634 lines
23 KiB
Text
634 lines
23 KiB
Text
/- Exact couples of graded (left-) R-modules. -/
|
||
|
||
-- Author: Floris van Doorn
|
||
|
||
import .graded ..homotopy.spectrum .product_group --types.int.order
|
||
|
||
open algebra is_trunc left_module is_equiv equiv eq function nat
|
||
|
||
-- move
|
||
section
|
||
open group int chain_complex pointed succ_str
|
||
definition LeftModule_int_of_AbGroup [constructor] (A : AbGroup) : LeftModule rℤ :=
|
||
LeftModule.mk A (left_module.mk sorry sorry sorry sorry 1 sorry sorry sorry sorry sorry sorry sorry sorry sorry)
|
||
|
||
definition lm_hom_int.mk [constructor] {A B : AbGroup} (φ : A →g B) :
|
||
LeftModule_int_of_AbGroup A →lm LeftModule_int_of_AbGroup B :=
|
||
homomorphism.mk φ sorry
|
||
|
||
definition is_exact_of_is_exact_at {N : succ_str} {A : chain_complex N} {n : N}
|
||
(H : is_exact_at A n) : is_exact (cc_to_fn A (S n)) (cc_to_fn A n) :=
|
||
is_exact.mk (cc_is_chain_complex A n) H
|
||
|
||
definition is_equiv_mul_right [constructor] {A : Group} (a : A) : is_equiv (λb, b * a) :=
|
||
adjointify _ (λb : A, b * a⁻¹) (λb, !inv_mul_cancel_right) (λb, !mul_inv_cancel_right)
|
||
|
||
definition right_action [constructor] {A : Group} (a : A) : A ≃ A :=
|
||
equiv.mk _ (is_equiv_mul_right a)
|
||
|
||
definition is_equiv_add_right [constructor] {A : AddGroup} (a : A) : is_equiv (λb, b + a) :=
|
||
adjointify _ (λb : A, b - a) (λb, !neg_add_cancel_right) (λb, !add_neg_cancel_right)
|
||
|
||
definition add_right_action [constructor] {A : AddGroup} (a : A) : A ≃ A :=
|
||
equiv.mk _ (is_equiv_add_right a)
|
||
|
||
|
||
section
|
||
variables {A B : Type} (f : A ≃ B) [ab_group A]
|
||
|
||
-- to group
|
||
definition group_equiv_mul_comm (b b' : B) : group_equiv_mul f b b' = group_equiv_mul f b' b :=
|
||
by rewrite [↑group_equiv_mul, mul.comm]
|
||
|
||
definition ab_group_equiv_closed : ab_group B :=
|
||
⦃ab_group, group_equiv_closed f,
|
||
mul_comm := group_equiv_mul_comm f⦄
|
||
end
|
||
|
||
definition ab_group_of_is_contr (A : Type) [is_contr A] : ab_group A :=
|
||
have ab_group unit, from ab_group_unit,
|
||
ab_group_equiv_closed (equiv_unit_of_is_contr A)⁻¹ᵉ
|
||
|
||
definition group_of_is_contr (A : Type) [is_contr A] : group A :=
|
||
have ab_group A, from ab_group_of_is_contr A, by apply _
|
||
|
||
definition ab_group_lift_unit : ab_group (lift unit) :=
|
||
ab_group_of_is_contr (lift unit)
|
||
|
||
definition trivial_ab_group_lift : AbGroup :=
|
||
AbGroup.mk _ ab_group_lift_unit
|
||
|
||
definition homomorphism_of_is_contr_right (A : Group) {B : Type} (H : is_contr B) :
|
||
A →g Group.mk B (group_of_is_contr B) :=
|
||
group.homomorphism.mk (λa, center _) (λa a', !is_prop.elim)
|
||
|
||
open trunc pointed is_conn
|
||
definition ab_group_homotopy_group_of_is_conn (n : ℕ) (A : Type*) [H : is_conn 1 A] : ab_group (π[n] A) :=
|
||
begin
|
||
have is_conn 0 A, from !is_conn_of_is_conn_succ,
|
||
cases n with n,
|
||
{ unfold [homotopy_group, ptrunc], apply ab_group_of_is_contr },
|
||
cases n with n,
|
||
{ unfold [homotopy_group, ptrunc], apply ab_group_of_is_contr },
|
||
exact ab_group_homotopy_group n A
|
||
end
|
||
|
||
end
|
||
|
||
namespace int /- move to int-/
|
||
definition max0 : ℤ → ℕ
|
||
| (of_nat n) := n
|
||
| (-[1+ n]) := 0
|
||
|
||
lemma le_max0 : Π(n : ℤ), n ≤ of_nat (max0 n)
|
||
| (of_nat n) := proof le.refl n qed
|
||
| (-[1+ n]) := proof unit.star qed
|
||
|
||
lemma le_of_max0_le {n : ℤ} {m : ℕ} (h : max0 n ≤ m) : n ≤ of_nat m :=
|
||
le.trans (le_max0 n) (of_nat_le_of_nat_of_le h)
|
||
|
||
end int
|
||
|
||
/- exact couples -/
|
||
|
||
namespace left_module
|
||
|
||
structure exact_couple (R : Ring) (I : Set) : Type :=
|
||
(D E : graded_module R I)
|
||
(i : D →gm D) (j : D →gm E) (k : E →gm D)
|
||
(ij : is_exact_gmod i j)
|
||
(jk : is_exact_gmod j k)
|
||
(ki : is_exact_gmod k i)
|
||
|
||
namespace derived_couple
|
||
section
|
||
open exact_couple
|
||
|
||
parameters {R : Ring} {I : Set} (X : exact_couple R I)
|
||
local abbreviation D := D X
|
||
local abbreviation E := E X
|
||
local abbreviation i := i X
|
||
local abbreviation j := j X
|
||
local abbreviation k := k X
|
||
local abbreviation ij := ij X
|
||
local abbreviation jk := jk X
|
||
local abbreviation ki := ki X
|
||
|
||
definition d : E →gm E := j ∘gm k
|
||
definition D' : graded_module R I := graded_image i
|
||
definition E' : graded_module R I := graded_homology d d
|
||
|
||
definition is_contr_E' {x : I} (H : is_contr (E x)) : is_contr (E' x) :=
|
||
!is_contr_homology
|
||
|
||
definition is_contr_D' {x : I} (H : is_contr (D x)) : is_contr (D' x) :=
|
||
!is_contr_image_module
|
||
|
||
definition i' : D' →gm D' :=
|
||
graded_image_lift i ∘gm graded_submodule_incl _
|
||
-- degree i + 0
|
||
|
||
lemma is_surjective_i' {x y : I} (p : deg i' x = y)
|
||
(H : Π⦃z⦄ (q : deg i z = x), is_surjective (i ↘ q)) : is_surjective (i' ↘ p) :=
|
||
begin
|
||
apply is_surjective_graded_hom_compose,
|
||
{ intro y q, apply is_surjective_graded_image_lift },
|
||
{ intro y q, apply is_surjective_of_is_equiv,
|
||
induction q,
|
||
exact to_is_equiv (equiv_of_isomorphism (image_module_isomorphism (i ← x) (H _)))
|
||
}
|
||
end
|
||
|
||
lemma j_lemma1 ⦃x : I⦄ (m : D x) : d ((deg j) x) (j x m) = 0 :=
|
||
begin
|
||
rewrite [graded_hom_compose_fn,↑d,graded_hom_compose_fn],
|
||
refine ap (graded_hom_fn j (deg k (deg j x))) _ ⬝
|
||
!to_respect_zero,
|
||
exact compose_constant.elim (gmod_im_in_ker (jk)) x m
|
||
end
|
||
|
||
lemma j_lemma2 : Π⦃x : I⦄ ⦃m : D x⦄ (p : i x m = 0),
|
||
(graded_quotient_map _ ∘gm graded_hom_lift j j_lemma1) x m = 0 :> E' _ :=
|
||
begin
|
||
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⦄ (p : i y m = 0), image (d ↘ r) (j y m),
|
||
begin
|
||
intros, induction s, induction q,
|
||
note m_in_im_k := is_exact.ker_in_im (ki idp _) _ p,
|
||
induction m_in_im_k with e q,
|
||
induction q,
|
||
apply image.mk e idp
|
||
end,
|
||
have Π⦃x : I⦄ ⦃m : D x⦄ (p : i x m = 0), image (d ← (deg j x)) (j x m),
|
||
begin
|
||
intros,
|
||
refine this _ _ _ p,
|
||
exact to_right_inv (deg k) _ ⬝ to_left_inv (deg j) x,
|
||
apply is_set.elim
|
||
-- rewrite [ap_con, -adj],
|
||
end,
|
||
intros,
|
||
rewrite [graded_hom_compose_fn],
|
||
exact quotient_map_eq_zero _ (this p)
|
||
end
|
||
|
||
definition j' : D' →gm E' :=
|
||
graded_image_elim (graded_homology_intro d d ∘gm graded_hom_lift j j_lemma1) j_lemma2
|
||
-- degree deg j - deg i
|
||
|
||
lemma k_lemma1 ⦃x : I⦄ (m : E x) : image (i ← (deg k x)) (k x m) :=
|
||
begin
|
||
exact sorry
|
||
end
|
||
|
||
lemma k_lemma2 : compose_constant (graded_hom_lift k k_lemma1 : E →gm D') d :=
|
||
begin
|
||
-- apply compose_constant.mk, intro x m,
|
||
-- rewrite [graded_hom_compose_fn],
|
||
-- refine ap (graded_hom_fn (graded_image_lift i) (deg k (deg d x))) _ ⬝ !to_respect_zero,
|
||
-- exact compose_constant.elim (gmod_im_in_ker jk) (deg k x) (k x m)
|
||
exact sorry
|
||
end
|
||
|
||
definition k' : E' →gm D' :=
|
||
graded_homology_elim (graded_hom_lift k k_lemma1) k_lemma2
|
||
|
||
definition deg_i' : deg i' ~ deg i := by reflexivity
|
||
definition deg_j' : deg j' ~ deg j ∘ (deg i)⁻¹ := by reflexivity
|
||
definition deg_k' : deg k' ~ deg k := by reflexivity
|
||
|
||
lemma i'j' : is_exact_gmod i' j' :=
|
||
begin
|
||
apply is_exact_gmod.mk,
|
||
{ intro x, refine total_image.rec _, intro m, exact sorry
|
||
-- exact calc
|
||
-- j' (deg i' x) (i' x ⟨(i ← x) m, image.mk m idp⟩)
|
||
-- = j' (deg i' x) (graded_image_lift i x ((i ← x) m)) : idp
|
||
-- ... = graded_homology_intro d d (deg j ((deg i)⁻¹ᵉ (deg i x)))
|
||
-- (graded_hom_lift j j_lemma1 ((deg i)⁻¹ᵉ (deg i x))
|
||
-- (i ↘ (!to_right_inv ⬝ !to_left_inv⁻¹) m)) : _
|
||
-- ... = graded_homology_intro d d (deg j ((deg i)⁻¹ᵉ (deg i x)))
|
||
-- (graded_hom_lift j j_lemma1 ((deg i)⁻¹ᵉ (deg i x))
|
||
-- (i ↘ (!to_right_inv ⬝ !to_left_inv⁻¹) m)) : _
|
||
-- ... = 0 : _
|
||
},
|
||
{ exact sorry }
|
||
end
|
||
|
||
lemma j'k' : is_exact_gmod j' k' :=
|
||
begin
|
||
apply is_exact_gmod.mk,
|
||
{ exact sorry },
|
||
{ exact sorry }
|
||
end
|
||
|
||
lemma k'i' : is_exact_gmod k' i' :=
|
||
begin
|
||
apply is_exact_gmod.mk,
|
||
{ intro x m, exact sorry },
|
||
{ exact sorry }
|
||
end
|
||
|
||
end
|
||
end derived_couple
|
||
|
||
section
|
||
open derived_couple exact_couple
|
||
|
||
definition derived_couple [constructor] {R : Ring} {I : Set}
|
||
(X : exact_couple R I) : exact_couple R I :=
|
||
⦃exact_couple, D := D' X, E := E' X, i := i' X, j := j' X, k := k' X,
|
||
ij := i'j' X, jk := j'k' X, ki := k'i' X⦄
|
||
|
||
structure is_bounded {R : Ring} {I : Set} (X : exact_couple R I) : Type :=
|
||
(B B' : I → ℕ)
|
||
(Dub : Π⦃x y⦄ ⦃s : ℕ⦄, (deg (i X))^[s] x = y → B x ≤ s → is_contr (D X y))
|
||
(Eub : Π⦃x y⦄ ⦃s : ℕ⦄, (deg (i X))^[s] x = y → B x ≤ s → is_contr (E X y))
|
||
(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))
|
||
(Elb : Π⦃x y⦄ ⦃s : ℕ⦄, (deg (i X))⁻¹ᵉ^[s] x = y → B x ≤ s → is_contr (E X y))
|
||
(deg_ik_commute : hsquare (deg (k X)) (deg (k X)) (deg (i X)) (deg (i X)))
|
||
(deg_ij_commute : hsquare (deg (j X)) (deg (j X)) (deg (i X)) (deg (i X)))
|
||
|
||
-- definition is_bounded.mk_commute {R : Ring} {I : Set} {X : exact_couple R I}
|
||
-- (B B' : I → ℕ)
|
||
-- (Dub : Π⦃x : I⦄ ⦃s : ℕ⦄, B x ≤ s → is_contr (D X ((deg (i X))^[s] x)))
|
||
-- (Eub : Π⦃x : I⦄ ⦃s : ℕ⦄, B x ≤ s → is_contr (E X ((deg (i X))^[s] x)))
|
||
-- (Dlb : Π⦃x : I⦄ ⦃s : ℕ⦄, B' x ≤ s → is_surjective (i X (((deg (i X))⁻¹ᵉ^[s + 1] x))))
|
||
-- (Elb : Π⦃x : I⦄ ⦃s : ℕ⦄, B x ≤ s → is_contr (E X ((deg (i X))⁻¹ᵉ^[s] x)))
|
||
-- (deg_ik_commute : deg (i X) ∘ deg (k X) ~ deg (k X) ∘ deg (i X))
|
||
-- (deg_ij_commute : deg (i X) ∘ deg (j X) ~ deg (j X) ∘ deg (i X)) : is_bounded X :=
|
||
-- begin
|
||
-- apply is_bounded.mk B B',
|
||
-- { intro x y s p h, induction p, exact Dub h },
|
||
-- { intro x y s p h, induction p,
|
||
-- refine @(is_contr_middle_of_is_exact (exact_couple.jk X (right_inv (deg (j X)) _) idp)) _ _ _,
|
||
|
||
-- --refine transport (λx, is_contr (E X x)) _ (Eub h), exact sorry
|
||
-- },
|
||
-- { exact sorry },
|
||
-- { exact sorry },
|
||
-- { assumption },
|
||
-- end
|
||
|
||
open is_bounded
|
||
parameters {R : Ring} {I : Set} (X : exact_couple R I) (HH : is_bounded X)
|
||
|
||
local abbreviation B := B HH
|
||
local abbreviation B' := B' HH
|
||
local abbreviation Dub := Dub HH
|
||
local abbreviation Eub := Eub HH
|
||
local abbreviation Dlb := Dlb HH
|
||
local abbreviation Elb := Elb HH
|
||
local abbreviation deg_ik_commute := deg_ik_commute HH
|
||
local abbreviation deg_ij_commute := deg_ij_commute HH
|
||
|
||
definition deg_iterate_ik_commute (n : ℕ) :
|
||
hsquare (deg (k X)) (deg (k X)) ((deg (i X))^[n]) ((deg (i X))^[n]) :=
|
||
iterate_commute n deg_ik_commute
|
||
|
||
definition deg_iterate_ij_commute (n : ℕ) :
|
||
hsquare (deg (j X)) (deg (j X)) ((deg (i X))⁻¹ᵉ^[n]) ((deg (i X))⁻¹ᵉ^[n]) :=
|
||
iterate_commute n (hvinverse deg_ij_commute)
|
||
|
||
-- we start counting pages at 0, not at 2.
|
||
definition page (r : ℕ) : exact_couple R I :=
|
||
iterate derived_couple r X
|
||
|
||
definition is_contr_E (r : ℕ) (x : I) (h : is_contr (E X x)) :
|
||
is_contr (E (page r) x) :=
|
||
by induction r with r IH; exact h; exact is_contr_E' (page r) IH
|
||
|
||
definition is_contr_D (r : ℕ) (x : I) (h : is_contr (D X x)) :
|
||
is_contr (D (page r) x) :=
|
||
by induction r with r IH; exact h; exact is_contr_D' (page r) IH
|
||
|
||
definition deg_i (r : ℕ) : deg (i (page r)) ~ deg (i X) :=
|
||
begin
|
||
induction r with r IH,
|
||
{ reflexivity },
|
||
{ exact IH }
|
||
end
|
||
|
||
definition deg_k (r : ℕ) : deg (k (page r)) ~ deg (k X) :=
|
||
begin
|
||
induction r with r IH,
|
||
{ reflexivity },
|
||
{ exact IH }
|
||
end
|
||
|
||
definition deg_j (r : ℕ) :
|
||
deg (j (page r)) ~ deg (j X) ∘ iterate (deg (i X))⁻¹ r :=
|
||
begin
|
||
induction r with r IH,
|
||
{ reflexivity },
|
||
{ refine hwhisker_left (deg (j (page r)))
|
||
(to_inv_homotopy_inv (deg_i r)) ⬝hty _,
|
||
refine hwhisker_right _ IH ⬝hty _,
|
||
apply hwhisker_left, symmetry, apply iterate_succ }
|
||
end
|
||
|
||
definition deg_j_inv (r : ℕ) :
|
||
(deg (j (page r)))⁻¹ ~ iterate (deg (i X)) r ∘ (deg (j X))⁻¹ :=
|
||
have H : deg (j (page r)) ~ iterate_equiv (deg (i X))⁻¹ᵉ r ⬝e deg (j X), from deg_j r,
|
||
λx, to_inv_homotopy_to_inv H x ⬝ iterate_inv (deg (i X))⁻¹ᵉ r ((deg (j X))⁻¹ x)
|
||
|
||
definition deg_d (r : ℕ) :
|
||
deg (d (page r)) ~ deg (j X) ∘ iterate (deg (i X))⁻¹ r ∘ deg (k X) :=
|
||
compose2 (deg_j r) (deg_k r)
|
||
|
||
definition deg_d_inv (r : ℕ) :
|
||
(deg (d (page r)))⁻¹ ~ (deg (k X))⁻¹ ∘ iterate (deg (i X)) r ∘ (deg (j X))⁻¹ :=
|
||
compose2 (to_inv_homotopy_to_inv (deg_k r)) (deg_j_inv r)
|
||
|
||
definition B2 (x : I) : ℕ :=
|
||
max (B (deg (j X) (deg (k X) x))) (B ((deg (k X))⁻¹ ((deg (j X))⁻¹ x)))
|
||
|
||
include Elb Eub
|
||
definition Estable {x : I} {r : ℕ} (H : B2 x ≤ r) :
|
||
E (page (r + 1)) x ≃lm E (page r) x :=
|
||
begin
|
||
change homology (d (page r) x) (d (page r) ← x) ≃lm E (page r) x,
|
||
apply homology_isomorphism: apply is_contr_E,
|
||
exact Eub (hhinverse (deg_iterate_ik_commute r) _ ⬝ (deg_d_inv r x)⁻¹)
|
||
(le.trans !le_max_right H),
|
||
exact Elb (deg_iterate_ij_commute r _ ⬝ (deg_d r x)⁻¹)
|
||
(le.trans !le_max_left H)
|
||
end
|
||
|
||
include Dlb
|
||
definition is_surjective_i {x y z : I} {r s : ℕ} (H : B' z ≤ s + r)
|
||
(p : deg (i (page r)) x = y) (q : iterate (deg (i X)) s y = z) :
|
||
is_surjective (i (page r) ↘ p) :=
|
||
begin
|
||
revert x y z s H p q, induction r with r IH: intro x y z s H p q,
|
||
{ exact Dlb p q H },
|
||
{ change is_surjective (i' (page r) ↘ p),
|
||
apply is_surjective_i', intro z' q',
|
||
refine IH _ _ _ _ (le.trans H (le_of_eq (succ_add s r)⁻¹)) _ _,
|
||
refine !iterate_succ ⬝ ap ((deg (i X))^[s]) _ ⬝ q,
|
||
exact !deg_i⁻¹ ⬝ p }
|
||
end
|
||
|
||
definition Dstable {x : I} {r : ℕ} (H : B' x ≤ r) :
|
||
D (page (r + 1)) x ≃lm D (page r) x :=
|
||
begin
|
||
change image_module (i (page r) ← x) ≃lm D (page r) x,
|
||
refine image_module_isomorphism (i (page r) ← x)
|
||
(is_surjective_i (le.trans H (le_of_eq !zero_add⁻¹)) _ _),
|
||
reflexivity
|
||
end
|
||
|
||
definition Einf : graded_module R I :=
|
||
λx, E (page (B2 x)) x
|
||
|
||
definition Dinf : graded_module R I :=
|
||
λx, D (page (B' x)) x
|
||
|
||
definition Einfstable {x y : I} {r : ℕ} (Hr : B2 y ≤ r) (p : x = y) : Einf y ≃lm E (page r) x :=
|
||
by symmetry; induction p; induction Hr with r Hr IH; reflexivity; exact Estable Hr ⬝lm IH
|
||
|
||
definition Dinfstable {x y : I} {r : ℕ} (Hr : B' y ≤ r) (p : x = y) : Dinf y ≃lm D (page r) x :=
|
||
by symmetry; induction p; induction Hr with r Hr IH; reflexivity; exact Dstable Hr ⬝lm IH
|
||
|
||
parameters {x : I}
|
||
|
||
definition r (n : ℕ) : ℕ :=
|
||
max (max (B (deg (j X) (deg (k X) x)) + n + 1) (B2 ((deg (i X))^[n] x)))
|
||
(max (B' (deg (k X) ((deg (i X))^[n] x)))
|
||
(max (B' (deg (k X) ((deg (i X))^[n+1] x))) (B ((deg (j X))⁻¹ ((deg (i X))^[n] x)))))
|
||
|
||
lemma rb0 (n : ℕ) : r n ≥ n + 1 :=
|
||
ge.trans !le_max_left (ge.trans !le_max_left !le_add_left)
|
||
lemma rb1 (n : ℕ) : B (deg (j X) (deg (k X) x)) ≤ r n - (n + 1) :=
|
||
le_sub_of_add_le (le.trans !le_max_left !le_max_left)
|
||
lemma rb2 (n : ℕ) : B2 ((deg (i X))^[n] x) ≤ r n :=
|
||
le.trans !le_max_right !le_max_left
|
||
lemma rb3 (n : ℕ) : B' (deg (k X) ((deg (i X))^[n] x)) ≤ r n :=
|
||
le.trans !le_max_left !le_max_right
|
||
lemma rb4 (n : ℕ) : B' (deg (k X) ((deg (i X))^[n+1] x)) ≤ r n :=
|
||
le.trans (le.trans !le_max_left !le_max_right) !le_max_right
|
||
lemma rb5 (n : ℕ) : B ((deg (j X))⁻¹ ((deg (i X))^[n] x)) ≤ r n :=
|
||
le.trans (le.trans !le_max_right !le_max_right) !le_max_right
|
||
|
||
definition Einfdiag : graded_module R ℕ :=
|
||
λn, Einf ((deg (i X))^[n] x)
|
||
|
||
definition Dinfdiag : graded_module R ℕ :=
|
||
λn, Dinf (deg (k X) ((deg (i X))^[n] x))
|
||
|
||
include deg_ik_commute Dub
|
||
definition short_exact_mod_page_r (n : ℕ) : short_exact_mod
|
||
(E (page (r n)) ((deg (i X))^[n] x))
|
||
(D (page (r n)) (deg (k (page (r n))) ((deg (i X))^[n] x)))
|
||
(D (page (r n)) (deg (i (page (r n))) (deg (k (page (r n))) ((deg (i X))^[n] x)))) :=
|
||
begin
|
||
fapply short_exact_mod_of_is_exact,
|
||
{ exact j (page (r n)) ← ((deg (i X))^[n] x) },
|
||
{ exact k (page (r n)) ((deg (i X))^[n] x) },
|
||
{ exact i (page (r n)) (deg (k (page (r n))) ((deg (i X))^[n] x)) },
|
||
{ exact j (page (r n)) _ },
|
||
{ apply is_contr_D, refine Dub !deg_j_inv⁻¹ (rb5 n) },
|
||
{ apply is_contr_E, refine Elb _ (rb1 n),
|
||
refine !deg_iterate_ij_commute ⬝ _,
|
||
refine ap (deg (j X)) _ ⬝ !deg_j⁻¹,
|
||
refine iterate_sub _ !rb0 _ ⬝ _, apply ap (_^[r n]),
|
||
exact ap (deg (i X)) (!deg_iterate_ik_commute ⬝ !deg_k⁻¹) ⬝ !deg_i⁻¹ },
|
||
{ apply jk (page (r n)) },
|
||
{ apply ki (page (r n)) },
|
||
{ apply ij (page (r n)) }
|
||
end
|
||
|
||
definition short_exact_mod_infpage (n : ℕ) :
|
||
short_exact_mod (Einfdiag n) (Dinfdiag n) (Dinfdiag (n+1)) :=
|
||
begin
|
||
refine short_exact_mod_isomorphism _ _ _ (short_exact_mod_page_r n),
|
||
{ exact Einfstable !rb2 idp },
|
||
{ exact Dinfstable !rb3 !deg_k },
|
||
{ exact Dinfstable !rb4 (!deg_i ⬝ ap (deg (i X)) !deg_k ⬝ !deg_ik_commute) }
|
||
end
|
||
|
||
definition Dinfdiag0 (bound_zero : B' (deg (k X) x) = 0) : Dinfdiag 0 ≃lm D X (deg (k X) x) :=
|
||
Dinfstable (le_of_eq bound_zero) idp
|
||
|
||
definition Dinfdiag_stable {s : ℕ} (h : B (deg (k X) x) ≤ s) : is_contr (Dinfdiag s) :=
|
||
is_contr_D _ _ (Dub !deg_iterate_ik_commute h)
|
||
|
||
end
|
||
|
||
|
||
end left_module
|
||
open left_module
|
||
namespace pointed
|
||
|
||
open pointed int group is_trunc trunc is_conn
|
||
|
||
definition homotopy_group_conn_nat (n : ℕ) (A : Type*[1]) : AbGroup :=
|
||
AbGroup.mk (π[n] A) (ab_group_homotopy_group_of_is_conn n A)
|
||
|
||
definition homotopy_group_conn : Π(n : ℤ) (A : Type*[1]), AbGroup
|
||
| (of_nat n) A := homotopy_group_conn_nat n A
|
||
| (-[1+ n]) A := trivial_ab_group_lift
|
||
|
||
notation `πag'[`:95 n:0 `]`:0 := homotopy_group_conn n
|
||
|
||
definition homotopy_group_conn_nat_functor (n : ℕ) {A B : Type*[1]} (f : A →* B) :
|
||
homotopy_group_conn_nat n A →g homotopy_group_conn_nat n B :=
|
||
begin
|
||
cases n with n, { apply homomorphism_of_is_contr_right },
|
||
cases n with n, { apply homomorphism_of_is_contr_right },
|
||
exact π→g[n+2] f
|
||
end
|
||
|
||
definition homotopy_group_conn_functor : Π(n : ℤ) {A B : Type*[1]} (f : A →* B), πag'[n] A →g πag'[n] B
|
||
| (of_nat n) A B f := homotopy_group_conn_nat_functor n f
|
||
| (-[1+ n]) A B f := homomorphism_of_is_contr_right _ _
|
||
|
||
notation `π→ag'[`:95 n:0 `]`:0 := homotopy_group_conn_functor n
|
||
|
||
section
|
||
open prod prod.ops fiber
|
||
parameters {A : ℤ → Type*[1]} (f : Π(n : ℤ), A n →* A (n - 1)) [Hf : Πn, is_conn_fun 1 (f n)]
|
||
include Hf
|
||
definition I [constructor] : Set := trunctype.mk (ℤ × ℤ) !is_trunc_prod
|
||
|
||
definition D_sequence : graded_module rℤ I :=
|
||
λv, LeftModule_int_of_AbGroup (πag'[v.2] (A (v.1)))
|
||
|
||
definition E_sequence : graded_module rℤ I :=
|
||
λv, LeftModule_int_of_AbGroup (πag'[v.2] (pconntype.mk (pfiber (f (v.1))) !Hf pt))
|
||
|
||
definition exact_couple_sequence : exact_couple rℤ I :=
|
||
exact_couple.mk D_sequence E_sequence sorry sorry sorry sorry sorry sorry
|
||
|
||
end
|
||
|
||
|
||
end pointed
|
||
|
||
namespace spectrum
|
||
open pointed int group is_trunc trunc is_conn prod prod.ops group fin chain_complex
|
||
section
|
||
|
||
parameters {A : ℤ → spectrum} (f : Π(s : ℤ), A s →ₛ A (s - 1))
|
||
|
||
definition I [constructor] : Set := trunctype.mk (gℤ ×g gℤ) !is_trunc_prod
|
||
|
||
definition D_sequence : graded_module rℤ I :=
|
||
λv, LeftModule_int_of_AbGroup (πₛ[v.1] (A (v.2)))
|
||
|
||
definition E_sequence : graded_module rℤ I :=
|
||
λv, LeftModule_int_of_AbGroup (πₛ[v.1] (sfiber (f (v.2))))
|
||
|
||
include f
|
||
definition i_sequence : D_sequence →gm D_sequence :=
|
||
begin
|
||
fapply graded_hom.mk, exact (prod_equiv_prod erfl (add_right_action (- 1))),
|
||
intro v, induction v with n s,
|
||
apply lm_hom_int.mk, esimp,
|
||
exact πₛ→[n] (f s)
|
||
end
|
||
|
||
definition j_sequence : D_sequence →gm E_sequence :=
|
||
begin
|
||
fapply graded_hom.mk_out',
|
||
exact (prod_equiv_prod (add_right_action 1) (add_right_action (- 1))),
|
||
intro v, induction v with n s,
|
||
apply lm_hom_int.mk, esimp,
|
||
rexact shomotopy_groups_fun (f s) (n, 2)
|
||
end
|
||
|
||
definition k_sequence : E_sequence →gm D_sequence :=
|
||
begin
|
||
fapply graded_hom.mk erfl,
|
||
intro v, induction v with n s,
|
||
apply lm_hom_int.mk, esimp,
|
||
exact πₛ→[n] (spoint (f s))
|
||
end
|
||
|
||
lemma ij_sequence : is_exact_gmod i_sequence j_sequence :=
|
||
begin
|
||
intro x y z p q,
|
||
revert y z q p,
|
||
refine eq.rec_right_inv (deg j_sequence) _,
|
||
intro y, induction x with n s, induction y with m t,
|
||
refine equiv_rect !dpair_eq_dpair_equiv⁻¹ᵉ _ _,
|
||
intro pq, esimp at pq, induction pq with p q,
|
||
revert t q, refine eq.rec_equiv (add_right_action (- 1)) _,
|
||
induction p using eq.rec_symm,
|
||
apply is_exact_homotopy homotopy.rfl,
|
||
{ symmetry, intro, apply graded_hom_mk_out'_left_inv },
|
||
rexact is_exact_of_is_exact_at (is_exact_LES_of_shomotopy_groups (f s) (m, 2)),
|
||
-- exact sorry
|
||
end
|
||
|
||
lemma jk_sequence : is_exact_gmod j_sequence k_sequence :=
|
||
begin
|
||
intro x y z p q, induction q,
|
||
revert x y p, refine eq.rec_right_inv (deg j_sequence) _,
|
||
intro x, induction x with n s,
|
||
apply is_exact_homotopy,
|
||
{ symmetry, intro, apply graded_hom_mk_out'_left_inv },
|
||
{ reflexivity },
|
||
rexact is_exact_of_is_exact_at (is_exact_LES_of_shomotopy_groups (f s) (n, 1)),
|
||
end
|
||
|
||
lemma ki_sequence : is_exact_gmod k_sequence i_sequence :=
|
||
begin
|
||
intro i j k p q, induction p, induction q, induction i with n s,
|
||
rexact is_exact_of_is_exact_at (is_exact_LES_of_shomotopy_groups (f s) (n, 0)),
|
||
end
|
||
|
||
definition exact_couple_sequence [constructor] : exact_couple rℤ I :=
|
||
exact_couple.mk D_sequence E_sequence i_sequence j_sequence k_sequence ij_sequence jk_sequence ki_sequence
|
||
|
||
open int
|
||
parameters (ub : ℤ) (lb : ℤ → ℤ)
|
||
(Aub : Πs n, s ≥ ub → is_contr (A s n))
|
||
(Alb : Πs n, s ≤ lb n → is_contr (πₛ[n] (A s)))
|
||
|
||
definition B : I → ℕ
|
||
| (n, s) := max0 (s - lb n)
|
||
|
||
definition B' : I → ℕ
|
||
| (n, s) := max0 (ub - s)
|
||
|
||
lemma iterate_deg_i (n s : ℤ) (m : ℕ) : (deg i_sequence)^[m] (n, s) = (n, s - m) :=
|
||
begin
|
||
induction m with m IH,
|
||
{ exact prod_eq idp !sub_zero⁻¹ },
|
||
{ exact ap (deg i_sequence) IH ⬝ (prod_eq idp !sub_sub) }
|
||
end
|
||
|
||
include Aub Alb
|
||
lemma Dub ⦃x : I⦄ ⦃t : ℕ⦄ (h : B x ≤ t) : is_contr (D_sequence ((deg i_sequence)^[t] x)) :=
|
||
begin
|
||
-- apply is_contr_homotopy_group_of_is_contr,
|
||
apply Alb, induction x with n s, rewrite [iterate_deg_i],
|
||
apply sub_le_of_sub_le,
|
||
exact le_of_max0_le h,
|
||
end
|
||
|
||
lemma Eub ⦃x : I⦄ ⦃s : ℕ⦄ (H : B x ≤ s) : is_contr (E_sequence ((deg i_sequence)^[s] x)) :=
|
||
begin
|
||
exact sorry
|
||
end
|
||
|
||
lemma Dlb ⦃x : I⦄ ⦃s : ℕ⦄ (H : B' x ≤ s) : is_surjective (i_sequence ((deg i_sequence)⁻¹ᵉ^[s+1] x)) :=
|
||
begin
|
||
exact sorry
|
||
end
|
||
|
||
lemma Elb ⦃x : I⦄ ⦃s : ℕ⦄ (H : B x ≤ s) : is_contr (E_sequence ((deg i_sequence)⁻¹ᵉ^[s] x)) :=
|
||
begin
|
||
exact sorry
|
||
end
|
||
|
||
-- definition is_bounded_sequence : is_bounded exact_couple_sequence :=
|
||
-- is_bounded.mk_commute B B' Dub Eub Dlb Elb (by reflexivity) sorry
|
||
|
||
end
|
||
|
||
|
||
|
||
|
||
end spectrum
|