start on convergence theorem
This commit is contained in:
Floris van Doorn
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1b4c40413e
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2 changed files with 164 additions and 48 deletions
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@ -2,7 +2,7 @@
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-- Author: Floris van Doorn
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import .left_module .direct_sum .submodule ..heq
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import .left_module .direct_sum .submodule --..heq
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open algebra eq left_module pointed function equiv is_equiv is_trunc prod group sigma
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@ -183,6 +183,9 @@ infixl ` ⬝epgm `:75 := graded_iso.eq_trans
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definition graded_hom_of_eq [constructor] {M₁ M₂ : graded_module R I} (p : M₁ ~ M₂) : M₁ →gm M₂ :=
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proof graded_iso_of_eq p qed
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definition foo {I : Set} (P : I → Type) {i j : I} (M : P i) (N : P j) := unit
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notation M ` ==[`:50 P:0 `] `:0 N:50 := foo P M N
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definition graded_homotopy (f g : M₁ →gm M₂) : Type :=
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Π⦃i j k⦄ (p : deg f i = j) (q : deg g i = k) (m : M₁ i), f ↘ p m ==[λi, M₂ i] g ↘ q m
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-- mk' :: (hd : deg f ~ deg g)
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@ -199,7 +202,7 @@ infix ` ~gm `:50 := graded_homotopy
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definition graded_homotopy.mk (h : Πi m, f i m ==[λi, M₂ i] g i m) : f ~gm g :=
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begin
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intros i j k p q m, induction q, induction p, exact h i m
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intros i j k p q m, induction q, induction p, constructor --exact h i m
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end
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-- definition graded_hom_compose_out {d₁ d₂ : I ≃ I} (f₂ : Πi, M₂ i →lm M₃ (d₂ i))
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@ -318,7 +321,8 @@ definition graded_submodule_incl [constructor] (S : Πi, submodule_rel (M i)) :
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graded_submodule S →gm M :=
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graded_hom.mk erfl (λi, submodule_incl (S i))
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definition graded_hom_lift [constructor] {S : Πi, submodule_rel (M₂ i)} (φ : M₁ →gm M₂)
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definition graded_hom_lift [constructor] {S : Πi, submodule_rel (M₂ i)}
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(φ : M₁ →gm M₂)
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(h : Π(i : I) (m : M₁ i), S (deg φ i) (φ i m)) : M₁ →gm graded_submodule S :=
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graded_hom.mk (deg φ) (λi, hom_lift (φ i) (h i))
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@ -361,7 +365,7 @@ theorem graded_image'_elim_compute {f : M₁ →gm M₂} {g : M₁ →gm M₃}
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graded_image'_elim g h ∘gm graded_image'_lift f ~gm g :=
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begin
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apply graded_homotopy.mk,
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intro i m, reflexivity
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intro i m, exact sorry --reflexivity
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end
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theorem graded_image_elim_compute {f : M₁ →gm M₂} {g : M₁ →gm M₃}
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@ -406,6 +410,7 @@ begin
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-- induction (is_prop.elim (λi, to_right_inv (deg f) (β i)) p),
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-- apply graded_homotopy.mk,
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-- intro i m, reflexivity
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exact sorry
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end
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definition graded_kernel (f : M₁ →gm M₂) : graded_module R I :=
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@ -443,26 +448,46 @@ begin intro i j k p q; induction p; induction q; split, apply h₁, apply h₂ e
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definition gmod_im_in_ker (h : is_exact_gmod f f') : compose_constant f' f :=
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λi j k p q, is_exact.im_in_ker (h p q)
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structure exact_couple (M₁ M₂ : graded_module R I) : Type :=
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(i : M₁ →gm M₁) (j : M₁ →gm M₂) (k : M₂ →gm M₁)
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(exact_ij : is_exact_gmod i j)
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(exact_jk : is_exact_gmod j k)
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(exact_ki : is_exact_gmod k i)
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-- structure exact_couple (M₁ M₂ : graded_module R I) : Type :=
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-- (i : M₁ →gm M₁) (j : M₁ →gm M₂) (k : M₂ →gm M₁)
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-- (exact_ij : is_exact_gmod i j)
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-- (exact_jk : is_exact_gmod j k)
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-- (exact_ki : is_exact_gmod k i)
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end left_module
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namespace left_module
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structure exact_couple (R : Ring) (I : Set) : Type :=
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(D E : graded_module R I)
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(i : D →gm D) (j : D →gm E) (k : E →gm D)
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(ij : is_exact_gmod i j)
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(jk : is_exact_gmod j k)
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(ki : is_exact_gmod k i)
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namespace derived_couple
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section
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parameters {R : Ring} {I : Set} {D E : graded_module R I}
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{i : D →gm D} {j : D →gm E} {k : E →gm D}
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(exact_ij : is_exact_gmod i j) (exact_jk : is_exact_gmod j k) (exact_ki : is_exact_gmod k i)
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open exact_couple
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parameters {R : Ring} {I : Set} (X : exact_couple R I)
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local abbreviation D := D X
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local abbreviation E := E X
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local abbreviation i := i X
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local abbreviation j := j X
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local abbreviation k := k X
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local abbreviation ij := ij X
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local abbreviation jk := jk X
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local abbreviation ki := ki X
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definition d : E →gm E := j ∘gm k
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definition D' : graded_module R I := graded_image i
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definition E' : graded_module R I := graded_homology d d
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include exact_jk exact_ki exact_ij
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definition is_contr_E' {x : I} (H : is_contr (E x)) : is_contr (E' x) :=
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sorry
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definition is_contr_D' {x : I} (H : is_contr (D x)) : is_contr (D' x) :=
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sorry
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definition i' : D' →gm D' :=
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graded_image_lift i ∘gm graded_submodule_incl _
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@ -471,18 +496,19 @@ namespace left_module
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theorem j_lemma1 ⦃x : I⦄ (m : D x) : d ((deg j) x) (j x m) = 0 :=
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begin
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rewrite [graded_hom_compose_fn,↑d,graded_hom_compose_fn],
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refine ap (graded_hom_fn j (deg k (deg j x))) _ ⬝ !to_respect_zero,
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exact compose_constant.elim (gmod_im_in_ker exact_jk) x m
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refine ap (graded_hom_fn j (deg k (deg j x))) _ ⬝
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!to_respect_zero,
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exact compose_constant.elim (gmod_im_in_ker (jk)) x m
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end
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theorem j_lemma2 : Π⦃x : I⦄ ⦃m : D x⦄ (p : i x m = 0),
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(graded_quotient_map _ ∘gm graded_hom_lift j j_lemma1) x m = 0 :> E' _ :=
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begin
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have Π⦃x y : I⦄ (q : deg k x = y) (r : deg d x = deg j y) (s : ap (deg j) q = r) ⦃m : D y⦄
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(p : i y m = 0), image (d ↘ r) (j y m),
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have Π⦃x y : I⦄ (q : deg k x = y) (r : deg d x = deg j y)
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(s : ap (deg j) q = r) ⦃m : D y⦄ (p : i y m = 0), image (d ↘ r) (j y m),
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begin
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intros, induction s, induction q,
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note m_in_im_k := is_exact.ker_in_im (exact_ki idp _) _ p,
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note m_in_im_k := is_exact.ker_in_im (ki idp _) _ p,
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induction m_in_im_k with e q,
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induction q,
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apply image.mk e idp
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@ -502,57 +528,132 @@ namespace left_module
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definition j' : D' →gm E' :=
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graded_image_elim (graded_homology_intro d d ∘gm graded_hom_lift j j_lemma1) j_lemma2
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-- degree -(deg i) + deg j
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-- degree deg j - deg i
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theorem k_lemma1 ⦃x : I⦄ (m : E x) : image (i ← (deg k x)) (k x m) :=
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begin
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exact sorry
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end
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theorem k_lemma2 : compose_constant (graded_hom_lift k k_lemma1 : E →gm D') d :=
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begin
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-- apply compose_constant.mk, intro x m,
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-- rewrite [graded_hom_compose_fn],
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-- refine ap (graded_hom_fn (graded_image_lift i) (deg k (deg d x))) _ ⬝ !to_respect_zero,
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-- exact compose_constant.elim (gmod_im_in_ker jk) (deg k x) (k x m)
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exact sorry
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end
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definition k' : E' →gm D' :=
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graded_homology_elim (graded_image_lift i ∘gm k)
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abstract begin
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apply compose_constant.mk, intro x m,
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rewrite [graded_hom_compose_fn],
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refine ap (graded_hom_fn (graded_image_lift i) (deg k (deg d x))) _ ⬝ !to_respect_zero,
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exact compose_constant.elim (gmod_im_in_ker exact_jk) (deg k x) (k x m)
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end end
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-- degree deg i + deg k
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graded_homology_elim (graded_hom_lift k k_lemma1) k_lemma2
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theorem is_exact_i'j' : is_exact_gmod i' j' :=
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definition deg_i' : deg i' ~ deg i := by reflexivity
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definition deg_j' : deg j' ~ deg j ∘ (deg i)⁻¹ := by reflexivity
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definition deg_k' : deg k' ~ deg k := by reflexivity
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theorem i'j' : is_exact_gmod i' j' :=
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begin
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apply is_exact_gmod.mk,
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{ intro x, refine total_image.rec _, intro m,
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exact calc
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j' (deg i' x) (i' x ⟨(i ← x) m, image.mk m idp⟩)
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= j' (deg i' x) (graded_image_lift i x ((i ← x) m)) : idp
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... = graded_homology_intro d d (deg j ((deg i)⁻¹ᵉ (deg i x)))
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(graded_hom_lift j j_lemma1 ((deg i)⁻¹ᵉ (deg i x))
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(i ↘ (!to_right_inv ⬝ !to_left_inv⁻¹) m)) : _
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... = graded_homology_intro d d (deg j ((deg i)⁻¹ᵉ (deg i x)))
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(graded_hom_lift j j_lemma1 ((deg i)⁻¹ᵉ (deg i x))
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(i ↘ (!to_right_inv ⬝ !to_left_inv⁻¹) m)) : _
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... = 0 : _
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{ intro x, refine total_image.rec _, intro m, exact sorry
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-- exact calc
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-- j' (deg i' x) (i' x ⟨(i ← x) m, image.mk m idp⟩)
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-- = j' (deg i' x) (graded_image_lift i x ((i ← x) m)) : idp
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-- ... = graded_homology_intro d d (deg j ((deg i)⁻¹ᵉ (deg i x)))
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-- (graded_hom_lift j j_lemma1 ((deg i)⁻¹ᵉ (deg i x))
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-- (i ↘ (!to_right_inv ⬝ !to_left_inv⁻¹) m)) : _
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-- ... = graded_homology_intro d d (deg j ((deg i)⁻¹ᵉ (deg i x)))
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-- (graded_hom_lift j j_lemma1 ((deg i)⁻¹ᵉ (deg i x))
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-- (i ↘ (!to_right_inv ⬝ !to_left_inv⁻¹) m)) : _
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-- ... = 0 : _
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},
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{ exact sorry }
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end
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theorem is_exact_j'k' : is_exact_gmod j' k' :=
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theorem j'k' : is_exact_gmod j' k' :=
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begin
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apply is_exact_gmod.mk,
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{ },
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{ exact sorry },
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{ exact sorry }
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end
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theorem is_exact_k'i' : is_exact_gmod k' i' :=
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theorem k'i' : is_exact_gmod k' i' :=
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begin
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apply is_exact_gmod.mk,
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{ intro x m, },
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{ intro x m, exact sorry },
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{ exact sorry }
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end
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end
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-- homomorphism_fn (graded_hom_fn j' (to_fun (deg i') i))
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-- (homomorphism_fn (graded_hom_fn i' i) m) = 0
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end derived_couple
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section
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open derived_couple exact_couple
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definition derived_couple [constructor] {R : Ring} {I : Set}
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(X : exact_couple R I) : exact_couple R I :=
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⦃exact_couple, D := D' X, E := E' X, i := i' X, j := j' X, k := k' X,
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ij := i'j' X, jk := j'k' X, ki := k'i' X⦄
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parameters {R : Ring} {I : Set} (X : exact_couple R I) (B : I → ℕ)
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(Dub : Π⦃x y⦄ ⦃s : ℕ⦄, B x ≤ s → deg (j X) (iterate (deg (i X)) s x) = y → is_contr (D X y))
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(Dlb : Π⦃x y⦄ ⦃s : ℕ⦄, B x ≤ s → deg (j X) (iterate (deg (i X))⁻¹ s x) = y → is_contr (D X y))
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(Eub : Π⦃x y⦄ ⦃s : ℕ⦄, B x ≤ s → deg (j X) (iterate (deg (i X)) s x) = y → is_contr (E X y))
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(Elb : Π⦃x y⦄ ⦃s : ℕ⦄, B x ≤ s → deg (j X) (iterate (deg (i X))⁻¹ s x) = y → is_contr (E X y))
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-- also need a single deg j and/or deg k here
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-- we start counting pages at 0, not at 2.
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definition page (r : ℕ) : exact_couple R I :=
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iterate derived_couple r X
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definition is_contr_E (r : ℕ) (x : I) (h : is_contr (E X x)) :
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is_contr (E (page r) x) :=
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by induction r with r IH; exact h; exact is_contr_E' (page r) IH
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definition is_contr_D (r : ℕ) (x : I) (h : is_contr (D X x)) :
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is_contr (D (page r) x) :=
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by induction r with r IH; exact h; exact is_contr_D' (page r) IH
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definition deg_i (r : ℕ) : deg (i (page r)) ~ deg (i X) :=
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begin
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induction r with r IH,
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{ reflexivity },
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{ exact IH }
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end
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definition deg_k (r : ℕ) : deg (k (page r)) ~ deg (k X) :=
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begin
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induction r with r IH,
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{ reflexivity },
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{ exact IH }
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end
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definition deg_j (r : ℕ) :
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deg (j (page r)) ~ deg (j X) ∘ iterate (deg (i X))⁻¹ r :=
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begin
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induction r with r IH,
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{ reflexivity },
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{ refine hwhisker_left (deg (j (page r)))
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(to_inv_homotopy_inv (deg_i r)) ⬝hty _,
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refine hwhisker_right _ IH ⬝hty _,
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apply hwhisker_left, symmetry, apply iterate_succ }
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end
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definition deg_d (r : ℕ) :
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deg (d (page r)) ~ deg (j X) ∘ iterate (deg (i X))⁻¹ r ∘ deg (k X) :=
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compose2 (deg_j r) (deg_k r)
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definition Eub' (x : I) (r : ℕ) (h : B (deg (k X) x) ≤ r) :
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is_contr (E (page r) (deg (d (page r)) x)) :=
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is_contr_E _ _ (Elb h (deg_d r x)⁻¹)
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definition Estable {x : I} {r : ℕ} (H : B (deg (k X) x) ≤ r) :
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E (page (r + 1)) x ≃ E (page r) x :=
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sorry
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definition inf_page : graded_module R I :=
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λx, E (page (B (deg (k X) x))) x
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end
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end left_module
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@ -305,6 +305,17 @@ end
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-- (p : f ~ g) (q : a = a') : natural_square p q = square_of_pathover (apd p q) :=
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-- idp
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definition inv_homotopy_inv {A B : Type} {f g : A → B} [is_equiv f] [is_equiv g]
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(p : f ~ g) : f⁻¹ ~ g⁻¹ :=
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λa, inv_eq_of_eq (p (g⁻¹ a) ⬝ right_inv g a)⁻¹
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definition to_inv_homotopy_inv {A B : Type} {f g : A ≃ B}
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(p : f ~ g) : f⁻¹ ~ g⁻¹ :=
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inv_homotopy_inv p
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definition compose2 {A B C : Type} {g g' : B → C} {f f' : A → B}
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(p : g ~ g') (q : f ~ f') : g ∘ f ~ g' ∘ f' :=
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hwhisker_right f p ⬝hty hwhisker_left g' q
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section hsquare
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variables {A₀₀ A₂₀ A₄₀ A₀₂ A₂₂ A₄₂ A₀₄ A₂₄ A₄₄ : Type}
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@ -1142,6 +1153,10 @@ section
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show a = 0, from is_injective_of_is_embedding this
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end
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definition iterate_succ {A : Type} (f : A → A) (n : ℕ) (x : A) :
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iterate f (succ n) x = iterate f n (f x) :=
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by induction n with n p; reflexivity; exact ap f p
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/- put somewhere in algebra -/
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structure Ring :=
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