reindex the AHSS and SSS
Now the 2nd page is the correct indexing.
This commit is contained in:
Floris van Doorn
parent
4d3053daff
commit
f9ce395b1c
2 changed files with 124 additions and 60 deletions
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@ -292,7 +292,8 @@ namespace left_module
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structure is_bounded {R : Ring} {I : AddAbGroup} (X : exact_couple R I) : Type :=
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mk' :: (B B' : I → ℕ)
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(Dub : Π⦃x y⦄ ⦃s : ℕ⦄, (deg (i X))^[s] x = y → B x ≤ s → is_contr (D X y))
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(Dlb : Π⦃x y z⦄ ⦃s : ℕ⦄ (p : deg (i X) x = y), (deg (i X))^[s] y = z → B' z ≤ s → is_surjective (i X ↘ p))
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(Dlb : Π⦃x y z⦄ ⦃s : ℕ⦄ (p : deg (i X) x = y), (deg (i X))^[s] y = z → B' z ≤ s →
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is_surjective (i X ↘ p))
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(Elb : Π⦃x y⦄ ⦃s : ℕ⦄, (deg (i X))⁻¹ᵉ^[s] x = y → B x ≤ s → is_contr (E X y))
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/- Note: Elb proves Dlb for some bound B', but we want tight control over when B' = 0 -/
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@ -495,22 +496,22 @@ namespace left_module
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parameters (x : I)
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definition r (n : ℕ) : ℕ :=
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definition rb (n : ℕ) : ℕ :=
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max (max (B (deg (j X) (deg (k X) x)) + n + 1) (B3 ((deg (i X))^[n] x)))
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(max (B' (deg (k X) ((deg (i X))^[n] x)))
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(max (B' (deg (k X) ((deg (i X))^[n+1] x))) (B ((deg (j X))⁻¹ ((deg (i X))^[n] x)))))
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lemma rb0 (n : ℕ) : r n ≥ n + 1 :=
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lemma rb0 (n : ℕ) : rb n ≥ n + 1 :=
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ge.trans !le_max_left (ge.trans !le_max_left !le_add_left)
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lemma rb1 (n : ℕ) : B (deg (j X) (deg (k X) x)) ≤ r n - (n + 1) :=
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lemma rb1 (n : ℕ) : B (deg (j X) (deg (k X) x)) ≤ rb n - (n + 1) :=
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nat.le_sub_of_add_le (le.trans !le_max_left !le_max_left)
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lemma rb2 (n : ℕ) : B3 ((deg (i X))^[n] x) ≤ r n :=
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lemma rb2 (n : ℕ) : B3 ((deg (i X))^[n] x) ≤ rb n :=
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le.trans !le_max_right !le_max_left
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lemma rb3 (n : ℕ) : B' (deg (k X) ((deg (i X))^[n] x)) ≤ r n :=
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lemma rb3 (n : ℕ) : B' (deg (k X) ((deg (i X))^[n] x)) ≤ rb n :=
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le.trans !le_max_left !le_max_right
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lemma rb4 (n : ℕ) : B' (deg (k X) ((deg (i X))^[n+1] x)) ≤ r n :=
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lemma rb4 (n : ℕ) : B' (deg (k X) ((deg (i X))^[n+1] x)) ≤ rb n :=
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le.trans (le.trans !le_max_left !le_max_right) !le_max_right
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lemma rb5 (n : ℕ) : B ((deg (j X))⁻¹ ((deg (i X))^[n] x)) ≤ r n :=
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lemma rb5 (n : ℕ) : B ((deg (j X))⁻¹ ((deg (i X))^[n] x)) ≤ rb n :=
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le.trans (le.trans !le_max_right !le_max_right) !le_max_right
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definition Einfdiag : graded_module R ℕ :=
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@ -520,24 +521,24 @@ namespace left_module
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λn, Dinf (deg (k X) ((deg (i X))^[n] x))
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definition short_exact_mod_page_r (n : ℕ) : short_exact_mod
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(E (page (r n)) ((deg (i X))^[n] x))
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(D (page (r n)) (deg (k (page (r n))) ((deg (i X))^[n] x)))
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(D (page (r n)) (deg (i (page (r n))) (deg (k (page (r n))) ((deg (i X))^[n] x)))) :=
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(E (page (rb n)) ((deg (i X))^[n] x))
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(D (page (rb n)) (deg (k (page (rb n))) ((deg (i X))^[n] x)))
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(D (page (rb n)) (deg (i (page (rb n))) (deg (k (page (rb n))) ((deg (i X))^[n] x)))) :=
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begin
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fapply short_exact_mod_of_is_exact,
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{ exact j (page (r n)) ← ((deg (i X))^[n] x) },
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{ exact k (page (r n)) ((deg (i X))^[n] x) },
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{ exact i (page (r n)) (deg (k (page (r n))) ((deg (i X))^[n] x)) },
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{ exact j (page (r n)) _ },
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{ exact j (page (rb n)) ← ((deg (i X))^[n] x) },
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{ exact k (page (rb n)) ((deg (i X))^[n] x) },
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{ exact i (page (rb n)) (deg (k (page (rb n))) ((deg (i X))^[n] x)) },
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{ exact j (page (rb n)) _ },
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{ apply is_contr_D, refine Dub !deg_j_inv⁻¹ (rb5 n) },
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{ apply is_contr_E, refine Elb _ (rb1 n),
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refine !deg_iterate_ij_commute ⬝ _,
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refine ap (deg (j X)) _ ⬝ !deg_j⁻¹,
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refine iterate_sub _ !rb0 _ ⬝ _, apply ap (_^[r n]),
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refine iterate_sub _ !rb0 _ ⬝ _, apply ap (_^[rb n]),
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exact ap (deg (i X)) (!deg_iterate_ik_commute ⬝ !deg_k⁻¹) ⬝ !deg_i⁻¹ },
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{ apply jk (page (r n)) },
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{ apply ki (page (r n)) },
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{ apply ij (page (r n)) }
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{ apply jk (page (rb n)) },
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{ apply ki (page (rb n)) },
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{ apply ij (page (rb n)) }
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end
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definition short_exact_mod_infpage (n : ℕ) :
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@ -573,6 +574,20 @@ namespace left_module
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(λs, Dinfdiag_stable)
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end
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definition deg_d_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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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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refine z _ ⬝ _, apply ap ((deg (i X))⁻¹ᵉ^[r]),
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exact to_right_inv (group.equiv_of_isomorphism e) _
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end
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end convergence_theorem
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open int group prod convergence_theorem prod.ops
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@ -592,12 +607,12 @@ 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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(s₀ : agℤ → agℤ)
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(HB : Π(n : agℤ), is_bounded.B' HH (deg (k X) (n, s₀ n)) = 0)
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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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(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) (n, s₀ n)) ≃lm Dinf n)
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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_eq : Π(r : ℕ) (n s : agℤ), deg (d (page X r)) (n, s) = (n, s) + (deg_d1 r, deg_d2 r))
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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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structure converging_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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@ -622,9 +637,14 @@ 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 s₀ := s₀ c
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local abbreviation Dinfdiag (n : agℤ) (k : ℕ) := Dinfdiag HH (n, s₀ n) k
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local abbreviation Einfdiag (n : agℤ) (k : ℕ) := Einfdiag HH (n, s₀ n) k
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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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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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!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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@ -632,16 +652,38 @@ namespace left_module
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definition converges_to_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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converges_to.mk X HH s₀ (HB c)
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converges_to.mk X HH s₁ s₂ (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 n s, deg_d_eq c r n s)
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(deg_d1 c) (deg_d2 c) (λr, deg_d_eq0 c r)
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include c
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definition converges_to_reindex (i : agℤ ×ag agℤ ≃g agℤ ×ag agℤ) :
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(λp q, E' (i (p, q)).1 (i (p, q)).2) ⟹ (λn, Dinf n) :=
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begin
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fapply converges_to.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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{ 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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{ 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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end
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omit c
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/- definition converges_to_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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converges_to.mk (exact_couple_reindex i X) (is_bounded_reindex i HH) s₀
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converges_to.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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@ -651,17 +693,17 @@ namespace left_module
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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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converges_to.mk (exact_couple_reindex (equiv_neg ×≃ equiv_neg) X) (is_bounded_reindex _ HH)
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(λn, -s₀ (-n))
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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_converges_to (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 (n, s₀ n) (HB c 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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/- TODO: reprove this using is_built_of -/
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theorem is_contr_converges_to_precise (n : agℤ)
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(H : Π(n : agℤ) (l : ℕ), is_contr (E' ((deg i)^[l] (n, s₀ n)).1 ((deg i)^[l] (n, s₀ n)).2)) :
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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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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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@ -669,13 +711,13 @@ 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) (n, s₀ n)) },
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{ exact is_bounded.B HH (deg (k X) (s₁ n, s₂ 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 (n, s₀ n) l)
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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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(H2 l) H }},
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refine is_trunc_equiv_closed _ _ H3,
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exact equiv_of_isomorphism (Dinfdiag0 HH (n, s₀ n) (HB c n) ⬝lm f c n)
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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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end
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theorem is_contr_converges_to (n : agℤ) (H : Π(n s : agℤ), is_contr (E' n s)) : is_contr (Dinf n) :=
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@ -686,7 +728,7 @@ namespace left_module
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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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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 c r n s)⁻¹ᵖ (H2 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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@ -699,7 +741,7 @@ namespace left_module
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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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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 c r n s)⁻¹ᵖ (H2 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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@ -921,15 +963,16 @@ namespace spectrum
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fapply converges_to.mk,
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{ exact exact_couple_sequence },
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{ exact is_bounded_sequence },
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{ exact id },
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{ exact ub },
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{ intro n, change max0 (ub n - ub n) = 0, exact ap max0 (sub_self (ub n)) },
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{ intro ns, reflexivity },
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{ intro n, reflexivity },
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{ intro r, exact - (1 : ℤ) },
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{ intro r, exact r + (1 : ℤ) },
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{ intro r n s, refine !convergence_theorem.deg_d ⬝ _,
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{ intro r, refine !convergence_theorem.deg_d ⬝ _,
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refine ap (deg j_sequence) !iterate_deg_i_inv ⬝ _,
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exact prod_eq idp (!add.assoc ⬝ ap (λx, s + (r + x)) !neg_neg) },
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exact prod_eq idp !zero_add }
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end
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end
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@ -1,7 +1,7 @@
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import ..algebra.spectral_sequence ..spectrum.trunc .basic
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open eq spectrum trunc is_trunc pointed int EM algebra left_module fiber lift equiv is_equiv
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cohomology group sigma unit is_conn
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cohomology group sigma unit is_conn prod
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set_option pp.binder_types true
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/- Eilenberg MacLane spaces are the fibers of the Postnikov system of a type -/
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@ -171,14 +171,14 @@ section atiyah_hirzebruch
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definition is_bounded_atiyah_hirzebruch : is_bounded atiyah_hirzebruch_exact_couple :=
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is_bounded_sequence _ (λn, s₀) (λn, n - 1) atiyah_hirzebruch_lb atiyah_hirzebruch_ub
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definition atiyah_hirzebruch_convergence' :
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definition atiyah_hirzebruch_convergence1 :
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(λn s, πₛ[n] (sfiber (spi_compose_left (λx, postnikov_smap (Y x) s)))) ⟹ᵍ
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(λn, πₛ[n] (spi X (λx, strunc s₀ (Y x)))) :=
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converges_to_sequence _ (λn, s₀) (λn, n - 1) atiyah_hirzebruch_lb atiyah_hirzebruch_ub
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definition atiyah_hirzebruch_convergence :
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definition atiyah_hirzebruch_convergence2 :
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(λn s, opH^-(n-s)[(x : X), πₛ[s] (Y x)]) ⟹ᵍ (λn, pH^-n[(x : X), Y x]) :=
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converges_to_g_isomorphism atiyah_hirzebruch_convergence'
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converges_to_g_isomorphism atiyah_hirzebruch_convergence1
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begin
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intro n s,
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refine _ ⬝g (parametrized_cohomology_isomorphism_shomotopy_group_spi _ idp)⁻¹ᵍ,
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@ -194,11 +194,32 @@ section atiyah_hirzebruch
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exact ppi_pequiv_right (λx, ptrunc_pequiv (maxm2 (s₀ + k)) (Y x k)),
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end
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-- set_option pp.metavar_args true
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-- definition atiyah_reindexed : (λp q, opH^p[(x : X), πₛ[q] (Y x)]) ⟹ᵍ (λn, pH^n[(x : X), Y x])
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-- :=
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-- converges_to_reindex atiyah_hirzebruch_convergence (λp q, -(p - q)) (λp q, q) (λp q, by reflexivity)
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-- (λn, -n) (λn, by reflexivity)
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open prod.ops
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definition atiyah_hirzebruch_base_change [constructor] : agℤ ×ag agℤ ≃g agℤ ×ag agℤ :=
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begin
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fapply group.isomorphism.mk,
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{ fapply group.homomorphism.mk, exact (λpq, (-(pq.1 + pq.2), -pq.2)),
|
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intro pq pq',
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induction pq with p q, induction pq' with p' q', esimp,
|
||||
exact prod_eq (ap neg !add.comm4 ⬝ !neg_add) !neg_add },
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||||
{ fapply adjointify,
|
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{ exact (λns, (ns.2 - ns.1, -ns.2)) },
|
||||
{ intro ns, esimp,
|
||||
exact prod_eq (ap neg (!add.comm ⬝ !neg_add_cancel_left) ⬝ !neg_neg) !neg_neg },
|
||||
{ intro pq, esimp,
|
||||
exact prod_eq (ap (λx, _ + x) !neg_neg ⬝ !add.comm ⬝ !add_neg_cancel_right) !neg_neg }}
|
||||
end
|
||||
|
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definition atiyah_hirzebruch_convergence :
|
||||
(λp q, opH^p[(x : X), πₛ[-q] (Y x)]) ⟹ᵍ (λn, pH^-n[(x : X), Y x]) :=
|
||||
begin
|
||||
note z := converges_to_reindex atiyah_hirzebruch_convergence2 atiyah_hirzebruch_base_change,
|
||||
refine converges_to_g_isomorphism z _ (λn, by reflexivity),
|
||||
intro p q,
|
||||
apply parametrized_cohomology_change_int,
|
||||
esimp,
|
||||
refine !neg_neg_sub_neg ⬝ !add_neg_cancel_right
|
||||
end
|
||||
|
||||
end atiyah_hirzebruch
|
||||
|
||||
|
|
@ -206,11 +227,11 @@ section unreduced_atiyah_hirzebruch
|
|||
|
||||
definition unreduced_atiyah_hirzebruch_convergence {X : Type} (Y : X → spectrum) (s₀ : ℤ)
|
||||
(H : Πx, is_strunc s₀ (Y x)) :
|
||||
(λn s, uopH^-(n-s)[(x : X), πₛ[s] (Y x)]) ⟹ᵍ (λn, upH^-n[(x : X), Y x]) :=
|
||||
(λp q, uopH^p[(x : X), πₛ[-q] (Y x)]) ⟹ᵍ (λn, upH^-n[(x : X), Y x]) :=
|
||||
converges_to_g_isomorphism
|
||||
(@atiyah_hirzebruch_convergence X₊ (add_point_spectrum Y) s₀ (is_strunc_add_point_spectrum H))
|
||||
begin
|
||||
intro n s, refine _ ⬝g !uopH_isomorphism_opH⁻¹ᵍ,
|
||||
intro p q, refine _ ⬝g !uopH_isomorphism_opH⁻¹ᵍ,
|
||||
apply ordinary_parametrized_cohomology_isomorphism_right,
|
||||
intro x,
|
||||
apply shomotopy_group_add_point_spectrum
|
||||
|
|
@ -227,17 +248,17 @@ section serre
|
|||
|
||||
include H
|
||||
definition serre_convergence :
|
||||
(λn s, uopH^-(n-s)[(b : B), uH^-s[F b, Y]]) ⟹ᵍ (λn, uH^-n[Σ(b : B), F b, Y]) :=
|
||||
(λp q, uopH^p[(b : B), uH^q[F b, Y]]) ⟹ᵍ (λn, uH^-n[Σ(b : B), F b, Y]) :=
|
||||
proof
|
||||
converges_to_g_isomorphism
|
||||
(unreduced_atiyah_hirzebruch_convergence
|
||||
(λx, sp_ucotensor (F x) Y) s₀
|
||||
(λx, is_strunc_sp_ucotensor s₀ (F x) H))
|
||||
begin
|
||||
intro n s,
|
||||
refine unreduced_ordinary_parametrized_cohomology_isomorphism_right _ (-(n-s)),
|
||||
intro p q,
|
||||
refine unreduced_ordinary_parametrized_cohomology_isomorphism_right _ p,
|
||||
intro x,
|
||||
exact (unreduced_cohomology_isomorphism_shomotopy_group_sp_ucotensor _ _ idp)⁻¹ᵍ
|
||||
exact (unreduced_cohomology_isomorphism_shomotopy_group_sp_ucotensor _ _ !neg_neg)⁻¹ᵍ
|
||||
end
|
||||
begin
|
||||
intro n,
|
||||
|
|
@ -249,29 +270,29 @@ section serre
|
|||
qed
|
||||
|
||||
definition serre_convergence_map :
|
||||
(λn s, uopH^-(n-s)[(b : B), uH^-s[fiber f b, Y]]) ⟹ᵍ (λn, uH^-n[X, Y]) :=
|
||||
(λp q, uopH^p[(b : B), uH^q[fiber f b, Y]]) ⟹ᵍ (λn, uH^-n[X, Y]) :=
|
||||
proof
|
||||
converges_to_g_isomorphism
|
||||
(serre_convergence (fiber f) Y s₀ H)
|
||||
begin intro n s, reflexivity end
|
||||
begin intro p q, reflexivity end
|
||||
begin intro n, apply unreduced_cohomology_isomorphism, exact !sigma_fiber_equiv⁻¹ᵉ end
|
||||
qed
|
||||
|
||||
definition serre_convergence_of_is_conn (H2 : is_conn 1 B) :
|
||||
(λn s, uoH^-(n-s)[B, uH^-s[F b₀, Y]]) ⟹ᵍ (λn, uH^-n[Σ(b : B), F b, Y]) :=
|
||||
(λp q, uoH^p[B, uH^q[F b₀, Y]]) ⟹ᵍ (λn, uH^-n[Σ(b : B), F b, Y]) :=
|
||||
proof
|
||||
converges_to_g_isomorphism
|
||||
(serre_convergence F Y s₀ H)
|
||||
begin intro n s, exact @uopH_isomorphism_uoH_of_is_conn (pointed.MK B b₀) _ _ H2 end
|
||||
begin intro p q, exact @uopH_isomorphism_uoH_of_is_conn (pointed.MK B b₀) _ _ H2 end
|
||||
begin intro n, reflexivity end
|
||||
qed
|
||||
|
||||
definition serre_convergence_map_of_is_conn (H2 : is_conn 1 B) :
|
||||
(λn s, uoH^-(n-s)[B, uH^-s[fiber f b₀, Y]]) ⟹ᵍ (λn, uH^-n[X, Y]) :=
|
||||
(λp q, uoH^p[B, uH^q[fiber f b₀, Y]]) ⟹ᵍ (λn, uH^-n[X, Y]) :=
|
||||
proof
|
||||
converges_to_g_isomorphism
|
||||
(serre_convergence_of_is_conn b₀ (fiber f) Y s₀ H H2)
|
||||
begin intro n s, reflexivity end
|
||||
begin intro p q, reflexivity end
|
||||
begin intro n, apply unreduced_cohomology_isomorphism, exact !sigma_fiber_equiv⁻¹ᵉ end
|
||||
qed
|
||||
|
||||
|
|
|
|||
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Reference in a new issue