finish proving that the gysin sequence consists of the correct groups
This commit is contained in:
Floris van Doorn
parent
ea402f56ea
commit
2913de520d
5 changed files with 140 additions and 41 deletions
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@ -14,18 +14,18 @@ structure module_chain_complex (R : Ring) (N : succ_str) : Type :=
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(is_chain_complex :
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Π (n : N) (x : mod (S (S n))), hom n (hom (S n) x) = 0)
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namespace module_chain_complex
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namespace left_module
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variables {R : Ring} {N : succ_str}
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definition mcc_mod [unfold 2] [coercion] (C : module_chain_complex R N) (n : N) :
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definition mcc_mod [unfold 3] [coercion] (C : module_chain_complex R N) (n : N) :
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LeftModule R :=
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module_chain_complex.mod C n
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definition mcc_carr [unfold 2] [coercion] (C : module_chain_complex R N) (n : N) :
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definition mcc_carr [unfold 3] [coercion] (C : module_chain_complex R N) (n : N) :
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Type :=
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C n
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definition mcc_pcarr [unfold 2] [coercion] (C : module_chain_complex R N) (n : N) :
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definition mcc_pcarr [unfold 3] [coercion] (C : module_chain_complex R N) (n : N) :
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Set* :=
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mcc_mod C n
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@ -46,7 +46,11 @@ namespace module_chain_complex
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-- maybe we don't even need this?
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definition is_exact_at_m (C : module_chain_complex R N) (n : N) : Type :=
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is_exact_at C n
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end module_chain_complex
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definition is_exact_m (C : module_chain_complex R N) : Type :=
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∀n, is_exact_at_m C n
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end left_module
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namespace left_module
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variable {R : Ring}
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@ -159,6 +159,10 @@ namespace spectral_sequence
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variable {c}
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variable (nc : is_normal c)
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include nc
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definition deg_d_normal_pr1 (r : ℕ) : (deg_d c r).1 = r+2 := ap pr1 (normal3 nc r)
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definition deg_d_normal_pr2 (r : ℕ) : (deg_d c r).2 = -(r+1) := ap pr2 (normal3 nc r)
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definition stable_range {n s : ℤ} {r : ℕ} (H1 : n < r + 2) (H2 : s < r + 1) :
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Einf c (n, s) ≃lm E c r (n, s) :=
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begin
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@ -1,10 +1,10 @@
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/- submodules and quotient modules -/
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-- Authors: Floris van Doorn, Jeremy Avigad
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import .left_module .quotient_group
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import .left_module .quotient_group .module_chain_complex
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open algebra eq group sigma sigma.ops is_trunc function trunc equiv is_equiv property
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universe variable u
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namespace left_module
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/- submodules -/
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variables {R : Ring} {M M₁ M₁' M₂ M₂' M₃ M₃' : LeftModule R} {m m₁ m₂ : M}
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@ -484,19 +484,17 @@ quotient_module_isomorphism_quotient_module
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begin intro m, reflexivity end
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open chain_complex fin nat
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definition LES_of_SESs.{u} {N : succ_str} (A B C : N → LeftModule.{_ u} R) (φ : Πn, A n →lm B n)
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definition LES_of_SESs {N : succ_str} (A B C : N → LeftModule.{_ u} R) (φ : Πn, A n →lm B n)
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(ses : Πn : N, short_exact_mod (cokernel_module (φ (succ_str.S n))) (C n) (kernel_module (φ n))) :
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chain_complex.{_ u} (stratified N 2) :=
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module_chain_complex.{_ _ u} R (stratified N 2) :=
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begin
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fapply chain_complex.mk,
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{ intro x, apply @pSet_of_LeftModule R,
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induction x with n k, induction k with k H, do 3 (cases k with k; rotate 1),
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fapply module_chain_complex.mk,
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{ intro x, induction x with n k, induction k with k H, do 3 (cases k with k; rotate 1),
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{ /-k≥3-/ exfalso, apply lt_le_antisymm H, apply le_add_left},
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{ /-k=0-/ exact B n },
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{ /-k=1-/ exact A n },
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{ /-k=2-/ exact C n }},
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{ intro x, apply @pmap_of_homomorphism R,
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induction x with n k, induction k with k H, do 3 (cases k with k; rotate 1),
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{ intro x, induction x with n k, induction k with k H, do 3 (cases k with k; rotate 1),
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{ /-k≥3-/ exfalso, apply lt_le_antisymm H, apply le_add_left},
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{ /-k=0-/ exact φ n },
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{ /-k=1-/ exact submodule_incl _ ∘lm short_exact_mod.g (ses n) },
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@ -504,14 +502,32 @@ begin
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{ intros x m, induction x with n k, induction k with k H, do 3 (cases k with k; rotate 1),
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{ exfalso, apply lt_le_antisymm H, apply le_add_left},
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{ exact (short_exact_mod.g (ses n) m).2 },
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{ exact ap pr1 (is_short_exact.im_in_ker (short_exact_mod.h (ses n)) (quotient_map _ m)) },
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{ exact ap (short_exact_mod.f (ses n)) (quotient_map_eq_zero _ _ (image.mk m idp)) ⬝
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to_respect_zero (short_exact_mod.f (ses n)) }}
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{ note h := is_short_exact.im_in_ker (short_exact_mod.h (ses n)) (quotient_map _ m),
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exact ap pr1 h },
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{ refine _ ⬝ to_respect_zero (short_exact_mod.f (ses n)),
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rexact ap (short_exact_mod.f (ses n)) (quotient_map_eq_zero _ _ (image.mk m idp)) }}
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end
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definition is_exact_LES_of_SESs.{u} {N : succ_str} (A B C : N → LeftModule.{_ u} R) (φ : Πn, A n →lm B n)
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open prod
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definition LES_of_SESs_zero {N : succ_str} {A B C : N → LeftModule.{_ u} R} (φ : Πn, A n →lm B n)
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(ses : Πn : N, short_exact_mod (cokernel_module (φ (succ_str.S n))) (C n) (kernel_module (φ n)))
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(n : N) : LES_of_SESs A B C φ ses (n, 0) ≃lm B n :=
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by reflexivity
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definition LES_of_SESs_one {N : succ_str} {A B C : N → LeftModule.{_ u} R} (φ : Πn, A n →lm B n)
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(ses : Πn : N, short_exact_mod (cokernel_module (φ (succ_str.S n))) (C n) (kernel_module (φ n)))
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(n : N) : LES_of_SESs A B C φ ses (n, 1) ≃lm A n :=
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by reflexivity
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definition LES_of_SESs_two {N : succ_str} {A B C : N → LeftModule.{_ u} R} (φ : Πn, A n →lm B n)
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(ses : Πn : N, short_exact_mod (cokernel_module (φ (succ_str.S n))) (C n) (kernel_module (φ n)))
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(n : N) : LES_of_SESs A B C φ ses (n, 2) ≃lm C n :=
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by reflexivity
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definition is_exact_LES_of_SESs {N : succ_str} {A B C : N → LeftModule.{_ u} R}
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(φ : Πn, A n →lm B n)
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(ses : Πn : N, short_exact_mod (cokernel_module (φ (succ_str.S n))) (C n) (kernel_module (φ n))) :
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is_exact (LES_of_SESs A B C φ ses) :=
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is_exact_m (LES_of_SESs A B C φ ses) :=
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begin
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intros x m p, induction x with n k, induction k with k H, do 3 (cases k with k; rotate 1),
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{ exfalso, apply lt_le_antisymm H, apply le_add_left},
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@ -395,14 +395,14 @@ definition unreduced_ordinary_cohomology_pbool_nonzero (G : AbGroup) (n : ℤ) (
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is_contr_equiv_closed_rev (equiv_of_isomorphism (unreduced_ordinary_cohomology_nonzero pbool G n H))
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(ordinary_cohomology_dimension G n H)
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definition unreduced_ordinary_cohomology_sphere_zero (G : AbGroup) (n : ℕ) (H : n ≠ 0) :
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definition unreduced_ordinary_cohomology_sphere_zero (G : AbGroup.{u}) (n : ℕ) (H : n ≠ 0) :
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uoH^0[sphere n, G] ≃g G :=
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unreduced_ordinary_cohomology_zero (sphere n) G ⬝g
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product_trivial_left _ _ (ordinary_cohomology_sphere_of_neq _ (λh, H h⁻¹))
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product_trivial_left _ _ (ordinary_cohomology_sphere_of_neq _ (λh, H (of_nat.inj h⁻¹)))
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definition unreduced_ordinary_cohomology_sphere (G : AbGroup) (n : ℕ) (H : n ≠ 0) :
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uoH^n[sphere n, G] ≃g G :=
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unreduced_ordinary_cohomology_nonzero (sphere n) G n H ⬝g
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unreduced_ordinary_cohomology_nonzero (sphere n) G n (λh, H (of_nat.inj h)) ⬝g
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ordinary_cohomology_sphere G n
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definition unreduced_ordinary_cohomology_sphere_of_neq (G : AbGroup) {n : ℤ} {k : ℕ} (p : n ≠ k)
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@ -415,6 +415,11 @@ definition unreduced_ordinary_cohomology_sphere_of_neq_nat (G : AbGroup) {n k :
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(q : n ≠ 0) : is_contr (uoH^n[sphere k, G]) :=
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unreduced_ordinary_cohomology_sphere_of_neq G (λh, p (of_nat.inj h)) (λh, q (of_nat.inj h))
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definition unreduced_ordinary_cohomology_sphere_of_gt (G : AbGroup) {n k : ℕ} (p : n > k) :
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is_contr (uoH^n[sphere k, G]) :=
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unreduced_ordinary_cohomology_sphere_of_neq_nat
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G (ne_of_gt p) (ne_of_gt (lt_of_le_of_lt (zero_le k) p))
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theorem is_contr_cohomology_of_is_contr_spectrum (n : ℤ) (X : Type*) (Y : spectrum)
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(H : is_contr (Y n)) : is_contr (H^n[X, Y]) :=
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begin
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@ -4,9 +4,10 @@
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import .serre
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open eq pointed is_trunc is_conn is_equiv equiv sphere fiber chain_complex left_module spectrum nat
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prod nat int algebra function spectral_sequence
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prod nat int algebra function spectral_sequence fin group
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namespace cohomology
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universe variable u
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/-
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We have maps:
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d_m = d_(m-1,n+1)^n : E_(m-1,n+1)^n → E_(m+n+1,0)^n
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@ -20,8 +21,30 @@ namespace cohomology
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... E_(m+n,0)^n → D_{m+n}^∞ → E_(m-1,n+1)^n → E_(m+n+1,0)^n → D_{m+n+1}^∞ ...
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-/
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definition gysin_sequence' {E B : Type*} {n : ℕ} (HB : is_conn 1 B) (f : E →* B)
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(e : pfiber f ≃* sphere (n+1)) (A : AbGroup) : chain_complex -3ℤ :=
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definition gysin_trivial_Epage {E B : pType.{u}} {n : ℕ} (HB : is_conn 1 B) (f : E →* B)
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(e : pfiber f ≃* sphere (n+1)) (A : AbGroup) (r : ℕ) (p q : ℤ) (hq : q ≠ 0)
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(hq' : q ≠ of_nat (n+1)):
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is_contr (convergent_spectral_sequence.E (serre_spectral_sequence_map_of_is_conn pt f
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(EM_spectrum A) 0 (is_strunc_EM_spectrum A) HB) r (p, q)) :=
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begin
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intros, apply is_contr_E, apply is_contr_ordinary_cohomology, esimp,
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refine is_contr_equiv_closed_rev _ (unreduced_ordinary_cohomology_sphere_of_neq A hq' hq),
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apply group.equiv_of_isomorphism, apply unreduced_ordinary_cohomology_isomorphism, exact e⁻¹ᵉ*
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end
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definition gysin_trivial_Epage2 {E B : pType.{u}} {n : ℕ} (HB : is_conn 1 B) (f : E →* B)
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(e : pfiber f ≃* sphere (n+1)) (A : AbGroup) (r : ℕ) (p q : ℤ) (hq : q > n+1) :
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is_contr (convergent_spectral_sequence.E (serre_spectral_sequence_map_of_is_conn pt f
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(EM_spectrum A) 0 (is_strunc_EM_spectrum A) HB) r (p, q)) :=
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begin
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intros, apply gysin_trivial_Epage HB f e A r p q,
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{ intro h, subst h, apply not_lt_zero (n+1), exact lt_of_of_nat_lt_of_nat hq },
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{ intro h, subst h, exact lt.irrefl _ hq }
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end
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definition gysin_sequence' {E B : pType.{u}} {n : ℕ} (HB : is_conn 1 B) (f : E →* B)
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(e : pfiber f ≃* sphere (n+1)) (A : AbGroup) : module_chain_complex rℤ -3ℤ :=
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let c := serre_spectral_sequence_map_of_is_conn pt f (EM_spectrum A) 0
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(is_strunc_EM_spectrum A) HB in
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let cn : is_normal c := !is_normal_serre_spectral_sequence_map_of_is_conn in
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@ -36,18 +59,10 @@ begin
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end,
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have trivial_E : Π(r : ℕ) (p q : ℤ) (hq : q ≠ 0) (hq' : q ≠ of_nat (n+1)),
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is_contr (convergent_spectral_sequence.E c r (p, q)),
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begin
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intros, apply is_contr_E, apply is_contr_ordinary_cohomology, esimp,
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refine is_contr_equiv_closed_rev _ (unreduced_ordinary_cohomology_sphere_of_neq A hq' hq),
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apply group.equiv_of_isomorphism, apply unreduced_ordinary_cohomology_isomorphism, exact e⁻¹ᵉ*
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end,
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from gysin_trivial_Epage HB f e A,
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have trivial_E' : Π(r : ℕ) (p q : ℤ) (hq : q > n+1),
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is_contr (convergent_spectral_sequence.E c r (p, q)),
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begin
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intros, apply trivial_E r p q,
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{ intro h, subst h, apply not_lt_zero (n+1), exact lt_of_of_nat_lt_of_nat hq },
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{ intro h, subst h, exact lt.irrefl _ hq }
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end,
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from gysin_trivial_Epage2 HB f e A,
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left_module.LES_of_SESs _ _ _ (λm, convergent_spectral_sequence.d c n (m - 1, n + 1))
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begin
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intro m,
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@ -96,13 +111,68 @@ left_module.LES_of_SESs _ _ _ (λm, convergent_spectral_sequence.d c n (m - 1, n
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refine ap (λx, x + _) !add.right_inv ⬝ !zero_add }}
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end
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-- open fin
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-- definition gysin_sequence'_zero_equiv {E B : Type*} {n : ℕ} (HB : is_conn 1 B) (f : E →* B)
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-- (e : pfiber f ≃* sphere (n+1)) (A : AbGroup) (m : ℤ) :
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-- gysin_sequence' HB f e A (m, 0) ≃ _ :=
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-- _
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-- set_option pp.universes true
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-- print unreduced_ordinary_cohomology_sphere_zero
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-- print unreduced_ordinary_cohomology_zero
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-- print ordinary_cohomology_sphere_of_neq
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-- set_option formatter.hide_full_terms false
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definition gysin_sequence'_zero {E B : Type*} {n : ℕ} (HB : is_conn 1 B) (f : E →* B)
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(e : pfiber f ≃* sphere (n+1)) (A : AbGroup) (m : ℤ) :
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gysin_sequence' HB f e A (m, 0) ≃lm LeftModule_int_of_AbGroup (uoH^m+n+1[B, A]) :=
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let c := serre_spectral_sequence_map_of_is_conn pt f (EM_spectrum A) 0
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(is_strunc_EM_spectrum A) HB in
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let cn : is_normal c := !is_normal_serre_spectral_sequence_map_of_is_conn in
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begin
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refine LES_of_SESs_zero _ _ m ⬝lm _,
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transitivity convergent_spectral_sequence.E c n (m+n+1, 0),
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exact isomorphism_of_eq (ap (convergent_spectral_sequence.E c n)
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(deg_d_normal_eq cn _ _ ⬝ prod_eq (add.comm4 m (- 1) n 2) (add.right_inv (n+1)))),
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refine E_isomorphism0 _ _ _ ⬝lm
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lm_iso_int.mk (unreduced_ordinary_cohomology_isomorphism_right _ _ _),
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{ intros r hr, apply is_contr_ordinary_cohomology,
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refine is_contr_equiv_closed_rev
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(equiv_of_isomorphism (cohomology_change_int _ _ _ ⬝g uoH^≃r+1[e⁻¹ᵉ*, A]))
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(unreduced_ordinary_cohomology_sphere_of_neq_nat A _ _),
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{ exact !zero_sub ⬝ ap neg (deg_d_normal_pr2 cn r) ⬝ !neg_neg },
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{ apply ne_of_lt, apply add_lt_add_right, exact hr },
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{ apply succ_ne_zero }},
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{ intros r hr, apply is_contr_ordinary_cohomology,
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refine is_contr_equiv_closed_rev
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(equiv_of_isomorphism (cohomology_change_int _ _ (!zero_add ⬝ deg_d_normal_pr2 cn r)))
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(is_contr_ordinary_cohomology_of_neg _ _ _),
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{ rewrite [-neg_zero], apply neg_lt_neg, apply of_nat_lt_of_nat_of_lt, apply zero_lt_succ }},
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{ exact uoH^≃ 0[e⁻¹ᵉ*, A] ⬝g unreduced_ordinary_cohomology_sphere_zero _ _ (succ_ne_zero n) }
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end
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definition gysin_sequence'_one {E B : Type*} {n : ℕ} (HB : is_conn 1 B) (f : E →* B)
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(e : pfiber f ≃* sphere (n+1)) (A : AbGroup) (m : ℤ) :
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gysin_sequence' HB f e A (m, 1) ≃lm LeftModule_int_of_AbGroup (uoH^m - 1[B, A]) :=
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let c := serre_spectral_sequence_map_of_is_conn pt f (EM_spectrum A) 0
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(is_strunc_EM_spectrum A) HB in
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let cn : is_normal c := !is_normal_serre_spectral_sequence_map_of_is_conn in
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begin
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refine LES_of_SESs_one _ _ m ⬝lm _,
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refine E_isomorphism0 _ _ _ ⬝lm
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lm_iso_int.mk (unreduced_ordinary_cohomology_isomorphism_right _ _ _),
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{ intros r hr, apply is_contr_ordinary_cohomology,
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refine is_contr_equiv_closed_rev
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(equiv_of_isomorphism uoH^≃_[e⁻¹ᵉ*, A])
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(unreduced_ordinary_cohomology_sphere_of_gt _ _),
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{ apply lt_add_of_pos_right, apply zero_lt_succ }},
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{ intros r hr, apply is_contr_ordinary_cohomology,
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refine is_contr_equiv_closed_rev
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(equiv_of_isomorphism (cohomology_change_int _ _ _ ⬝g uoH^≃n-r[e⁻¹ᵉ*, A]))
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(unreduced_ordinary_cohomology_sphere_of_neq_nat A _ _),
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{ refine ap (add _) (deg_d_normal_pr2 cn r) ⬝ add_sub_comm n 1 r 1 ⬝ !add_zero ⬝ _,
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symmetry, apply of_nat_sub, exact le_of_lt hr },
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{ apply ne_of_lt, exact lt_of_le_of_lt (nat.sub_le n r) (lt.base n) },
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{ apply ne_of_gt, exact nat.sub_pos_of_lt hr }},
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{ refine uoH^≃n+1[e⁻¹ᵉ*, A] ⬝g unreduced_ordinary_cohomology_sphere _ _ (succ_ne_zero n) }
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end
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definition gysin_sequence'_two {E B : Type*} {n : ℕ} (HB : is_conn 1 B) (f : E →* B)
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(e : pfiber f ≃* sphere (n+1)) (A : AbGroup) (m : ℤ) :
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gysin_sequence' HB f e A (m, 2) ≃lm LeftModule_int_of_AbGroup (uoH^n+m[E, A]) :=
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LES_of_SESs_two _ _ m
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end cohomology
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Loading…
Reference in a new issue