some more algebra
This commit is contained in:
Floris van Doorn
parent
5c9927ce2d
commit
94066a6ba8
8 changed files with 173 additions and 103 deletions
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@ -671,8 +671,8 @@ namespace pointed
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definition homotopy_group_conn_nat_functor (n : ℕ) {A B : Type*[1]} (f : A →* B) :
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homotopy_group_conn_nat n A →g homotopy_group_conn_nat n B :=
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begin
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cases n with n, { apply trivial_homomorphism },
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cases n with n, { apply trivial_homomorphism },
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cases n with n, { apply group.trivial_homomorphism },
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cases n with n, { apply group.trivial_homomorphism },
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exact π→g[n+2] f
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end
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@ -243,37 +243,6 @@ definition is_contr_left_of_is_short_exact {A B : Type} {C : Type*} {f : A → B
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(H : is_short_exact f g) (HB : is_contr B) (a₀ : A) : is_contr A :=
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is_contr_of_is_embedding f (is_short_exact.is_emb H) _ a₀
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/- TODO: move and remove other versions -/
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definition is_surjective_qg_map {A : Group} (N : property A) [is_normal_subgroup A N] :
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is_surjective (qg_map N) :=
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begin
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intro x, induction x,
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fapply image.mk,
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exact a, reflexivity,
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apply is_prop.elimo
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end
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definition is_surjective_ab_qg_map {A : AbGroup} (N : property A) [is_normal_subgroup A N] :
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is_surjective (ab_qg_map N) :=
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is_surjective_ab_qg_map _
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definition qg_map_eq_one {A : Group} {K : property A} [is_normal_subgroup A K] (g : A)
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(H : g ∈ K) :
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qg_map K g = 1 :=
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begin
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apply set_quotient.eq_of_rel,
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have e : g * 1⁻¹ = g,
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from calc
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g * 1⁻¹ = g * 1 : one_inv
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... = g : mul_one,
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exact transport (λx, K x) e⁻¹ H
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end
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definition ab_qg_map_eq_one {A : AbGroup} {K : property A} [is_subgroup A K] (g : A)
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(H : g ∈ K) :
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ab_qg_map K g = 1 :=
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ab_qg_map_eq_one g H
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definition is_short_exact_normal_subgroup {G : Group} (S : property G) [is_normal_subgroup G S] :
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is_short_exact (incl_of_subgroup S) (qg_map S) :=
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@ -287,4 +256,5 @@ begin
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{ exact is_surjective_qg_map S },
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end
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end algebra
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@ -183,6 +183,10 @@ definition left_module_AddAbGroup_of_LeftModule [instance] {R : Ring} (M : LeftM
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left_module R (AddAbGroup_of_LeftModule M) :=
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LeftModule.struct M
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definition left_module_AddAbGroup_of_LeftModule2 [instance] {R : Ring} (M : LeftModule R) :
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left_module R (Group_of_AbGroup (AddAbGroup_of_LeftModule M)) :=
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LeftModule.struct M
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definition left_module_of_ab_group {G : Type} [gG : add_ab_group G] {R : Type} [ring R]
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(smul : R → G → G)
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(h1 : Π (r : R) (x y : G), smul r (x + y) = (smul r x + smul r y))
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@ -423,6 +427,15 @@ section
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(g : B →lm C)
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(h : @is_short_exact A B C f g)
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structure five_exact_mod (A B C D E : LeftModule R) :=
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(f₁ : A →lm B)
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(f₂ : B →lm C)
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(f₃ : C →lm D)
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(f₄ : D →lm E)
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(h₁ : @is_exact A B C f₁ f₂)
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(h₂ : @is_exact B C D f₂ f₃)
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(h₃ : @is_exact C D E f₃ f₄)
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local abbreviation g_of_lm := @group_homomorphism_of_lm_homomorphism
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definition short_exact_mod_of_is_exact {X A B C Y : LeftModule R}
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(k : X →lm A) (f : A →lm B) (g : B →lm C) (l : C →lm Y)
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@ -610,4 +623,5 @@ group_isomorphism_of_lm_isomorphism φ
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end int
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end left_module
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@ -150,7 +150,8 @@ namespace group
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definition ab_qg_map {G : AbGroup} (N : property G) [is_subgroup G N] : G →g quotient_ab_group N :=
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@qg_map _ N (is_normal_subgroup_ab _)
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definition is_surjective_ab_qg_map {A : AbGroup} (N : property A) [is_subgroup A N] : is_surjective (ab_qg_map N) :=
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definition is_surjective_qg_map {A : Group} (N : property A) [is_normal_subgroup A N] :
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is_surjective (qg_map N) :=
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begin
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intro x, induction x,
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fapply image.mk,
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@ -158,6 +159,10 @@ namespace group
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apply is_prop.elimo
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end
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definition is_surjective_ab_qg_map {A : AbGroup} (N : property A) [is_subgroup A N] :
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is_surjective (ab_qg_map N) :=
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@is_surjective_qg_map _ _ _
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namespace quotient
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notation `⟦`:max a `⟧`:0 := qg_map _ a
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end quotient
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@ -165,25 +170,20 @@ namespace group
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open quotient
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variables {N N'}
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definition qg_map_eq_one (g : G) (H : N g) : qg_map N g = 1 :=
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definition qg_map_eq_one {A : Group} {K : property A} [is_normal_subgroup A K] (g : A)
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(H : g ∈ K) : qg_map K g = 1 :=
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begin
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apply eq_of_rel,
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have e : (g * 1⁻¹ = g),
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apply set_quotient.eq_of_rel,
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have e : g * 1⁻¹ = g,
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from calc
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g * 1⁻¹ = g * 1 : one_inv
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... = g : mul_one,
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unfold quotient_rel, rewrite e, exact H
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exact transport (λx, K x) e⁻¹ H
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end
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definition ab_qg_map_eq_one {K : property A} [is_subgroup A K] (g :A) (H : K g) : ab_qg_map K g = 1 :=
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begin
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apply eq_of_rel,
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have e : (g * 1⁻¹ = g),
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from calc
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g * 1⁻¹ = g * 1 : one_inv
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... = g : mul_one,
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unfold quotient_rel, xrewrite e, exact H
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end
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definition ab_qg_map_eq_one {A : AbGroup} {K : property A} [is_subgroup A K] (g : A) (H : g ∈ K) :
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ab_qg_map K g = 1 :=
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@qg_map_eq_one _ _ _ g H
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--- there should be a smarter way to do this!! Please have a look, Floris.
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definition rel_of_qg_map_eq_one (g : G) (H : qg_map N g = 1) : g ∈ N :=
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@ -454,16 +454,14 @@ namespace group
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definition kernel_quotient_extension {A B : AbGroup} (f : A →g B) : quotient_ab_group (kernel f) →g B :=
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begin
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unfold quotient_ab_group,
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fapply @quotient_group_elim A B _ (@is_normal_subgroup_ab _ (kernel f) _) f,
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apply quotient_ab_group_elim f,
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intro a, intro p, exact p
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end
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definition kernel_quotient_extension_triangle {A B : AbGroup} (f : A →g B) :
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kernel_quotient_extension f ∘ ab_qg_map (kernel f) ~ f :=
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begin
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intro a,
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apply @quotient_group_compute _ _ _ (@is_normal_subgroup_ab _ (kernel f) _)
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intro a, reflexivity
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end
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definition is_embedding_kernel_quotient_extension {A B : AbGroup} (f : A →g B) :
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@ -474,7 +472,7 @@ definition is_embedding_kernel_quotient_extension {A B : AbGroup} (f : A →g B)
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note H := is_surjective_ab_qg_map (kernel f) x,
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induction H, induction p,
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intro q,
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apply @qg_map_eq_one _ _ (@is_normal_subgroup_ab _ (kernel f) _),
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apply ab_qg_map_eq_one,
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refine _ ⬝ q,
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symmetry,
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rexact kernel_quotient_extension_triangle f a
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@ -698,6 +696,27 @@ namespace group
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exact ap set_quotient.class_of (p g)
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end
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definition quotient_group_isomorphism_quotient_group [constructor] (φ : G ≃g H)
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(h : Πg, g ∈ R ↔ φ g ∈ S) : quotient_group R ≃g quotient_group S :=
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begin
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refine isomorphism.MK (quotient_group_functor φ (λg, iff.mp (h g)))
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(quotient_group_functor φ⁻¹ᵍ (λg gS, iff.mpr (h _) (transport S (right_inv φ g)⁻¹ gS))) _ _,
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{ refine quotient_group_functor_compose _ _ _ _ ⬝hty
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quotient_group_functor_homotopy _ _ proof right_inv φ qed ⬝hty
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quotient_group_functor_gid },
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{ refine quotient_group_functor_compose _ _ _ _ ⬝hty
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quotient_group_functor_homotopy _ _ proof left_inv φ qed ⬝hty
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quotient_group_functor_gid }
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end
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definition is_equiv_qg_map {G : Group} (H : property G) [is_normal_subgroup G H]
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(H₂ : Π⦃g⦄, g ∈ H → g = 1) : is_equiv (qg_map H) :=
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set_quotient.is_equiv_class_of _ (λg h r, eq_of_mul_inv_eq_one (H₂ r))
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definition quotient_group_isomorphism [constructor] {G : Group} (H : property G)
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[is_normal_subgroup G H] (h : Πg, g ∈ H → g = 1) : quotient_group H ≃g G :=
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(isomorphism.mk _ (is_equiv_qg_map H h))⁻¹ᵍ
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end group
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namespace group
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@ -735,4 +754,16 @@ namespace group
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quotient_ab_group_functor φ hφ ~ quotient_ab_group_functor ψ hψ :=
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@quotient_group_functor_homotopy G H R _ S _ ψ φ hψ hφ p
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definition is_equiv_ab_qg_map {G : AbGroup} (H : property G) [is_subgroup G H]
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(h : Π⦃g⦄, g ∈ H → g = 1) : is_equiv (ab_qg_map H) :=
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proof @is_equiv_qg_map G H (is_normal_subgroup_ab _) h qed
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definition ab_quotient_group_isomorphism [constructor] {G : AbGroup} (H : property G)
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[is_subgroup G H] (h : Πg, H g → g = 1) : quotient_ab_group H ≃g G :=
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(isomorphism.mk _ (is_equiv_ab_qg_map H h))⁻¹ᵍ
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definition quotient_ab_group_isomorphism_quotient_ab_group [constructor] (φ : G ≃g H)
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(h : Πg, g ∈ R ↔ φ g ∈ S) : quotient_ab_group R ≃g quotient_ab_group S :=
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@quotient_group_isomorphism_quotient_group _ _ _ _ _ _ φ h
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end group
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@ -140,7 +140,16 @@ namespace spectral_sequence
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Einf c (n, s) ≃lm E' n s :=
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E_isomorphism0 c (λr Hr, H1 r) (λr Hr, H2 r)
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/- we call a spectral sequence normal if it is a first-quadrant spectral sequence and the degree of d is what we expect -/
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definition convergence_0 (n : ℤ) (H : Πm, lb c m = 0) :
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is_built_from (Dinf n) (λ(k : ℕ), Einf c (n - k, k)) :=
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is_built_from_isomorphism isomorphism.rfl
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(λk, left_module.isomorphism_of_eq (ap (Einf c)
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(prod_eq (ap (sub n) (ap (add _) !H ⬝ add_zero _)) (ap (add _) !H ⬝ add_zero _))))
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(HDinf c n)
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/- we call a spectral sequence normal if it is a first-quadrant spectral sequence and
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the degree of d on page r (for r ≥ 2) is (r, -(r-1)).
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The indexing is different, because we start counting pages at 2. -/
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include c
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structure is_normal : Type :=
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(normal1 : Π{n} s, n < 0 → is_contr (E' n s))
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@ -166,10 +175,7 @@ namespace spectral_sequence
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refine lt_of_lt_of_le H2 (le.trans _ (le_of_eq !add_zero⁻¹)),
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exact add_le_add_right (of_nat_le_of_nat_of_le Hr') 1 },
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end
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/- some properties which use the degree of the spectral sequence we construct. For the AHSS and SSS the hypothesis is by reflexivity -/
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-- definition foo
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omit d
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end spectral_sequence
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@ -79,7 +79,7 @@ namespace group
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variables {G₁ G₂ : Group}
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-- TODO: maybe define this in more generality for pointed propertys?
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-- TODO: maybe define this in more generality for pointed sets?
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definition kernel [constructor] (φ : G₁ →g G₂) : property G₁ := { g | trunctype.mk (φ g = 1) _}
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theorem mul_mem_kernel (φ : G₁ →g G₂) (g h : G₁) (H₁ : g ∈ kernel φ) (H₂ : h ∈ kernel φ) : g * h ∈ kernel φ :=
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@ -555,6 +555,17 @@ end
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intro r, induction r, reflexivity
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end
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definition is_equiv_incl_of_subgroup {G : Group} (H : property G) [is_subgroup G H]
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(h : Πg, g ∈ H) : is_equiv (incl_of_subgroup H) :=
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have is_surjective (incl_of_subgroup H),
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begin intro g, exact image.mk ⟨g, h g⟩ idp end,
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have is_embedding (incl_of_subgroup H), from is_embedding_incl_of_subgroup H,
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function.is_equiv_of_is_surjective_of_is_embedding (incl_of_subgroup H)
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definition subgroup_isomorphism [constructor] {G : Group} (H : property G) [is_subgroup G H]
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(h : Πg, g ∈ H) : subgroup H ≃g G :=
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isomorphism.mk _ (is_equiv_incl_of_subgroup H h)
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end group
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open group
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@ -5,41 +5,6 @@ import .left_module .quotient_group
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open algebra eq group sigma sigma.ops is_trunc function trunc equiv is_equiv property
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definition group_homomorphism_of_add_group_homomorphism [constructor] {G₁ G₂ : AddGroup}
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(φ : G₁ →a G₂) : G₁ →g G₂ :=
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φ
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-- move to subgroup
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-- attribute normal_subgroup_rel._trans_of_to_subgroup_rel [unfold 2]
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-- attribute normal_subgroup_rel.to_subgroup_rel [constructor]
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definition is_equiv_incl_of_subgroup {G : Group} (H : property G) [is_subgroup G H] (h : Πg, g ∈ H) :
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is_equiv (incl_of_subgroup H) :=
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have is_surjective (incl_of_subgroup H),
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begin intro g, exact image.mk ⟨g, h g⟩ idp end,
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have is_embedding (incl_of_subgroup H), from is_embedding_incl_of_subgroup H,
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function.is_equiv_of_is_surjective_of_is_embedding (incl_of_subgroup H)
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definition subgroup_isomorphism [constructor] {G : Group} (H : property G) [is_subgroup G H] (h : Πg, g ∈ H) :
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subgroup H ≃g G :=
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isomorphism.mk _ (is_equiv_incl_of_subgroup H h)
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definition is_equiv_qg_map {G : Group} (H : property G) [is_normal_subgroup G H] (H₂ : Π⦃g⦄, g ∈ H → g = 1) :
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is_equiv (qg_map H) :=
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set_quotient.is_equiv_class_of _ (λg h r, eq_of_mul_inv_eq_one (H₂ r))
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definition quotient_group_isomorphism [constructor] {G : Group} (H : property G) [is_normal_subgroup G H]
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(h : Πg, g ∈ H → g = 1) : quotient_group H ≃g G :=
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(isomorphism.mk _ (is_equiv_qg_map H h))⁻¹ᵍ
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definition is_equiv_ab_qg_map {G : AbGroup} (H : property G) [is_subgroup G H] (h : Π⦃g⦄, g ∈ H → g = 1) :
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is_equiv (ab_qg_map H) :=
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proof @is_equiv_qg_map G H (is_normal_subgroup_ab _) h qed
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definition ab_quotient_group_isomorphism [constructor] {G : AbGroup} (H : property G) [is_subgroup G H]
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(h : Πg, H g → g = 1) : quotient_ab_group H ≃g G :=
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(isomorphism.mk _ (is_equiv_ab_qg_map H h))⁻¹ᵍ
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namespace left_module
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/- submodules -/
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variables {R : Ring} {M M₁ M₂ M₃ : LeftModule R} {m m₁ m₂ : M}
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@ -61,7 +26,7 @@ transport (λx, S x) (by rewrite [add.comm, neg_add_cancel_left]) H
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-- open is_submodule
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variables {S : property M} [is_submodule M S]
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variables {S : property M} [is_submodule M S] {S₂ : property M₂} [is_submodule M₂ S₂]
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definition is_subgroup_of_is_submodule [instance] (S : property M) [is_submodule M S] :
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is_subgroup (AddGroup_of_AddAbGroup M) S :=
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@ -221,23 +186,30 @@ definition quotient_map [constructor] : M →lm quotient_module S :=
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lm_homomorphism_of_group_homomorphism (ab_qg_map _) (λr g, idp)
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definition quotient_map_eq_zero (m : M) (H : S m) : quotient_map S m = 0 :=
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@qg_map_eq_one _ _ (is_normal_subgroup_ab _) _ H
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@ab_qg_map_eq_one _ _ _ _ H
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definition rel_of_quotient_map_eq_zero (m : M) (H : quotient_map S m = 0) : S m :=
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@rel_of_qg_map_eq_one _ _ (is_normal_subgroup_ab _) m H
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@rel_of_qg_map_eq_one _ _ _ m H
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variable {S}
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definition quotient_elim [constructor] (φ : M →lm M₂) (H : Π⦃m⦄, m ∈ S → φ m = 0) :
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quotient_module S →lm M₂ :=
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lm_homomorphism_of_group_homomorphism
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(quotient_ab_group_elim (group_homomorphism_of_lm_homomorphism φ) H)
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definition respect_smul_quotient_elim [constructor] (φ : M →lm M₂) (H : Π⦃m⦄, m ∈ S → φ m = 0)
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(r : R) (m : quotient_module S) :
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quotient_ab_group_elim (group_homomorphism_of_lm_homomorphism φ) H
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(@has_scalar.smul _ (quotient_module S) _ r m) =
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r • quotient_ab_group_elim (group_homomorphism_of_lm_homomorphism φ) H m :=
|
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begin
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intro r, esimp,
|
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revert m,
|
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refine @set_quotient.rec_prop _ _ _ (λ x, !is_trunc_eq) _,
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intro m,
|
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exact to_respect_smul φ r m
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end
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definition quotient_elim [constructor] (φ : M →lm M₂) (H : Π⦃m⦄, m ∈ S → φ m = 0) :
|
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quotient_module S →lm M₂ :=
|
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lm_homomorphism_of_group_homomorphism
|
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(quotient_ab_group_elim (group_homomorphism_of_lm_homomorphism φ) H)
|
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(respect_smul_quotient_elim φ H)
|
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|
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definition is_prop_quotient_module (S : property M) [is_submodule M S] [H : is_prop M] : is_prop (quotient_module S) :=
|
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begin apply @set_quotient.is_trunc_set_quotient, exact H end
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|
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|
|
@ -255,6 +227,17 @@ definition quotient_module_isomorphism [constructor] (S : property M) [is_submod
|
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quotient_module S ≃lm M :=
|
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(isomorphism.mk (quotient_map S) (is_equiv_ab_qg_map S h))⁻¹ˡᵐ
|
||||
|
||||
definition quotient_module_functor [constructor] (φ : M →lm M₂) (h : Πg, g ∈ S → φ g ∈ S₂) :
|
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quotient_module S →lm quotient_module S₂ :=
|
||||
quotient_elim (quotient_map S₂ ∘lm φ)
|
||||
begin intros m Hm, rexact quotient_map_eq_zero S₂ (φ m) (h m Hm) end
|
||||
|
||||
definition quotient_module_isomorphism_quotient_module [constructor] (φ : M ≃lm M₂)
|
||||
(h : Πm, m ∈ S ↔ φ m ∈ S₂) : quotient_module S ≃lm quotient_module S₂ :=
|
||||
lm_isomorphism_of_group_isomorphism
|
||||
(quotient_ab_group_isomorphism_quotient_ab_group (group_isomorphism_of_lm_isomorphism φ) h)
|
||||
(to_respect_smul (quotient_module_functor φ (λg, iff.mp (h g))))
|
||||
|
||||
/- specific submodules -/
|
||||
definition has_scalar_image (φ : M₁ →lm M₂) ⦃m : M₂⦄ (r : R)
|
||||
(h : image φ m) : image φ (r • m) :=
|
||||
|
|
@ -455,6 +438,57 @@ definition is_surjective_of_is_contr_homology_of_is_contr (ψ : M₂ →lm M₃)
|
|||
(H₁ : is_contr (homology ψ φ)) (H₂ : is_contr M₃) : is_surjective φ :=
|
||||
is_surjective_of_is_contr_homology_of_constant ψ H₁ (λm, @eq_of_is_contr _ H₂ _ _)
|
||||
|
||||
definition cokernel_module (φ : M₁ →lm M₂) : LeftModule R :=
|
||||
quotient_module (image φ)
|
||||
|
||||
definition cokernel_module_isomorphism_homology (φ : M₁ →lm M₂) :
|
||||
homology (trivial_homomorphism M₂ (trivial_LeftModule R)) φ ≃lm cokernel_module φ :=
|
||||
quotient_module_isomorphism_quotient_module
|
||||
(submodule_isomorphism _ (λm, idp))
|
||||
begin intro m, reflexivity end
|
||||
|
||||
open chain_complex fin nat
|
||||
definition LES_of_SESs.{u} {N : succ_str} (A B C : N → LeftModule.{_ u} R) (φ : Πn, A n →lm B n)
|
||||
(ses : Πn : N, short_exact_mod (cokernel_module (φ (succ_str.S n))) (C n) (kernel_module (φ n))) :
|
||||
chain_complex.{_ u} (stratified N 2) :=
|
||||
begin
|
||||
fapply chain_complex.mk,
|
||||
{ intro x, apply @pSet_of_LeftModule R,
|
||||
induction x with n k, induction k with k H, do 3 (cases k with k; rotate 1),
|
||||
{ /-k≥3-/ exfalso, apply lt_le_antisymm H, apply le_add_left},
|
||||
{ /-k=0-/ exact B n },
|
||||
{ /-k=1-/ exact A n },
|
||||
{ /-k=2-/ exact C n }},
|
||||
{ intro x, apply @pmap_of_homomorphism R,
|
||||
induction x with n k, induction k with k H, do 3 (cases k with k; rotate 1),
|
||||
{ /-k≥3-/ exfalso, apply lt_le_antisymm H, apply le_add_left},
|
||||
{ /-k=0-/ exact φ n },
|
||||
{ /-k=1-/ exact submodule_incl _ ∘lm short_exact_mod.g (ses n) },
|
||||
{ /-k=2-/ change B (succ_str.S n) →lm C n, exact short_exact_mod.f (ses n) ∘lm !quotient_map }},
|
||||
{ intros x m, induction x with n k, induction k with k H, do 3 (cases k with k; rotate 1),
|
||||
{ exfalso, apply lt_le_antisymm H, apply le_add_left},
|
||||
{ exact (short_exact_mod.g (ses n) m).2 },
|
||||
{ exact ap pr1 (is_short_exact.im_in_ker (short_exact_mod.h (ses n)) (quotient_map _ m)) },
|
||||
{ exact ap (short_exact_mod.f (ses n)) (quotient_map_eq_zero _ _ (image.mk m idp)) ⬝
|
||||
to_respect_zero (short_exact_mod.f (ses n)) }}
|
||||
end
|
||||
|
||||
definition is_exact_LES_of_SESs.{u} {N : succ_str} (A B C : N → LeftModule.{_ u} R) (φ : Πn, A n →lm B n)
|
||||
(ses : Πn : N, short_exact_mod (cokernel_module (φ (succ_str.S n))) (C n) (kernel_module (φ n))) :
|
||||
is_exact (LES_of_SESs A B C φ ses) :=
|
||||
begin
|
||||
intros x m p, induction x with n k, induction k with k H, do 3 (cases k with k; rotate 1),
|
||||
{ exfalso, apply lt_le_antisymm H, apply le_add_left},
|
||||
{ induction is_short_exact.is_surj (short_exact_mod.h (ses n)) ⟨m, p⟩ with m' q,
|
||||
exact image.mk m' (ap pr1 q) },
|
||||
{ induction is_short_exact.ker_in_im (short_exact_mod.h (ses n)) m (subtype_eq p) with m' q,
|
||||
induction m' using set_quotient.rec_prop with m',
|
||||
exact image.mk m' q },
|
||||
{ apply rel_of_quotient_map_eq_zero (image (φ (succ_str.S n))) m,
|
||||
apply @is_injective_of_is_embedding _ _ _ (is_short_exact.is_emb (short_exact_mod.h (ses n))),
|
||||
exact p ⬝ (to_respect_zero (short_exact_mod.f (ses n)))⁻¹ }
|
||||
end
|
||||
|
||||
-- remove:
|
||||
|
||||
-- definition homology.rec (P : homology ψ φ → Type)
|
||||
|
|
|
|||
|
|
@ -157,6 +157,10 @@ end sigma open sigma
|
|||
|
||||
namespace group
|
||||
|
||||
definition group_homomorphism_of_add_group_homomorphism [constructor] {G₁ G₂ : AddGroup}
|
||||
(φ : G₁ →a G₂) : G₁ →g G₂ :=
|
||||
φ
|
||||
|
||||
|
||||
-- definition is_equiv_isomorphism
|
||||
|
||||
|
|
|
|||
Loading…
Reference in a new issue