.html#def:inflation map","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.cochain_infl","lean_decl_name":"groupCohomology.cochain_infl","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"ClassFieldTheory/Cohomology/Functors/Inflation.lean","lean4_code":"noncomputable def cochain_infl :\n    quotientToInvariantsFunctor' surj ⋙ cochainsFunctor R Q ⟶ cochainsFunctor R G where\n  app M := cochainsMap φ <| res_quotientToInvariantsFunctor'_ι surj M\n  naturality _ _ _ := rfl\n","line":87,"end_line":90,"proof_status":"formalized"}
{"id":"def:coind₁'","label":"coind₁'","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(G\\) be a group and \\(M\\) a representation of \\(G\\) over a commutative ring \\(R\\). There is a representation \\(\\mathrm{coind}_1' (M)\\) on the \\(R\\)-module of functions \\(G \\to M\\). The action of an element \\(g \\in G\\) on a function \\(f : G \\to M\\) is given by \\[ (g \\bullet f)(x) = g \\bullet (f (xg)). \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Representation.coind₁'","latex_link":"sect0002.html#def:coind₁'","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Representation.coind₁'","lean_decl_name":"Representation.coind₁'","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/IndCoind/Finite.lean","lean4_code":"/--\nGiven a representation `ρ` of `G` on `V`, `coind₁' ρ` is the representation of `G`\non `G → V`, where the action of `G` is `(g f) x = ρ g (f (x * g))`.\n-/\n@[simps] def coind₁' : Representation R G (G → V) where\n  toFun g := {\n    toFun f x := ρ g (f (x * g))\n    map_add' _ _ := by ext; simp\n    map_smul' _ _ := by ext; simp\n  }\n  map_one' := by ext; simp\n  map_mul' g₁ g₂ := by ext; simp [mul_assoc]","line":138,"end_line":149,"proof_status":"formalized"}
{"id":"lem:coind₁' iso coind₁","label":"coind₁' iso coind₁","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:coind₁'","def:induced"],"nlp_text":"The representations \\(\\mathrm{coind}_1'(M)\\) and \\(\\mathrm{coind}_1(G,M)\\) of \\(G\\) are isomorphic. More precisely there is an isomorphism of functors \\(\\mathrm{coind}_1' \\cong \\mathrm{coind}_1(G,-) \\circ \\mathbf{forget}\\), where \\(\\mathbf{forget}: \\mathbf{Rep}(R,G) \\to \\mathbf{Mod}(R)\\) is the forgetful functor.","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.coind₁'_obj_iso_coind₁","latex_link":"sect0002.html#lem:coind₁' iso coind₁","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.coind₁'_obj_iso_coind₁","lean_decl_name":"Rep.coind₁'_obj_iso_coind₁","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/IndCoind/Finite.lean","lean4_code":"@[simps] def coind₁'_obj_iso_coind₁ : coind₁'.obj M ≅ (coind₁ G).obj M.V where\n  hom.hom := ofHom M.ρ.coind₁'_lequiv_coind₁.toLinearMap\n  inv.hom := ofHom M.ρ.coind₁'_lequiv_coind₁.symm.toLinearMap\n  hom.comm g := by ext : 1; exact M.ρ.coind₁'_lequiv_coind₁_comm g\n  inv.comm g := by\n    ext f\n    simp only [Functor.comp_obj, coindFunctor_obj, trivialFunctor_obj_V,\n      RingHom.toMonoidHom_eq_coe, RingEquiv.toRingHom_eq_coe, MonoidHom.coe_comp,\n      MonoidHom.coe_coe, RingHom.coe_coe, Function.comp_apply, coind_apply, ModuleCat.hom_comp,\n      ModuleCat.hom_ofHom, LinearMap.coe_comp, ρ_hom]\n    simp [coind₁', Representation.coind₁', coind₁'_lequiv_coind₁]\n    congr","line":464,"end_line":475,"proof_status":"formalized"}
{"id":"def:up","label":"up","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:coind₁'"],"nlp_text":"There is an injective morphism \\(M \\to \\mathrm{coind}_1'(M)\\) which takes a vector \\(m \\in M\\) to the constant function on \\(G\\) with value \\(m\\). We define a representation \\(\\mathrm{up}(M)\\) to be the cokernel of this map, so that we have a short exact sequence \\[ 0 \\to M \\to \\mathrm{coind}_1'(M) \\to \\mathrm{up}(M) \\to 0. \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Representation.coind₁'_ι","latex_link":"sect0002.html#def:up","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Representation.coind₁'_ι","lean_decl_name":"Representation.coind₁'_ι","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/IndCoind/Finite.lean","lean4_code":"/--\nThe linear map from `V` to `G → V` taking a vector `v : V` to the comstant function\nwith value `V`. If `ρ` is a representation of `G` on `V`, then this map intertwines\n`ρ` and `ρ.coind₁'`.\n-/\n@[simps] def coind₁'_ι : V →ₗ[R] (G → V) where\n  toFun     := Function.const G\n  map_add'  := by simp\n  map_smul' := by simp","line":169,"end_line":177,"proof_status":"formalized"}
{"id":"cor:coind₁' invariants trivial cohomology","label":"coind₁' invariants trivial cohomology","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:coind₁' iso coind₁","lem:coind₁ invariants"],"nlp_text":"Let \\(S\\) be a normal subgroup of \\(G\\). Then \\(\\mathrm{coind}_1'(M)^S\\) has trivial cohomology as a representation of \\(G/S\\).","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.coind₁'_quotientToInvariants_trivialCohomology","latex_link":"sect0002.html#cor:coind₁' invariants trivial cohomology","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.coind₁'_quotientToInvariants_trivialCohomology","lean_decl_name":"Rep.coind₁'_quotientToInvariants_trivialCohomology","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/IndCoind/TrivialCohomology.lean","lean4_code":"instance coind₁'_quotientToInvariants_trivialCohomology {Q : Type} [Group Q] {φ : G →* Q}\n    (surj : Function.Surjective φ) : ((coind₁'.obj M) ↑ surj).TrivialCohomology := by\n  have iso := (quotientToInvariantsFunctor' surj).mapIso (coind₁'_obj_iso_coind₁ M)\n  have : ((quotientToInvariantsFunctor' surj).obj ((coind₁ G).obj M.V)).TrivialCohomology :=\n    coind₁_quotientToInvariants_trivialCohomology M.V surj\n  exact .of_iso iso","line":213,"end_line":218,"proof_status":"formalized"}
{"id":"lem:splits in splitting module","label":"splits in splitting module","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:splitting module"],"nlp_text":"The image of \\(\\sigma '\\) in \\(H^2(G,\\mathrm{split}(\\sigma ))\\) is zero.","doc_url":null,"latex_link":"sect0002.html#lem:splits in splitting module","lean_link":null,"lean_decl_name":"splits in splitting module","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":"thm:splitting module trivial","label":"splitting module trivial","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:splits in splitting module","thm:triviality criterion","lem:restriction fundamental class generates","lem:linear injective of surjective"],"nlp_text":"Let \\(\\sigma '\\) be a 2-cocycle representing a fundamental class in \\(H^2(G,M)\\). Then \\(\\mathrm{split}(\\sigma ')\\) has trivial cohomology.","doc_url":null,"latex_link":"sect0002.html#thm:splitting module trivial","lean_link":null,"lean_decl_name":"splitting module trivial","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":"lem:Dirichlet density degree one","label":"Dirichlet density degree one","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:Dirichlet density union","lem:Dirichlet density top"],"nlp_text":"The set of primes of \\({\\mathcal O}_l\\) of degree one has Dirichlet density \\(1\\).","doc_url":null,"latex_link":"sect0004.html#lem:Dirichlet density degree one","lean_link":null,"lean_decl_name":"Dirichlet density degree one","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":"lem:Dirichlet density split","label":"Dirichlet density split","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:Dirichlet density degree one"],"nlp_text":"Let \\(l/k\\) be a finite Galois extension of number fields. Then the set of degree \\(1\\) primes of \\(k\\) which split completely in \\(l\\) has density \\(\\frac{1}{[l:k]}\\).","doc_url":null,"latex_link":"sect0004.html#lem:Dirichlet density split","lean_link":null,"lean_decl_name":"Dirichlet density split","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":"def:ind'","label":"ind'","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"There is a representation \\(\\mathrm{ind}_1' (M)\\) on the \\(R\\)-module of finitely supported functions \\(G \\to _0 M\\). The action of an element \\(g \\in G\\) on a function \\(f : G \\to _0 M\\) is given by \\[ (g \\bullet f)(x) = g \\bullet (f (xg)),\\qquad \\text{i.e. } g \\bullet \\mathrm{single}(g,m) = \\mathrm{single}(xg^{-1}, g\\bullet m). \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Representation.ind₁'","latex_link":"sect0002.html#def:ind'","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Representation.ind₁'","lean_decl_name":"Representation.ind₁'","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/IndCoind/Finite.lean","lean4_code":"/--\n`ind₁' ρ` is the representation of `G` on `G →₀ V`, where the action is defined by\n`ρ.ind₁' g f x = ρ g (f (x * g))`.\n-/\n@[simps] def ind₁' : Representation R G (G →₀ V) where\n  toFun g := lmapDomain _ _ (fun x ↦ x * g⁻¹) ∘ₗ mapRange.linearMap (ρ g)\n  map_one' := by ext; simp\n  map_mul' _ _ := by ext; simp [mul_assoc]","line":187,"end_line":194,"proof_status":"formalized"}
{"id":"lem:map2 image","label":"map2 image","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:ind'"],"nlp_text":"The image of \\(\\mathrm{map}_2 : \\mathrm{ind}_1'(M) \\to \\mathrm{ind}_1'(M)\\) is precisely the set of functions \\(G \\to _0 M\\) which sum to zero. This is the kernel of the map \\(\\mathrm{ind}_1'(M) \\to M\\), which we are calling \\(\\mathrm{down}(M)\\).","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Representation.map₂_range","latex_link":"sect0002.html#lem:map2 image","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Representation.map₂_range","lean_decl_name":"Representation.map₂_range","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"ClassFieldTheory/Cohomology/FiniteCyclic/UpDown.lean","lean4_code":"lemma map₂_range [Fintype G] :\n    LinearMap.range (map₂ (R := R) (G := G) (A := A)) = LinearMap.ker ind₁'_π := by\n  classical\n  ext w\n  constructor\n  · rintro ⟨y, rfl⟩\n    have := ind₁'_π_comp_map₂ (R := R) (G := G) (A := A)\n    apply_fun (· y) at this\n    exact this\n  · intro hw_ker\n    let f : G → A := fun g ↦ ∑ i ∈ Finset.Icc 0 (unique_gen_pow g).choose, w (gen G ^ i)\n    have hf_apply (k : ℤ) : f (gen G ^ k) = ∑ i ∈ Finset.Icc 0 (k.natMod (Fintype.card G)),\n        w (gen G ^ i) := by\n      simp only [f]\n      congr\n      rw [((unique_gen_pow (gen G ^ k)).choose_spec.right (k.natMod (Fintype.card G))\n        ⟨?_, ?_⟩).symm]\n      · exact Int.natMod_lt Fintype.card_ne_zero\n      · simp [← zpow_natCast, Int.natMod, Int.ofNat_toNat, Int.emod_nonneg]\n    have hf_apply_of_lt (k : ℕ) (hk : k < Fintype.card G) :\n        f (gen G ^ k) = ∑ i ∈ Finset.Icc 0 k, w (gen G ^ i) := by\n      convert hf_apply k\n      · simp\n      · zify\n        rw [Int.natMod, Int.toNat_of_nonneg, Int.emod_eq_of_lt]\n        · positivity\n        · exact_mod_cast hk\n        · apply Int.emod_nonneg\n          simp\n    use equivFunOnFinite.symm f\n    ext g\n    rw [map₂_apply]\n    change f g - f ((gen G)⁻¹ * g) = w g\n    obtain ⟨k, ⟨hk_lt, rfl⟩, hk_unique⟩ := unique_gen_pow g\n    by_cases hk : k = 0\n    · rw [hk, hf_apply_of_lt, pow_zero, mul_one]\n      · have : (gen G)⁻¹ = gen G ^ (Fintype.card G - 1 : ℕ) := by\n          rw [inv_eq_iff_mul_eq_one, ← pow_succ', Nat.sub_add_cancel Fintype.card_pos,\n            pow_card_eq_one]\n        rw [this, hf_apply_of_lt]\n        · simp only [Finset.Icc_self, Finset.sum_singleton, pow_zero, sub_eq_self]\n          calc\n          _ = ∑ i ∈ Finset.Ico 0 (Fintype.card G), w (gen G ^ i) := by\n            congr\n            apply Finset.Icc_sub_one_right_eq_Ico_of_not_isMin\n            rw [isMin_iff_forall_not_lt]\n            push_neg\n            use 0, Fintype.card_pos\n          _ = ∑ x ∈ (Finset.Ico 0 (Fintype.card G)).image (gen G ^ ·), w x := by\n            rw [Finset.sum_image]\n            intro x hx y hy h\n            simp only [Nat.Ico_zero_eq_range, Finset.coe_range, Set.mem_Iio] at hx hy h\n            simp only at hk_unique\n            have := (unique_gen_pow (gen G ^ x)).choose_spec.right\n            rw [this x, this y]\n            · simp only [hy, h, and_self]\n            · simp only [hx, and_self]\n          _ = ∑ x, w x := by\n            congr\n            rw [Finset.eq_univ_iff_forall]\n            intro x\n            simp only [Nat.Ico_zero_eq_range, Finset.mem_image, Finset.mem_range]\n            obtain ⟨a, ha, ha'⟩ := unique_gen_pow x\n            use a, ha.left, ha.right.symm\n          _ = 0 := by\n            simpa [Finsupp.sum_fintype] using hw_ker\n        · simpa using Fintype.card_pos\n      · exact Fintype.card_pos\n    · rw [← zpow_neg_one, hf_apply_of_lt, ← zpow_natCast, ← zpow_add, neg_add_eq_sub,\n        show (k : ℤ) - 1 = (k - 1 : ℕ) by omega, zpow_natCast, hf_apply_of_lt]\n      · nth_rw 1 [show k = k - 1 + 1 by omega]\n        rw [Finset.sum_Icc_succ_top, add_sub_cancel_left, zpow_natCast,\n          Nat.sub_add_cancel (by omega)]\n        omega\n      all_goals omega\n","line":150,"end_line":224,"proof_status":"formalized"}
{"id":"lem:cohomology S-ideles decomp","label":"cohomology S-ideles decomp","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:unramified units trivial","lem:semi-local iso coind"],"nlp_text":"There are isomorphisms for all \\(n {\\gt} 0\\) \\[ H^n(l/k, {\\mathbb A}_{S,l}^\\times ) \\cong \\prod _{v \\in S} H^n(l_{\\hat v} / k_v, l_{\\hat v}^\\times ). \\]","doc_url":null,"latex_link":"sect0004.html#lem:cohomology S-ideles decomp","lean_link":null,"lean_decl_name":"cohomology S-ideles decomp","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":"cor:cohomology unramified iso cohomology Z","label":"cohomology unramified iso cohomology Z","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:unramified units trivial"],"nlp_text":"Let \\(l/k\\) be an unramified extension of local fields. Then there are isomorphisms \\[ H^\\bullet _{\\mathrm{Tate}}(l/k,l^\\times ) \\cong H^\\bullet _{\\mathrm{Tate}} (l/k,{\\mathbb Z}) \\]","doc_url":null,"latex_link":"sect0003.html#cor:cohomology unramified iso cohomology Z","lean_link":null,"lean_decl_name":"cohomology unramified iso cohomology Z","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":"def:unramified local inv","label":"unramified local inv","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:unramified units trivial","def:local inv"],"nlp_text":"The local invariant \\(\\mathrm{inv}_{l/k} : H^2(l/k, l^\\times ) \\cong \\frac{1}{[l:k]}{\\mathbb Z}/ {\\mathbb Z}\\) is the composition of the isomorphism \\(H^2(l/k, l^\\times ) \\cong H^2(l/k, {\\mathbb Z})\\) from 88 with the local invariant \\(\\mathrm{inv}_{\\mathrm{Gal}(l/k)} : H^2(l/k, {\\mathbb Z}) \\cong \\frac{1}{[l:k]}{\\mathbb Z}/ {\\mathbb Z}\\) from 54 with generator the Frobenius.","doc_url":null,"latex_link":"sect0003.html#def:unramified local inv","lean_link":null,"lean_decl_name":"unramified local inv","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem:unramified fundamental class","label":"unramified fundamental class","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["cor:cohomology unramified iso cohomology Z","lem:H2 cyclic Z"],"nlp_text":"Let \\(l/k\\) be an unramified cyclic extension of local fields. Then \\(H^2(l/k,l^\\times )\\) is cyclic of order \\([l:k]\\). It is generated by the cohomology class of the following 2-cocycle \\[ \\sigma _{l/k} (F_k^r, F_k^s) = \\begin{cases} 1 & r + s {\\lt} [l:k], \\\\ \\pi _k & r + s \\ge [l:k]. \\end{cases} \\]","doc_url":null,"latex_link":"sect0003.html#lem:unramified fundamental class","lean_link":null,"lean_decl_name":"unramified fundamental class","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":"thm:local Kronecker Weber","label":"local Kronecker Weber","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:local cyclotomic norms"],"nlp_text":"Let \\(l/{\\mathbb Q}_p\\) be a finite abelian extension. Then \\(l\\) is isomorphic to a subfield of a cyclotomic extension.","doc_url":null,"latex_link":"sect0003.html#thm:local Kronecker Weber","lean_link":null,"lean_decl_name":"local Kronecker Weber","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":"lem:Dirichlet density union","label":"Dirichlet density union","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:Dirichlet density"],"nlp_text":"Suppose \\(M_1\\) and \\(M_2\\) are disjoint sets of primes of \\({\\mathcal O}_l\\). If two of the sets \\(M_1, M_2, M_1 \\cup M_2\\) have a Dirichlet density, then so does the third and we have \\[ \\mathrm{density}(M_1 \\cup M_2) = \\mathrm{density}(M_1) + \\mathrm{density}(M_2). \\]","doc_url":null,"latex_link":"sect0004.html#lem:Dirichlet density union","lean_link":null,"lean_decl_name":"Dirichlet density union","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":"thm:triviality criterion solvable","label":"triviality criterion solvable","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(M\\) be a representation of a finite solvable group \\(G\\) (note that the solvability condition will be removed in the next theorem). Suppose we have positive natural numbers \\(e\\) and \\(o\\) with \\(e\\) even and \\(o\\) odd, such that for all subgroups \\(S\\) of \\(G\\) we have \\[ H^e(S,M) \\cong 0, \\qquad H^o(S,M) \\cong 0. \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.trivialCohomology_of_even_of_odd_of_solvable","latex_link":"sect0002.html#thm:triviality criterion solvable","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.trivialCohomology_of_even_of_odd_of_solvable","lean_decl_name":"groupCohomology.trivialCohomology_of_even_of_odd_of_solvable","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/TrivialityCriterion.lean","lean4_code":"/--\nIf `H²ⁿ⁺²(H,M)` and `H²ᵐ⁺¹(H,M)` are both zero for every subgroup `H` of `G` then `M` is acyclic.\n-/\ntheorem groupCohomology.trivialCohomology_of_even_of_odd_of_solvable [Fintype G] [IsSolvable G]\n    (M : Rep R G) (n m : ℕ)\n    -- todo: don't quantify over all types\n    (h_even : ∀ (H : Type) [Group H] {φ : H →* G} (_ : Function.Injective φ),\n      IsZero (groupCohomology (M ↓ φ) (2 * n + 2)))\n    (h_odd : ∀ (H : Type) [Group H] {φ : H →* G} (_ : Function.Injective φ),\n      IsZero (groupCohomology (M ↓ φ) (2 * m + 1))) :\n    M.TrivialCohomology where\n  isZero H := by\n    classical\n    induction H using solvable_ind with\n    | bot =>\n      intro n\n      exact isZero_groupCohomology_succ_of_subsingleton ..\n    | ind K H h12 h1 h2 h3 =>\n    have IH : ∀ i, IsZero (groupCohomology (M ↓ H.subtype ↓\n        (QuotientGroup.mk' (K.subgroupOf H)).ker.subtype) (i + 1)) := by\n      refine fun i ↦ .of_iso (h3 (n := i)) <| groupCohomology.mapIso ((MulEquiv.subgroupCongr <|\n        QuotientGroup.ker_mk' _).trans <| Subgroup.subgroupOfEquivOfLe h12)\n        (by exact Iso.refl _) (by simp [res]) _\n    have : ∀ n, IsIso ((infl (QuotientGroup.mk'_surjective\n        (K.subgroupOf H)) (n + 1)).app (M ↓ H.subtype)) := by\n      intro n\n      apply (config := { allowSynthFailures := true }) isIso_of_mono_of_epi\n      · exact inflation_restriction_mono (R := R)\n          (QuotientGroup.mk'_surjective (K.subgroupOf H)) n (M := M ↓ H.subtype) (fun i _ ↦ IH i)\n      · exact (inflation_restriction_exact (R := R)\n          (QuotientGroup.mk'_surjective (K.subgroupOf H)) n (M := M ↓ H.subtype) (fun i _ ↦ IH i)).epi_f\n          (IsZero.eq_zero_of_tgt (IH _) _)\n    have : ∀ n : ℕ, groupCohomology ((M ↓ H.subtype) ↑\n      (QuotientGroup.mk'_surjective (K.subgroupOf H))) (n + 1) ≅\n      groupCohomology (M ↓ H.subtype) (n + 1) := fun n ↦ asIso ((infl (QuotientGroup.mk'_surjective\n        (K.subgroupOf H)) (n + 1)).app (M ↓ H.subtype))\n    specialize h_even H H.subtype_injective\n    specialize h_odd H H.subtype_injective\n    have zero1 := IsZero.of_iso h_even <| this (2 * n + 1)\n    have zero2 := IsZero.of_iso h_odd <| this (2 * m)\n    intro k\n    refine .of_iso (Rep.isZero_ofEven_odd ?_ ?_ zero1 zero2 _) <| this k |>.symm <;>\n      simp [parity_simps]","line":36,"end_line":78,"proof_status":"statement_formalized"}
{"id":"cor:up and down trivial cohomology","label":"up and down trivial cohomology","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["thm:triviality criterion"],"nlp_text":"If \\(M\\) is a representation of a finite group \\(G\\) and \\(M\\) has trivial cohomology then \\(\\mathrm{up}(M)\\) and \\(\\mathrm{down}(M)\\) have trivial cohomology.","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.dimensionShift.up_trivialCohomology","latex_link":"sect0002.html#cor:up and down trivial cohomology","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.dimensionShift.up_trivialCohomology","lean_decl_name":"Rep.dimensionShift.up_trivialCohomology","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/TrivialityCriterion.lean","lean4_code":"instance Rep.dimensionShift.up_trivialCohomology [Finite G] (M : Rep R G) [M.TrivialCohomology] :\n    (up.obj M).TrivialCohomology := open scoped Classical in\n  trivialCohomology_of_even_of_odd (up.obj M) 37 9\n    (fun H _ _ hφ ↦ .of_iso (isZero_of_injective M _ 77 (by decide) hφ) (δUpResIso M hφ 76))\n    (fun H _ _ hφ ↦ .of_iso (isZero_of_injective M _ 20 (by decide) hφ) (δUpResIso M hφ 19))","line":155,"end_line":159,"proof_status":"statement_formalized"}
{"id":"lem:local inv iso","label":"local inv iso","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:H2 cyclic Z"],"nlp_text":"Let \\(G\\) be a finite cyclic group of order \\(n\\) generated by an element \\(\\mathrm{gen}\\). The local invariant \\(\\mathrm{inv}_G : H^2(G,{\\mathbb Z}) \\to {\\mathbb Z}/n{\\mathbb Z}\\) is an isomorphism. The pre-image of \\(1 \\in {\\mathbb Z}/n{\\mathbb Z}\\) is the cohomology class of the cocycle \\[ \\sigma _G (\\mathrm{gen}^i, \\mathrm{gen}^j) = \\begin{cases} 1 & i+j \\ge n \\\\ 0 & i+j {\\lt} n, \\end{cases} \\qquad 0 \\le i,j {\\lt} n. \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/carry","latex_link":"sect0002.html#lem:local inv iso","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/carry","lean_decl_name":"carry","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"ClassFieldTheory/Cohomology/LocalInv.lean","lean4_code":"/-- The carry cocycle. -/\ndef carry (i j : ZMod n) : ℤ := (i.cast + j.cast - (i + j).cast) / n\n","line":26,"end_line":27,"proof_status":"formalized"}
{"id":"lem:herbrand L_S","label":"herbrand L_S","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:H2 cyclic Z"],"nlp_text":"If \\(l/k\\) is a cyclic extension then \\(h(l/k,L_S) = \\prod _{v \\in S} |D_{\\hat v}|\\).","doc_url":null,"latex_link":"sect0004.html#lem:herbrand L_S","lean_link":null,"lean_decl_name":"herbrand L_S","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":"lem:local unramified norms","label":"local unramified norms","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:unramified fundamental class"],"nlp_text":"Let \\(l/k\\) be an unramified extension of local fields of degree \\(f\\). Then \\(N(l^\\times ) = \\pi _k^{f{\\mathbb Z}} \\times {\\mathcal O}_k^\\times \\).","doc_url":null,"latex_link":"sect0003.html#lem:local unramified norms","lean_link":null,"lean_decl_name":"local unramified norms","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":"lem:local unram rest","label":"local unram rest","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:unramified fundamental class"],"nlp_text":"Let \\(m / l / k\\) be an unramified tower of extensions of local fields Then the restriction to \\(m/l\\) of \\(\\sigma _{m/k}\\) is \\(\\sigma _{m/l}\\).","doc_url":null,"latex_link":"sect0003.html#lem:local unram rest","lean_link":null,"lean_decl_name":"local unram rest","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":"def:trivial cohomology","label":"trivial cohomology","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:inflation restriction functors","def:group homology"],"nlp_text":"Let \\(M\\) be a representation of \\(G\\) over a ring \\(R\\). \\(M\\) is said to have trivial cohomology if for every subgroup \\(S \\le G\\) and every \\(n {\\gt} 0\\), \\(H^n(S,M) \\cong 0\\). \\(M\\) is said to have trivial homology if for every subgroup \\(S \\le G\\) and every \\(n {\\gt} 0\\), \\(H_n(S,M) \\cong 0\\). Suppose the group \\(G\\) is finite. Then \\(M\\) is said to have trivial Tate cohomology if for every subgroup \\(S \\le G\\) and every \\(n \\in {\\mathbb Z}\\), \\(H^n_{\\mathrm{Tate}}(S,M) \\cong 0\\). (We will later see that for a finite group \\(G\\), the three concepts are equivalent.)","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.TrivialCohomology","latex_link":"sect0002.html#def:trivial cohomology","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.TrivialCohomology","lean_decl_name":"Rep.TrivialCohomology","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"./ClassFieldTheory/Cohomology/IndCoind/TrivialCohomology.lean","lean4_code":"instance trivialCohomology_coind₁AsPi : TrivialCohomology (coind₁AsPi G A) := by\n  classical\n  -- Let `H` be a subgroup of `G`.\n  refine ⟨fun H n ↦ ?_⟩\n  -- The restriction to `H` of `coind₁ᴳ A` is isomorphic (after choosing coset representatives)\n  -- to `coind₁^H (G ⧸ H → A)`.\n  -- By Shapiro's lemma, this has trivial cohomology.\n  exact (isZero_of_trivialCohomology ..).of_iso  <| ((groupCohomology.functor _ _ _).mapIso <|\n    (resCoind₁AsPiIso H).trans (coind₁AsPiIso _)).trans (groupCohomology.coindIso ..)\n\n","line":92,"end_line":101,"proof_status":"formalized"}
{"id":"lem:coind₁ trivial cohomology","label":"coind₁ trivial cohomology","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:trivial cohomology","lem:Shapiro","lem:Hn unit","def:induced"],"nlp_text":"The representation \\(\\mathrm{coind}_1(G,A)\\) has trivial cohomology.","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.trivialCohomology_coind₁","latex_link":"sect0002.html#lem:coind₁ trivial cohomology","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.trivialCohomology_coind₁","lean_decl_name":"Rep.trivialCohomology_coind₁","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/IndCoind/TrivialCohomology.lean","lean4_code":"instance trivialCohomology_coind₁ (A : ModuleCat R) : TrivialCohomology ((coind₁ G).obj A) :=\n  .of_iso (coind₁AsPiIso _).symm","line":105,"end_line":106,"proof_status":"formalized"}
{"id":"lem:coind₁ invariants","label":"coind₁ invariants","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:coind₁ trivial cohomology"],"nlp_text":"Let \\(S\\) be a normal subgroup of \\(G\\). Then \\(\\mathrm{coind}_1(G,A)^S\\) is isomorphic to \\(\\mathrm{coind}_1(G/S,A)\\). In particular \\(\\mathrm{coind}_1(G,A)^S\\) is has trivial cohomology as a representation of \\(G/S\\).","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.coind₁_quotientToInvariants_iso","latex_link":"sect0002.html#lem:coind₁ invariants","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.coind₁_quotientToInvariants_iso","lean_decl_name":"Rep.coind₁_quotientToInvariants_iso","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"./obsolete/_4_DimensionShift.lean","lean4_code":"def coind₁_quotientToInvariants_iso (A : ModuleCat R) (H : Subgroup G) [H.Normal] :\n    ((coind₁ G).obj A).quotientToInvariants H ≅ (coind₁ (G ⧸ H)).obj A :=\n  /-\n  Use the isomorphism `Rep.coind₁_iso` on the left.\n  Then the `H`-invariants on the left hand side are just functions `G/H ⟶ M` with the action\n  of `G/H` by translation on `G/H`. This is exactly the right hand side.\n\n  Since `quotientToInvariants` is a current PR, this will have to wait.\n  -/\n  sorry\n\n/--\nThe `H`-invariants of `(coind₁ G).obj A` form an acyclic representation of `G ⧸ H`.\n-/\n","line":103,"end_line":116,"proof_status":"formalized"}
{"id":"thm:hilbert 90","label":"hilbert 90","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(l/k\\) be a finite Galois extension of fields. Then \\(H^1(l/k, l^\\times ) \\cong 0\\).","doc_url":null,"latex_link":"sect0003.html#thm:hilbert 90","lean_link":null,"lean_decl_name":"hilbert 90","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem:idele class invariants","label":"idele class invariants","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["thm:hilbert 90"],"nlp_text":"Let \\(l/k\\) be a finite Galois extension of number fields (or even global fields). The map \\(\\mathrm{Cl}_k \\to \\mathrm{Cl}_l ^{\\mathrm{Gal}(l/k)}\\) is an isomorphism.","doc_url":null,"latex_link":"sect0004.html#lem:idele class invariants","lean_link":null,"lean_decl_name":"idele class invariants","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":"lem:Shapiro","label":"Shapiro","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(S\\) be a subgroup of \\(G\\). Then there are isomorphisms for all \\(n \\ge 0\\): \\[ H^n(G,\\mathrm{coind}_S(M)) \\cong H^n(S,M), \\qquad H_n(G,\\mathrm{ind}_S(M)) \\cong H_n(S,M), \\qquad \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.coindIso","latex_link":"sect0002.html#lem:Shapiro","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.coindIso","lean_decl_name":"groupCohomology.coindIso","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/RepresentationTheory/Coinduced.lean","lean4_code":"noncomputable def coindIso : coind φ A ≅ coind' φ A :=\n  Action.mkIso (coindVEquiv φ A).toModuleIso fun h => by\n    ext\n    simp [ModuleCat.endRingEquiv, leftRegularHomEquiv_symm_apply (leftRegular k H)]\n\n/-- Given a monoid homomorphism `φ : G →* H`, the coinduction functors `Rep k G ⥤ Rep k H` given by\n`coindFunctor k φ` and `coindFunctor' k φ` are naturally isomorphic, with isomorphism on objects\ngiven by `coindIso φ`. -/\n","line":204,"end_line":211,"proof_status":"formalized"}
{"id":"lem:local isomorphism","label":"local isomorphism","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Assume \\(l/k\\) is a finite Galois extension of characteristic zero local fields, and let \\(P\\) be the maximal ideal of \\({\\mathcal O}_l\\). For \\(n \\in {\\mathbb N}\\) sufficiently large, the exponential map gives an isomorphism \\[ (P^n,+) \\cong (1 + P^n, \\times ). \\]","doc_url":null,"latex_link":"sect0003.html#lem:local isomorphism","lean_link":null,"lean_decl_name":"local isomorphism","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":"lem:semi-local iso coind","label":"semi-local iso coind","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Then there are isomorphisms \\[ \\prod _{w | v} l_w^\\times \\cong \\mathrm{coind}_{D_{\\hat v}}^{\\mathrm{Gal}(l/k)} l_{\\hat v}^\\times , \\qquad \\prod _{w | v} {\\mathcal O}_w^\\times \\cong \\mathrm{coind}_{D_{\\hat v}}^{\\mathrm{Gal}(l/k)} {\\mathcal O}_{\\hat v}^\\times . \\]","doc_url":null,"latex_link":"sect0004.html#lem:semi-local iso coind","lean_link":null,"lean_decl_name":"semi-local iso coind","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":"thm:Dirichlet unit theorem","label":"Dirichlet unit theorem","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(\\log _S({\\mathcal O}_S^\\times )\\) has zero intersection with \\(\\mathrm{Span}(1,1,\\ldots ,1)\\). The direct sum of these subrepresentations is a lattice in \\(V_S\\).","doc_url":null,"latex_link":"sect0004.html#thm:Dirichlet unit theorem","lean_link":null,"lean_decl_name":"Dirichlet unit theorem","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":"lem:Hn unit","label":"Hn unit","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(G\\) is the trivial group then for all \\(n{\\gt}0\\), \\(H^n(G,M)\\cong 0\\).","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.instTrivialCohomologyOfSubsingleton","latex_link":"sect0002.html#lem:Hn unit","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.instTrivialCohomologyOfSubsingleton","lean_decl_name":"Rep.instTrivialCohomologyOfSubsingleton","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/TrivialCohomology.lean","lean4_code":"instance [Subsingleton G] {M : Rep R G} : M.TrivialCohomology where\n  isZero H n := by\n    apply isZero_groupCohomology_succ_of_subsingleton","line":166,"end_line":168,"proof_status":"formalized"}
{"id":"lem:norm comm","label":"norm comm","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:norm"],"nlp_text":"For any \\(g \\in G\\) and \\(m \\in M\\) we have \\(g \\bullet N_G (m) = N_G (m)\\) and \\(N_G (g \\bullet m) = N_G (m)\\).","doc_url":null,"latex_link":"sect0002.html#lem:norm comm","lean_link":null,"lean_decl_name":"norm comm","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem:norm comp d","label":"norm comp d","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["lem:norm comm"],"nlp_text":"The composition \\(d^0 \\circ N_G\\) is zero.","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.tateNorm_comp_d","latex_link":"sect0002.html#lem:norm comp d","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.tateNorm_comp_d","lean_decl_name":"groupCohomology.tateNorm_comp_d","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"ClassFieldTheory/Cohomology/TateCohomology.lean","lean4_code":"lemma tateNorm_comp_d (M : Rep R G) : tateNorm M ≫ (inhomogeneousCochains M).d 0 1 = 0 := by\n  simp [tateNorm, -norm_hom]\n","line":57,"end_line":58,"proof_status":"formalized"}
{"id":"def:norm","label":"norm","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:group homology"],"nlp_text":"Let \\(G\\) be a finite group and \\(M\\) a representation of \\(G\\) over a commutative ring \\(R\\). There is a canonical linear map \\(N_G : M \\to M\\) called the norm, defined by \\[ N_G(m) = \\sum _{g \\in G} g \\bullet m. \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Representation.norm","latex_link":"sect0002.html#def:norm","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Representation.norm","lean_decl_name":"Representation.norm","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"./obsolete/HerbrandQuotient_old.lean","lean4_code":"def norm : A →ₗ[R] A := ∑ g : G, ρ g\n\n","line":116,"end_line":117,"proof_status":"formalized"}
{"id":"lem:norm naturality","label":"norm naturality","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:norm"],"nlp_text":"For every map \\(f : A \\to B\\) in \\(\\mathbf{Rep}(R,G)\\) we have a commuting square: \\[ \\begin{array}{rcl} A & \\stackrel{f}\\to & B \\\\[2mm] N_G\\downarrow & & \\downarrow N_G \\\\ A & \\stackrel{f}\\to & B \\end{array} . \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.tateComplex.norm_comm","latex_link":"sect0002.html#lem:norm naturality","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.tateComplex.norm_comm","lean_decl_name":"groupCohomology.tateComplex.norm_comm","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/RepresentationTheory/Rep.lean","lean4_code":"lemma norm_comm {A B : Rep k G} (f : A ⟶ B) : f ≫ norm B = norm A ≫ f := by\n  ext\n  simp [Representation.norm, hom_comm_apply]\n\n/-- Given a representation `A` of a finite group `G`, the norm map `A ⟶ A` defined by\n`x ↦ ∑ A.ρ g x` for `g` in `G` defines a natural endomorphism of the identity functor. -/\n","line":686,"end_line":691,"proof_status":"formalized"}
{"id":"def:splitting module","label":"splitting module","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The splitting module of \\(\\sigma '\\) is the \\(R\\)-module \\(M \\times \\mathrm{aug}(R,G)\\), with the action of an element \\(g \\in G\\) given by \\[ g \\bullet (m,f) = \\left(g \\bullet m + \\sum _{x \\in G} f(x) \\sigma '(g,x) , g \\bullet f\\right), \\qquad m \\in M, \\; f \\in \\mathrm{aug}(R,G). \\]","doc_url":null,"latex_link":"sect0002.html#def:splitting module","lean_link":null,"lean_decl_name":"splitting module","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":"def:restriction map","label":"restriction map","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:inflation restriction functors"],"nlp_text":"If \\(S\\) is a subgroup of \\(G\\), then we write \\(H^n(S,M)\\) for the cohomology groups of the restricted representation \\(M \\downarrow S\\). If \\(f : G^n \\to M\\) is an element of \\(C^n(G,M)\\), then we may restrict \\(f\\) to a function \\(S^n \\to M\\). Restricting functions in this way defines a map of cochain complexes \\(C^\\bullet (G,M) \\to C^\\bullet (S,M)\\), and hence a map of cohomology groups \\[ \\mathrm{rest}: H^n(G,M) \\to H^n(S,M). \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.rest","latex_link":"sect0002.html#def:restriction map","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupCohomology.rest","lean_decl_name":"groupCohomology.rest","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"ClassFieldTheory/Cohomology/Functors/Restriction.lean","lean4_code":"/--\nThe restriction map `Hⁿ(G,M) ⟶ Hⁿ(H,M)`, defined as a morphism of functors\n-/\ndef rest (n : ℕ) : functor R G n ⟶ Rep.res φ ⋙ functor R S n  where\n  app M               := map φ (𝟙 (M ↓ φ)) n\n  naturality M₁ M₂ f  := by\n    simp only [functor_obj, Functor.comp_obj, functor_map, Functor.comp_map]\n    rw [←map_comp, ←map_comp]\n    congr 1\n","line":169,"end_line":177,"proof_status":"formalized"}
{"id":"def:induced","label":"induced","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(G\\) be a group, \\(R\\) a commutative ring and \\(A\\) an \\(R\\)-module. There is a representation of \\(G\\) over \\(R\\) on the space of all functions \\(f : G \\to A\\). The action of an element \\(g \\in G\\) on \\(f\\) is defined by \\[ (g \\bullet f) (x) = f(xg). \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Representation.coind₁","latex_link":"sect0002.html#def:induced","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Representation.coind₁","lean_decl_name":"Representation.coind₁","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"./obsolete/_4_DimensionShift.lean","lean4_code":"theorem coind₁_isAcyclic (A : ModuleCat R) : ((coind₁ G).obj A).IsAcyclic :=\n  /-\n  There are many ways to prove this. The following method uses none of the\n  technology of homological algebra, so it should be fairly easy to formalize.\n\n  Fix a subgroup `H` of `G` and let `{gᵢ}` be a set of coset representatives for `H \\ G`.\n  Recall that a homogeneous `n + 1`-cochain on `H` with values in `G → A`\n  is a function `σ : H^{n+2} → (G → A)` satisfying\n\n    `σ (h * h₀, ... , h * h_{n+1}) (h * g) = σ (h₀,...,hₙ).`\n\n  The cochain `σ` is a cocycle if it satisfies the relation\n\n    `∑ᵢ (-1)ⁱ * σ (h₀, ... , (not hᵢ), ... , h_{n+2}) (g) = 0`.\n\n  Given a homogeneous `n + 1`-cocycle `σ`, we'll define a homogeneous `n`-cochain `τ` by\n\n    `τ (h₀,...,hₙ) (h * gᵢ) = σ (h,h₀,...,hₙ) (h * gᵢ)`.\n\n  The cocycle relation for `σ` implies `∂ τ = σ`, so `σ` is a coboundary.\n\n  Let's rephrase this in terms of inhomogeneous cocycles. The inhomogeneous cocycle\n  corresponding to `σ` is\n\n    `σ' (h₀,...,hₙ) (h * gᵢ) = σ (1,h₁,h₁*h₂,..., h₁*...*hₙ) (h * gᵢ)`\n\n  and the inhomogeneous cochain corresponding to `τ` is\n\n    `τ' (h₁,...,hₙ) (h * gᵢ)  = τ (1,h₁,... , h₁*...*hₙ) (h * gᵢ)`\n    `                         = σ (h, 1, h₁, h₁*h₂, ..., h₁*...*hₙ) (h * gᵢ)`\n    `                         = σ (1, h⁻¹, h⁻¹*h₁, h⁻¹*h₁*h₂, ..., h⁻¹* h₁*...*hₙ) (gᵢ)`\n    `                         = σ' (h⁻¹,h₁,...,hₙ) (gᵢ)`.\n\n  The last formula above defines an inhomogeneous cochain `τ'`, such that `∂ τ' = σ'`.\n  -/\n  sorry\n\n\n","line":65,"end_line":102,"proof_status":"formalized"}
{"id":"def:local inv","label":"local inv","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(G\\) be a finite cyclic group of order \\(n\\) generated by an element \\(\\mathrm{gen}\\). Then the map \\begin{align*} \\mathrm{inv}_G : H^2(G,{\\mathbb Z}) & \\to {\\mathbb Z}/n{\\mathbb Z}\\\\ \\sigma & \\mapsto \\sum _{i=0}^{n-1} \\sigma (\\mathrm{gen}^i,\\mathrm{gen}) \\end{align*}","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/localInv","latex_link":"sect0002.html#def:local inv","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/localInv","lean_decl_name":"localInv","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"ClassFieldTheory/Cohomology/LocalInv.lean","lean4_code":"/-- `ℤ` as a trivial `ℤ/nℤ`-module has second cohomology group `ℤ/nℤ`, with the forward map given by\nthe local invariant.\n\nSee `groupCohomology.evenTrivialInt` for the basis-free computation in the case of a trivial\ntorsion-free representation of a finite cyclic group. -/\ndef localInv : H2 (.trivial ℤ (ZMod n)ᵐ ℤ) ≃+ ZMod n where\n  toFun := localInvAux n\n  invFun := carryH2Aux\n  left_inv := leftInverse_carryH2Aux_localInvAux\n  right_inv := rightInverse_carryH2Aux_localInvAux\n  map_add' := by simp\n","line":136,"end_line":146,"proof_status":"formalized"}
{"id":"def:fundamental class","label":"fundamental class","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"In this section \\(G\\) is a finite group and \\(R\\) is a commutative ring. If \\(M\\) is a representation of \\(G\\) over \\(R\\) then we shall call \\(M\\) (or more precisely the triple \\((R,G,M)\\)) a finite class formation if: The ring \\(R\\) has no additive torsion. This implies for all subgroups \\(S\\) of \\(G\\): \\[ H^2(S,\\mathrm{aug}(R,G)) \\cong H^1(S,R) \\cong \\mathrm{Hom}(S,R) = 0. \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.split.FiniteClassFormation","latex_link":"sect0002.html#def:fundamental class","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.split.FiniteClassFormation","lean_decl_name":"Rep.split.FiniteClassFormation","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"ClassFieldTheory/Cohomology/SplittingModule.lean","lean4_code":"/--\nIf `M` is an `R[G]`-module and `σ : H²(G,M)`, we say `σ` is a *finite class formation* if\nfor all subgroups `H` of `G`,\n1) `H¹(H,M|H)=0`;\n2) The `R`-module `H²(H,M|H)` is spanned as an `R`-module by `Res(σ)`;\n3) The kernel of `R → H²(H,M|H)` is `|H|R`\n\nIn other words, for all subgroups H of G, H¹(H,M)=0 and H²(H,M)=R/|H|R\nwith the isomorphism given by sending 1 ∈ R/|H|R to σ.\n-/\nclass FiniteClassFormation (σ : H2 M) where\n  isZero_H1 {H : Type} [Group H] {φ : H →* G} (inj : Function.Injective φ) : IsZero (H1 (M ↓ φ))\n  hypothesis₂ (σ) {H : Type} [Group H] {φ : H →* G} (inj : Function.Injective φ) :\n    Submodule.span R {σ ↡ φ} = ⊤\n  hypothesis₂' (σ) {H : Type} [Group H] {φ : H →* G} (inj : Function.Injective φ) :\n    LinearMap.ker (LinearMap.toSpanSingleton R (H2 (M ↓ φ)) (σ ↡ φ)) = Ideal.span {(Nat.card H : R)}\n","line":234,"end_line":249,"proof_status":"formalized"}
{"id":"lem:restriction fundamental class generates","label":"restriction fundamental class generates","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:fundamental class"],"nlp_text":"Let \\(\\sigma \\in H^2(G,M)\\) be a fundamental class. Then the restriction of \\(\\sigma \\) to any subgroup \\(S\\) of \\(G\\) is a generator for \\(H^2(S,M)\\).","doc_url":null,"latex_link":"sect0002.html#lem:restriction fundamental class generates","lean_link":null,"lean_decl_name":"restriction fundamental class generates","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":"def:Dirichlet density","label":"Dirichlet density","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(M\\) be a set of primes of \\({\\mathcal O}_k\\). We’ll say that \\(M\\) has a Dirichlet density \\(c \\in {\\mathbb R}\\) if \\[ \\sum _{P \\in M} N(P)^{-s} \\stackrel{s \\to 1+}\\sim c \\cdot \\log \\left(\\frac{1}{s-1}\\right). \\]","doc_url":null,"latex_link":"sect0004.html#def:Dirichlet density","lean_link":null,"lean_decl_name":"Dirichlet density","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":"lem:Dirichlet density top","label":"Dirichlet density top","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":["def:Dirichlet density"],"nlp_text":"The set of all primes of \\({\\mathcal O}_k\\) has Dirichlet density \\(1\\).","doc_url":null,"latex_link":"sect0004.html#lem:Dirichlet density top","lean_link":null,"lean_decl_name":"Dirichlet density top","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":"def:inflation restriction functors","label":"inflation restriction functors","background_color":"#B0ECA3","border_color":"darkgreen","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The map \\(M \\mapsto M \\downarrow S\\) defines a functor \\(\\mathbf{res}: \\mathbf{Rep}(R,G) \\to \\mathbf{Rep}(R,S)\\). If \\(S\\) is a normal subgroup of \\(G\\) then then map \\(M \\mapsto M^S\\) defines a functor \\(\\mathbf{invar}: \\mathbf{Rep}(R,G) \\to \\mathbf{Rep}(R, G/S)\\). These functors are called restriction and inflation respectively. These are defined in Mathlib as Rep.res and Rep.quotientToInvariantsFunctor.","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.quotientToInvariantsFunctor","latex_link":"sect0002.html#def:inflation restriction functors","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.quotientToInvariantsFunctor","lean_decl_name":"Rep.quotientToInvariantsFunctor","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"ClassFieldTheory","source_path":"ClassFieldTheory/Cohomology/Functors/Inflation.lean","lean4_code":"@[simps! (isSimp := false) obj_V] noncomputable def quotientToInvariantsFunctor' :\n    Rep R G ⥤ Rep R Q :=\n  quotientToInvariantsFunctor R φ.ker ⋙\n    (Rep.resEquiv R (QuotientGroup.quotientKerEquivOfSurjective _ surj)).inverse\n","line":46,"end_line":49,"proof_status":"formalized"}
{"id":"lem:linear injective of surjective","label":"linear injective of surjective","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(I\\) be an ideal of a commutative ring \\(R\\) and let \\(f:R/I \\to R/I\\) be a surjective \\(R\\)-linear map. Then \\(f\\) is injective.","doc_url":null,"latex_link":"sect0002.html#lem:linear injective of surjective","lean_link":null,"lean_decl_name":"linear injective of surjective","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem:serre_approx","label":"serre_approx","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(G\\) be a group and let \\(M\\) be a \\(G\\)-module. Say we have a decreasing sequence \\(M=M_0\\supseteq M_1\\supseteq M_2\\supseteq \\cdots \\) of \\(G\\)-submodules with the property that the natural map \\(M\\to \\varprojlim (M/M_i)\\) is an isomorphism. Assume for some natural \\(q\\) that the group cohomology \\(H^{q+1}(G,M_i/M_{i+1})=0\\) for all \\(i\\). Then \\(H^{q+1}(G,M)=0.\\)","doc_url":null,"latex_link":"sect0003.html#lem:serre_approx","lean_link":null,"lean_decl_name":"serre_approx","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":"def:augmentation module","label":"augmentation module","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"As an example we consider the case of the trivial representation \\(R\\). The induced representation is then the group ring \\(RG\\), which is referred to Mathlib as Rep.leftRegular R G; this is a free \\(R\\)-module with basis \\(\\mathrm{single}(g,1)\\) for \\(g \\in G\\). For simplicity we shall write \\([g]\\) for the basis vector \\(\\mathrm{single}(g,1)\\). The map \\(\\mathrm{ind}_1'(R) \\to R\\) takes \\(\\sum _{g\\in G} x_g [g]\\) to \\(\\sum _{g \\in G} x_g\\). This map is commonly called the augmentation, and its kernel \\(\\mathrm{down}(R)\\) the augmentation module. We shall write \\(\\mathrm{aug}(R,G)\\) for this kernel. The kernel \\(\\mathrm{aug}(R,G)\\) is spanned by the elements \\([g]-[1]\\) for \\(g \\in G\\).","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.aug","latex_link":"sect0002.html#def:augmentation module","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/Rep.aug","lean_decl_name":"Rep.aug","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/kbuzzard/ClassFieldTheory/573dab0b2d85d633fd42230b31b9a27aa28d2d98/ClassFieldTheory/Cohomology/AugmentationModule.lean","lean4_code":"/--\nThe augmentation module `aug R G` is the kernel of the augmentation map\n\n  `ε : leftRegular R G ⟶ trivial R G R`.\n\n-/\nabbrev Rep.aug : Rep R G := kernel (ε R G)","line":43,"end_line":49,"proof_status":"formalized"}
{"id":"def:group homology","label":"group homology","background_color":"#B0ECA3","border_color":"darkgreen","kind":"box","dep_graph_url":"https://kbuzzard.github.io/ClassFieldTheory/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"There is also a chain complex of \\(R\\)-modules: \\[ \\cdots \\stackrel{d_2}\\to C_2(G,M) \\stackrel{d_1}\\to C_1(G,M) \\stackrel{d_0}\\to C_0(G,M) \\]","doc_url":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupHomology.inhomogeneousChains","latex_link":"sect0002.html#def:group homology","lean_link":"https://kbuzzard.github.io/ClassFieldTheory/docs/find/#doc/groupHomology.inhomogeneousChains","lean_decl_name":"groupHomology.inhomogeneousChains","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean","lean4_code":"theorem inhomogeneousChains.ext {M : ModuleCat k} {x y : (inhomogeneousChains A).X n ⟶ M}\n    (h : ∀ g, ModuleCat.ofHom (lsingle g) ≫ x = ModuleCat.ofHom (lsingle g) ≫ y) :\n    x = y := ModuleCat.hom_ext <| lhom_ext' fun g => ModuleCat.hom_ext_iff.1 (h g)\n\n","line":175,"end_line":178,"proof_status":"formalized"}