define spectral sequence from exact couple
this also defines the actual spectral sequences for the Atiyah-Hirzebruch and Serre spectral sequences. We need to reindex convergent_exact_couple_sequence to get a spectral sequence with the correct abutment from it.
This commit is contained in:
Floris van Doorn
parent
96e11300ed
commit
4481935a83
2 changed files with 143 additions and 100 deletions
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@ -382,11 +382,11 @@ namespace left_module
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have H : deg (j (page r)) ~ iterate_equiv (deg (i X))⁻¹ᵉ r ⬝e deg (j X), from deg_j r,
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λx, to_inv_homotopy_inv H x ⬝ iterate_inv (deg (i X))⁻¹ᵉ r ((deg (j X))⁻¹ x)
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definition deg_d (r : ℕ) :
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definition deg_dr (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 deg_d_inv (r : ℕ) :
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definition deg_dr_inv (r : ℕ) :
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(deg (d (page r)))⁻¹ ~ (deg (k X))⁻¹ ∘ iterate (deg (i X)) r ∘ (deg (j X))⁻¹ :=
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compose2 (to_inv_homotopy_inv (deg_k r)) (deg_j_inv r)
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@ -437,9 +437,9 @@ namespace left_module
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begin
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change homology (d (page r) x) (d (page r) ← x) ≃lm E (page r) x,
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apply homology_isomorphism: apply is_contr_E,
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exact Eub (hhinverse (deg_iterate_ik_commute r) _ ⬝ (deg_d_inv r x)⁻¹)
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exact Eub (hhinverse (deg_iterate_ik_commute r) _ ⬝ (deg_dr_inv r x)⁻¹)
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(le.trans !le_max_right H),
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exact Elb (deg_iterate_ij_commute r _ ⬝ (deg_d r x)⁻¹)
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exact Elb (deg_iterate_ij_commute r _ ⬝ (deg_dr r x)⁻¹)
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(le.trans !le_max_left H)
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end
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@ -575,12 +575,12 @@ namespace left_module
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end
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definition deg_d_reindex {R : Ring} {I J : AddAbGroup}
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definition deg_dr_reindex {R : Ring} {I J : AddAbGroup}
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(e : AddGroup_of_AddAbGroup J ≃g AddGroup_of_AddAbGroup I) (X : exact_couple R I)
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(r : ℕ) : deg (d (page (exact_couple_reindex e X) r)) ~
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e⁻¹ᵍ ∘ deg (d (page X r)) ∘ e :=
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begin
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intro x, refine !deg_d ⬝ _ ⬝ ap e⁻¹ᵍ !deg_d⁻¹,
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intro x, refine !deg_dr ⬝ _ ⬝ ap e⁻¹ᵍ !deg_dr⁻¹,
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apply ap (e⁻¹ᵍ ∘ deg (j X)),
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note z := @iterate_hsquare _ _ (deg (i (exact_couple_reindex e X)))⁻¹ᵉ (deg (i X))⁻¹ᵉ e
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(λx, proof to_right_inv (group.equiv_of_isomorphism e) _ qed)⁻¹ʰᵗʸʰ r,
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@ -598,18 +598,37 @@ namespace left_module
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(Dinf : agℤ → LeftModule.{u w} R) : Type.{max u (v+1) (w+1)} :=
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(X : exact_couple.{u 0 w v} R Z2)
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(HH : is_bounded X)
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(s₁ : agℤ → agℤ) (s₂ : agℤ → agℤ)
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(HB : Π(n : agℤ), is_bounded.B' HH (deg (k X) (s₁ n, s₂ n)) = 0)
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(st : agℤ → Z2)
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(HB : Π(n : agℤ), is_bounded.B' HH (deg (k X) (st n)) = 0)
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(e : Π(x : Z2), exact_couple.E X x ≃lm E' x.1 x.2)
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(f : Π(n : agℤ), exact_couple.D X (deg (k X) (s₁ n, s₂ n)) ≃lm Dinf n)
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(deg_d1 : ℕ → agℤ) (deg_d2 : ℕ → agℤ)
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(deg_d_eq0 : Π(r : ℕ), deg (d (page X r)) 0 = (deg_d1 r, deg_d2 r))
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(f : Π(n : agℤ), exact_couple.D X (deg (k X) (st n)) ≃lm Dinf n)
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(deg_d : ℕ → Z2)
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(deg_d_eq0 : Π(r : ℕ), deg (d (page X r)) 0 = deg_d r)
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infix ` ⟹ `:25 := convergent_exact_couple
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structure convergent_spectral_sequence.{u v w} {R : Ring} (E' : agℤ → agℤ → LeftModule.{u v} R)
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(Dinf : agℤ → LeftModule.{u w} R) : Type.{max u (v+1) (w+1)} :=
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(E : ℕ → graded_module.{u 0 v} R Z2)
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(d : Π(r : ℕ), E r →gm E r)
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(deg_d : ℕ → Z2)
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(deg_d_eq0 : Π(r : ℕ), deg (d r) 0 = deg_d r)
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(α : Π(r : ℕ) (x : Z2), E (r+1) x ≃lm graded_homology (d r) (d r) x)
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(e : Π(x : Z2), E 0 x ≃lm E' x.1 x.2)
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(s₀ : Z2 → ℕ)
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(f : Π{r : ℕ} {x : Z2} (h : s₀ x ≤ r), E (s₀ x) x ≃lm E r x)
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(lb : ℤ → ℤ)
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(HDinf : Π(n : agℤ), is_built_from (Dinf n)
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(λ(k : ℕ), (λx, E (s₀ x) x) (n - (k + lb n), k + lb n)))
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infix ` ⟹ `:25 := convergent_exact_couple -- todo: maybe this should define convergent_spectral_sequence
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definition convergent_exact_couple_g [reducible] (E' : agℤ → agℤ → AbGroup) (Dinf : agℤ → AbGroup) : Type :=
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(λn s, LeftModule_int_of_AbGroup (E' n s)) ⟹ (λn, LeftModule_int_of_AbGroup (Dinf n))
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definition convergent_spectral_sequence_g [reducible] (E' : agℤ → agℤ → AbGroup) (Dinf : agℤ → AbGroup) :
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Type :=
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convergent_spectral_sequence (λn s, LeftModule_int_of_AbGroup (E' n s))
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(λn, LeftModule_int_of_AbGroup (Dinf n))
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infix ` ⟹ᵍ `:25 := convergent_exact_couple_g
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section
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@ -618,76 +637,52 @@ namespace left_module
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local abbreviation X := X c
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local abbreviation i := i X
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local abbreviation HH := HH c
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local abbreviation s₁ := s₁ c
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local abbreviation s₂ := s₂ c
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local abbreviation Dinfdiag (n : agℤ) (k : ℕ) := Dinfdiag HH (s₁ n, s₂ n) k
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local abbreviation Einfdiag (n : agℤ) (k : ℕ) := Einfdiag HH (s₁ n, s₂ n) k
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local abbreviation st := st c
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local abbreviation Dinfdiag (n : agℤ) (k : ℕ) := Dinfdiag HH (st n) k
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local abbreviation Einfdiag (n : agℤ) (k : ℕ) := Einfdiag HH (st n) k
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definition deg_d_eq (r : ℕ) (n s : agℤ) :
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(deg (d (page X r))) (n, s) = (n + deg_d1 c r, s + deg_d2 c r) :=
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definition deg_d_eq (r : ℕ) (ns : Z2) : (deg (d (page X r))) ns = ns + deg_d c r :=
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!deg_eq ⬝ ap (λx, _ + x) (deg_d_eq0 c r)
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definition deg_d_inv_eq (r : ℕ) (n s : agℤ) :
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(deg (d (page X r)))⁻¹ᵉ (n, s) = (n - deg_d1 c r, s - deg_d2 c r) :=
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inv_eq_of_eq (!deg_d_eq ⬝ prod_eq !sub_add_cancel !sub_add_cancel)⁻¹
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definition deg_d_inv_eq (r : ℕ) (ns : Z2) :
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(deg (d (page X r)))⁻¹ᵉ ns = ns - deg_d c r :=
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inv_eq_of_eq (!deg_d_eq ⬝ proof !neg_add_cancel_right qed)⁻¹
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definition convergent_exact_couple_isomorphism {E'' : agℤ → agℤ → LeftModule R} {Dinf' : graded_module R agℤ}
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(e' : Πn s, E' n s ≃lm E'' n s) (f' : Πn, Dinf n ≃lm Dinf' n) : E'' ⟹ Dinf' :=
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convergent_exact_couple.mk X HH s₁ s₂ (HB c)
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convergent_exact_couple.mk X HH st (HB c)
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begin intro x, induction x with n s, exact e c (n, s) ⬝lm e' n s end
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(λn, f c n ⬝lm f' n)
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(deg_d1 c) (deg_d2 c) (λr, deg_d_eq0 c r)
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(λn, f c n ⬝lm f' n) (deg_d c) (deg_d_eq0 c)
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include c
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definition convergent_exact_couple_reindex (i : agℤ ×ag agℤ ≃g agℤ ×ag agℤ) :
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(λp q, E' (i (p, q)).1 (i (p, q)).2) ⟹ Dinf :=
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begin
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fapply convergent_exact_couple.mk (exact_couple_reindex i X) (is_bounded_reindex i HH),
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{ exact λn, (i⁻¹ᵍ (s₁ n, s₂ n)).1 },
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{ exact λn, (i⁻¹ᵍ (s₁ n, s₂ n)).2 },
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{ exact λn, i⁻¹ᵍ (st n) },
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{ intro n, refine ap (B' HH) _ ⬝ HB c n,
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refine to_right_inv (group.equiv_of_isomorphism i) _ ⬝ _,
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apply ap (deg (k X)), exact ap i !prod.eta ⬝ to_right_inv (group.equiv_of_isomorphism i) _ },
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apply ap (deg (k X)), exact to_right_inv (group.equiv_of_isomorphism i) _ },
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{ intro x, induction x with p q, exact e c (i (p, q)) },
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{ intro n, refine _ ⬝lm f c n, refine isomorphism_of_eq (ap (D X) _),
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refine to_right_inv (group.equiv_of_isomorphism i) _ ⬝ _,
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apply ap (deg (k X)), exact ap i !prod.eta ⬝ to_right_inv (group.equiv_of_isomorphism i) _ },
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{ exact λn, (i⁻¹ᵍ (deg_d1 c n, deg_d2 c n)).1 },
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{ exact λn, (i⁻¹ᵍ (deg_d1 c n, deg_d2 c n)).2 },
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{ intro r, esimp, refine !deg_d_reindex ⬝ ap i⁻¹ᵍ (ap (deg (d _)) (group.to_respect_zero i) ⬝
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deg_d_eq0 c r) ⬝ !prod.eta⁻¹ }
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apply ap (deg (k X)), exact to_right_inv (group.equiv_of_isomorphism i) _ },
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{ exact λn, i⁻¹ᵍ (deg_d c n) },
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{ intro r, esimp, refine !deg_dr_reindex ⬝ ap i⁻¹ᵍ (ap (deg (d _)) (group.to_respect_zero i) ⬝
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deg_d_eq0 c r) }
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end
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definition convergent_exact_couple_negate_inf : E' ⟹ (λn, Dinf (-n)) :=
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convergent_exact_couple.mk X HH (s₁ ∘ neg) (s₂ ∘ neg) (λn, HB c (-n)) (e c) (λn, f c (-n))
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(deg_d1 c) (deg_d2 c) (deg_d_eq0 c)
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definition convergent_exact_couple_negate_abutment : E' ⟹ (λn, Dinf (-n)) :=
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convergent_exact_couple.mk X HH (st ∘ neg) (λn, HB c (-n)) (e c) (λn, f c (-n))
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(deg_d c) (deg_d_eq0 c)
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omit c
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/- definition convergent_exact_couple_reindex {E'' : agℤ → agℤ → LeftModule R} {Dinf' : graded_module R agℤ}
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(i : agℤ ×g agℤ ≃ agℤ × agℤ) (e' : Πp q, E' p q ≃lm E'' (i (p, q)).1 (i (p, q)).2)
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(i2 : agℤ ≃ agℤ) (f' : Πn, Dinf n ≃lm Dinf' (i2 n)) :
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(λp q, E'' p q) ⟹ λn, Dinf' n :=
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convergent_exact_couple.mk (exact_couple_reindex i X) (is_bounded_reindex i HH) s₁ s₂
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sorry --(λn, ap (B' HH) (to_right_inv i _ ⬝ begin end) ⬝ HB c n)
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sorry --begin intro x, induction x with p q, exact e c (p, q) ⬝lm e' p q end
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sorry-/
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/- definition convergent_exact_couple_reindex_neg {E'' : agℤ → agℤ → LeftModule R} {Dinf' : graded_module R agℤ}
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(e' : Πp q, E' p q ≃lm E'' (-p) (-q))
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(f' : Πn, Dinf n ≃lm Dinf' (-n)) :
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(λp q, E'' p q) ⟹ λn, Dinf' n :=
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convergent_exact_couple.mk (exact_couple_reindex (equiv_neg ×≃ equiv_neg) X) (is_bounded_reindex _ HH)
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(λn, -s₂ (-n))
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(λn, ap (B' HH) (begin esimp, end) ⬝ HB c n)
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sorry --begin intro x, induction x with p q, exact e c (p, q) ⬝lm e' p q end
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sorry-/
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definition is_built_from_of_convergent_exact_couple (n : ℤ) : is_built_from (Dinf n) (Einfdiag n) :=
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is_built_from_isomorphism_left (f c n) (is_built_from_infpage HH (s₁ n, s₂ n) (HB c n))
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is_built_from_isomorphism_left (f c n) (is_built_from_infpage HH (st n) (HB c n))
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/- TODO: reprove this using is_built_of -/
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theorem is_contr_convergent_exact_couple_precise (n : agℤ)
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(H : Π(n : agℤ) (l : ℕ), is_contr (E' ((deg i)^[l] (s₁ n, s₂ n)).1 ((deg i)^[l] (s₁ n, s₂ n)).2)) :
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(H : Π(n : agℤ) (l : ℕ), is_contr (E' ((deg i)^[l] (st n)).1 ((deg i)^[l] (st n)).2)) :
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is_contr (Dinf n) :=
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begin
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assert H2 : Π(l : ℕ), is_contr (Einfdiag n l),
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@ -695,71 +690,70 @@ namespace left_module
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refine is_trunc_equiv_closed_rev -2 (equiv_of_isomorphism (e c _)) (H n l) },
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assert H3 : is_contr (Dinfdiag n 0),
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{ fapply nat.rec_down (λk, is_contr (Dinfdiag n k)),
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{ exact is_bounded.B HH (deg (k X) (s₁ n, s₂ n)) },
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{ exact is_bounded.B HH (deg (k X) (st n)) },
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{ apply Dinfdiag_stable, reflexivity },
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{ intro l H,
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exact is_contr_middle_of_short_exact_mod (short_exact_mod_infpage HH (s₁ n, s₂ n) l)
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exact is_contr_middle_of_short_exact_mod (short_exact_mod_infpage HH (st n) l)
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(H2 l) H }},
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refine is_trunc_equiv_closed _ _ H3,
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exact equiv_of_isomorphism (Dinfdiag0 HH (s₁ n, s₂ n) (HB c n) ⬝lm f c n)
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exact equiv_of_isomorphism (Dinfdiag0 HH (st n) (HB c n) ⬝lm f c n)
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end
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theorem is_contr_convergent_exact_couple (n : agℤ) (H : Π(n s : agℤ), is_contr (E' n s)) : is_contr (Dinf n) :=
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is_contr_convergent_exact_couple_precise n (λn s, !H)
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definition E_isomorphism {r₁ r₂ : ℕ} {n s : agℤ} (H : r₁ ≤ r₂)
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(H1 : Π⦃r⦄, r₁ ≤ r → r < r₂ → is_contr (E X (n - deg_d1 c r, s - deg_d2 c r)))
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(H2 : Π⦃r⦄, r₁ ≤ r → r < r₂ → is_contr (E X (n + deg_d1 c r, s + deg_d2 c r))) :
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(H1 : Π⦃r⦄, r₁ ≤ r → r < r₂ → is_contr (E X ((n, s) - deg_d c r)))
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(H2 : Π⦃r⦄, r₁ ≤ r → r < r₂ → is_contr (E X ((n, s) + deg_d c r))) :
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E (page X r₂) (n, s) ≃lm E (page X r₁) (n, s) :=
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E_isomorphism' X H (λr Hr₁ Hr₂, transport (is_contr ∘ E X) (deg_d_inv_eq r n s)⁻¹ᵖ (H1 Hr₁ Hr₂))
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(λr Hr₁ Hr₂, transport (is_contr ∘ E X) (deg_d_eq r n s)⁻¹ᵖ (H2 Hr₁ Hr₂))
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E_isomorphism' X H
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(λr Hr₁ Hr₂, transport (is_contr ∘ E X) (deg_d_inv_eq r (n, s))⁻¹ᵖ (H1 Hr₁ Hr₂))
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(λr Hr₁ Hr₂, transport (is_contr ∘ E X) (deg_d_eq r (n, s))⁻¹ᵖ (H2 Hr₁ Hr₂))
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definition E_isomorphism0 {r : ℕ} {n s : agℤ}
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(H1 : Πr, is_contr (E X (n - deg_d1 c r, s - deg_d2 c r)))
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(H2 : Πr, is_contr (E X (n + deg_d1 c r, s + deg_d2 c r))) :
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E (page X r) (n, s) ≃lm E' n s :=
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definition E_isomorphism0 {r : ℕ} {n s : agℤ} (H1 : Πr, is_contr (E X ((n, s) - deg_d c r)))
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(H2 : Πr, is_contr (E X ((n, s) + deg_d c r))) : E (page X r) (n, s) ≃lm E' n s :=
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E_isomorphism (zero_le r) _ _ ⬝lm e c (n, s)
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definition Einf_isomorphism (r₁ : ℕ) {n s : agℤ}
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(H1 : Π⦃r⦄, r₁ ≤ r → is_contr (E X (n - deg_d1 c r, s - deg_d2 c r)))
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(H2 : Π⦃r⦄, r₁ ≤ r → is_contr (E X (n + deg_d1 c r, s + deg_d2 c r))) :
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(H1 : Π⦃r⦄, r₁ ≤ r → is_contr (E X ((n,s) - deg_d c r)))
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(H2 : Π⦃r⦄, r₁ ≤ r → is_contr (E X ((n,s) + deg_d c r))) :
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Einf HH (n, s) ≃lm E (page X r₁) (n, s) :=
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Einf_isomorphism' HH r₁ (λr Hr₁, transport (is_contr ∘ E X) (deg_d_inv_eq r n s)⁻¹ᵖ (H1 Hr₁))
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(λr Hr₁, transport (is_contr ∘ E X) (deg_d_eq r n s)⁻¹ᵖ (H2 Hr₁))
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Einf_isomorphism' HH r₁ (λr Hr₁, transport (is_contr ∘ E X) (deg_d_inv_eq r (n, s))⁻¹ᵖ (H1 Hr₁))
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(λr Hr₁, transport (is_contr ∘ E X) (deg_d_eq r (n, s))⁻¹ᵖ (H2 Hr₁))
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definition Einf_isomorphism0 {n s : agℤ}
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(H1 : Π⦃r⦄, is_contr (E X (n - deg_d1 c r, s - deg_d2 c r)))
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(H2 : Π⦃r⦄, is_contr (E X (n + deg_d1 c r, s + deg_d2 c r))) :
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(H1 : Π⦃r⦄, is_contr (E X ((n,s) - deg_d c r)))
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(H2 : Π⦃r⦄, is_contr (E X ((n,s) + deg_d c r))) :
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Einf HH (n, s) ≃lm E' n s :=
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E_isomorphism0 _ _
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definition convergent_spectral_sequence_of_exact_couple
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(st_eq : Πn, (st n).1 + (st n).2 = n) (deg_i_eq : deg i 0 = (-(1 : ℤ), (1 : ℤ))) :
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convergent_spectral_sequence E' Dinf :=
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convergent_spectral_sequence.mk (λr, E (page X r)) (λr, d (page X r)) (deg_d c) (deg_d_eq0 c)
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(λr ns, by reflexivity) (e c) (B3 HH) (λr ns Hr, Einfstable HH Hr idp)
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(λn, (st n).2)
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begin
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intro n,
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||||
refine is_built_from_isomorphism (f c n) _ (is_built_from_infpage HH (st n) (HB c n)),
|
||||
intro p, apply isomorphism_of_eq, apply ap (λx, E (page X (B3 HH x)) x),
|
||||
induction p with p IH,
|
||||
{ exact !prod.eta⁻¹ ⬝ prod_eq (eq_sub_of_add_eq (ap (add _) !zero_add ⬝ st_eq n)) (zero_add (st n).2)⁻¹ },
|
||||
{ assert H1 : Π(a : ℤ), n - (p + a) - (1 : ℤ) = n - (succ p + a),
|
||||
exact λa, !sub_add_eq_sub_sub⁻¹ ⬝ ap (sub n) (add_comm_middle p a (1 : ℤ) ⬝ proof idp qed),
|
||||
assert H2 : Π(a : ℤ), p + a + 1 = succ p + a,
|
||||
exact λa, add_comm_middle p a 1,
|
||||
refine ap (deg i) IH ⬝ !deg_eq ⬝ ap (add _) deg_i_eq ⬝ prod_eq !H1 !H2 }
|
||||
end
|
||||
end
|
||||
|
||||
variables {E' : agℤ → agℤ → AbGroup} {Dinf : agℤ → AbGroup} (c : E' ⟹ᵍ Dinf)
|
||||
definition convergent_exact_couple_g_isomorphism {E'' : agℤ → agℤ → AbGroup} {Dinf' : agℤ → AbGroup}
|
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variables {E' : agℤ → agℤ → AbGroup} {Dinf : agℤ → AbGroup}
|
||||
definition convergent_exact_couple_g_isomorphism {E'' : agℤ → agℤ → AbGroup}
|
||||
{Dinf' : agℤ → AbGroup} (c : E' ⟹ᵍ Dinf)
|
||||
(e' : Πn s, E' n s ≃g E'' n s) (f' : Πn, Dinf n ≃g Dinf' n) : E'' ⟹ᵍ Dinf' :=
|
||||
convergent_exact_couple_isomorphism c (λn s, lm_iso_int.mk (e' n s)) (λn, lm_iso_int.mk (f' n))
|
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|
||||
|
||||
structure convergent_spectral_sequence.{u v w} {R : Ring} (E' : agℤ → agℤ → LeftModule.{u v} R)
|
||||
(Dinf : agℤ → LeftModule.{u w} R) : Type.{max u (v+1) (w+1)} :=
|
||||
(E : ℕ → graded_module.{u 0 v} R Z2)
|
||||
(d : Π(n : ℕ), E n →gm E n)
|
||||
(α : Π(n : ℕ) (x : Z2), E (n+1) x ≃lm graded_homology (d n) (d n) x)
|
||||
(e : Π(n : ℕ) (x : Z2), E 0 x ≃lm E' x.1 x.2)
|
||||
(s₀ : Z2 → ℕ)
|
||||
(f : Π{n : ℕ} {x : Z2} (h : s₀ x ≤ n), E (s₀ x) x ≃lm E n x)
|
||||
(lb : ℤ → ℤ)
|
||||
(HDinf : Π(n : agℤ), is_built_from (Dinf n) (λ(k : ℕ), (λx, E (s₀ x) x) (n - (k + lb n), k + lb n)))
|
||||
|
||||
/-
|
||||
todo:
|
||||
- double-check the definition of converging_spectral_sequence
|
||||
- construct converging spectral sequence from a converging exact couple,
|
||||
- This requires the additional hypothesis that the degree of i is (-1, 1), which is true by reflexivity for the spectral sequences we construct
|
||||
- redefine `⟹` to be a converging spectral sequence
|
||||
-/
|
||||
|
||||
end left_module
|
||||
open left_module
|
||||
namespace pointed
|
||||
|
|
@ -966,14 +960,12 @@ namespace spectrum
|
|||
fapply convergent_exact_couple.mk,
|
||||
{ exact exact_couple_sequence },
|
||||
{ exact is_bounded_sequence },
|
||||
{ exact id },
|
||||
{ exact ub },
|
||||
{ exact λn, (n, ub n) },
|
||||
{ intro n, change max0 (ub n - ub n) = 0, exact ap max0 (sub_self (ub n)) },
|
||||
{ intro ns, reflexivity },
|
||||
{ intro n, reflexivity },
|
||||
{ intro r, exact - (1 : ℤ) },
|
||||
{ intro r, exact r + (1 : ℤ) },
|
||||
{ intro r, refine !convergence_theorem.deg_d ⬝ _,
|
||||
{ intro r, exact (-(1 : ℤ), r + (1 : ℤ)) },
|
||||
{ intro r, refine !convergence_theorem.deg_dr ⬝ _,
|
||||
refine ap (deg j_sequence) !iterate_deg_i_inv ⬝ _,
|
||||
exact prod_eq idp !zero_add }
|
||||
end
|
||||
|
|
|
|||
|
|
@ -178,7 +178,8 @@ section atiyah_hirzebruch
|
|||
|
||||
definition atiyah_hirzebruch_convergence2 :
|
||||
(λn s, opH^-(n-s)[(x : X), πₛ[s] (Y x)]) ⟹ᵍ (λn, pH^n[(x : X), Y x]) :=
|
||||
convergent_exact_couple_g_isomorphism (convergent_exact_couple_negate_inf atiyah_hirzebruch_convergence1)
|
||||
convergent_exact_couple_g_isomorphism
|
||||
(convergent_exact_couple_negate_abutment atiyah_hirzebruch_convergence1)
|
||||
begin
|
||||
intro n s,
|
||||
refine _ ⬝g (parametrized_cohomology_isomorphism_shomotopy_group_spi _ idp)⁻¹ᵍ,
|
||||
|
|
@ -221,6 +222,14 @@ section atiyah_hirzebruch
|
|||
refine !neg_neg_sub_neg ⬝ !add_neg_cancel_right
|
||||
end
|
||||
|
||||
definition atiyah_hirzebruch_spectral_sequence :
|
||||
convergent_spectral_sequence_g (λp q, opH^p[(x : X), πₛ[-q] (Y x)]) (λn, pH^n[(x : X), Y x]) :=
|
||||
begin
|
||||
apply convergent_spectral_sequence_of_exact_couple atiyah_hirzebruch_convergence,
|
||||
{ intro n, exact add.comm (s₀ - -n) (-s₀) ⬝ !neg_add_cancel_left ⬝ !neg_neg },
|
||||
{ reflexivity }
|
||||
end
|
||||
|
||||
end atiyah_hirzebruch
|
||||
|
||||
section unreduced_atiyah_hirzebruch
|
||||
|
|
@ -239,6 +248,16 @@ convergent_exact_couple_g_isomorphism
|
|||
begin
|
||||
intro n, reflexivity
|
||||
end
|
||||
|
||||
definition unreduced_atiyah_hirzebruch_spectral_sequence {X : Type} (Y : X → spectrum) (s₀ : ℤ)
|
||||
(H : Πx, is_strunc s₀ (Y x)) :
|
||||
convergent_spectral_sequence_g (λp q, uopH^p[(x : X), πₛ[-q] (Y x)]) (λn, upH^n[(x : X), Y x]) :=
|
||||
begin
|
||||
apply convergent_spectral_sequence_of_exact_couple (unreduced_atiyah_hirzebruch_convergence Y s₀ H),
|
||||
{ intro n, exact add.comm (s₀ - -n) (-s₀) ⬝ !neg_add_cancel_left ⬝ !neg_neg },
|
||||
{ reflexivity }
|
||||
end
|
||||
|
||||
end unreduced_atiyah_hirzebruch
|
||||
|
||||
section serre
|
||||
|
|
@ -269,6 +288,14 @@ section serre
|
|||
end
|
||||
qed
|
||||
|
||||
definition serre_spectral_sequence :
|
||||
convergent_spectral_sequence_g (λp q, uopH^p[(b : B), uH^q[F b, Y]]) (λn, uH^n[Σ(b : B), F b, Y]) :=
|
||||
begin
|
||||
apply convergent_spectral_sequence_of_exact_couple (serre_convergence F Y s₀ H),
|
||||
{ intro n, exact add.comm (s₀ - -n) (-s₀) ⬝ !neg_add_cancel_left ⬝ !neg_neg },
|
||||
{ reflexivity }
|
||||
end
|
||||
|
||||
definition serre_convergence_map :
|
||||
(λp q, uopH^p[(b : B), uH^q[fiber f b, Y]]) ⟹ᵍ (λn, uH^n[X, Y]) :=
|
||||
proof
|
||||
|
|
@ -278,6 +305,14 @@ section serre
|
|||
begin intro n, apply unreduced_cohomology_isomorphism, exact !sigma_fiber_equiv⁻¹ᵉ end
|
||||
qed
|
||||
|
||||
definition serre_spectral_sequence_map :
|
||||
convergent_spectral_sequence_g (λp q, uopH^p[(b : B), uH^q[fiber f b, Y]]) (λn, uH^n[X, Y]) :=
|
||||
begin
|
||||
apply convergent_spectral_sequence_of_exact_couple (serre_convergence_map f Y s₀ H),
|
||||
{ intro n, exact add.comm (s₀ - -n) (-s₀) ⬝ !neg_add_cancel_left ⬝ !neg_neg },
|
||||
{ reflexivity }
|
||||
end
|
||||
|
||||
definition serre_convergence_of_is_conn (H2 : is_conn 1 B) :
|
||||
(λp q, uoH^p[B, uH^q[F b₀, Y]]) ⟹ᵍ (λn, uH^n[Σ(b : B), F b, Y]) :=
|
||||
proof
|
||||
|
|
@ -287,6 +322,14 @@ section serre
|
|||
begin intro n, reflexivity end
|
||||
qed
|
||||
|
||||
definition serre_spectral_sequence_of_is_conn (H2 : is_conn 1 B) :
|
||||
convergent_spectral_sequence_g (λp q, uoH^p[B, uH^q[F b₀, Y]]) (λn, uH^n[Σ(b : B), F b, Y]) :=
|
||||
begin
|
||||
apply convergent_spectral_sequence_of_exact_couple (serre_convergence_of_is_conn b₀ F Y s₀ H H2),
|
||||
{ intro n, exact add.comm (s₀ - -n) (-s₀) ⬝ !neg_add_cancel_left ⬝ !neg_neg },
|
||||
{ reflexivity }
|
||||
end
|
||||
|
||||
definition serre_convergence_map_of_is_conn (H2 : is_conn 1 B) :
|
||||
(λp q, uoH^p[B, uH^q[fiber f b₀, Y]]) ⟹ᵍ (λn, uH^n[X, Y]) :=
|
||||
proof
|
||||
|
|
@ -296,6 +339,14 @@ section serre
|
|||
begin intro n, apply unreduced_cohomology_isomorphism, exact !sigma_fiber_equiv⁻¹ᵉ end
|
||||
qed
|
||||
|
||||
definition serre_spectral_sequence_map_of_is_conn (H2 : is_conn 1 B) :
|
||||
convergent_spectral_sequence_g (λp q, uoH^p[B, uH^q[fiber f b₀, Y]]) (λn, uH^n[X, Y]) :=
|
||||
begin
|
||||
apply convergent_spectral_sequence_of_exact_couple (serre_convergence_map_of_is_conn b₀ f Y s₀ H H2),
|
||||
{ intro n, exact add.comm (s₀ - -n) (-s₀) ⬝ !neg_add_cancel_left ⬝ !neg_neg },
|
||||
{ reflexivity }
|
||||
end
|
||||
|
||||
end serre
|
||||
|
||||
end spectrum
|
||||
|
|
|
|||
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Reference in a new issue