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continue with derived couple of graded R-modules, almost finish defining the maps

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Floris van Doorn 2017-04-20 22:58:19 -04:00
parent c313d33b03
commit bb209af2e8
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288
algebra/graded.hlean
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@ -2,9 +2,9 @@
-- Author: Floris van Doorn
import .left_module .direct_sum
import .left_module .direct_sum .submodule
open algebra eq left_module pointed function equiv is_equiv is_trunc prod group
open algebra eq left_module pointed function equiv is_equiv is_trunc prod group sigma
namespace left_module
@ -32,43 +32,81 @@ variables {R : Ring} {I : Type} {M M₁ M₂ M₃ : graded_module R I}
deg f i = j
we get a function M₁ i → M₂ j.
-/
definition graded_hom_of_deg (d : I ≃ I) (M₁ M₂ : graded_module R I) : Type :=
Π⦃i j : I⦄ (p : d i = j), M₁ i →lm M₂ j
definition gmd_constant [constructor] (d : I ≃ I) (M₁ M₂ : graded_module R I) : graded_hom_of_deg d M₁ M₂ :=
λi j p, lm_constant (M₁ i) (M₂ j)
definition gmd0 [constructor] {d : I ≃ I} {M₁ M₂ : graded_module R I} : graded_hom_of_deg d M₁ M₂ :=
gmd_constant d M₁ M₂
structure graded_hom (M₁ M₂ : graded_module R I) : Type :=
mk' :: (d : I → I)
(fn' : Π⦃i j : I⦄ (p : d i = j), M₁ i →lm M₂ j)
mk' :: (d : I ≃ I)
(fn' : graded_hom_of_deg d M₁ M₂)
notation M₁ ` →gm ` M₂ := graded_hom M₁ M₂
abbreviation deg [unfold 5] := @graded_hom.d
notation `↘` := graded_hom.fn' -- there is probably a better character for this?
postfix ` ↘`:(max+10) := graded_hom.fn' -- there is probably a better character for this? Maybe ↷?
definition graded_hom_fn [unfold 5] [coercion] (f : M₁ →gm M₂) (i : I) : M₁ i →lm M₂ (deg f i) :=
↘f idp
definition graded_hom_fn [reducible] [unfold 5] [coercion] (f : M₁ →gm M₂) (i : I) : M₁ i →lm M₂ (deg f i) :=
f ↘ idp
definition graded_hom.mk [constructor] {M₁ M₂ : graded_module R I} (d : I → I)
definition graded_hom_fn_in [reducible] [unfold 5] (f : M₁ →gm M₂) (i : I) : M₁ ((deg f)⁻¹ i) →lm M₂ i :=
f ↘ (to_right_inv (deg f) i)
infix ` ← `:101 := graded_hom_fn_in -- todo: change notation
definition graded_hom.mk [constructor] {M₁ M₂ : graded_module R I} (d : I ≃ I)
(fn : Πi, M₁ i →lm M₂ (d i)) : M₁ →gm M₂ :=
graded_hom.mk' d (λi j p, homomorphism_of_eq (ap M₂ p) ∘lm fn i)
variables {f' : M₂ →gm M₃} {f : M₁ →gm M₂}
definition graded_hom.mk_in [constructor] {M₁ M₂ : graded_module R I} (d : I ≃ I)
(fn : Πi, M₁ (d⁻¹ i) →lm M₂ i) : M₁ →gm M₂ :=
graded_hom.mk' d (λi j p, fn j ∘lm homomorphism_of_eq (ap M₁ (eq_inv_of_eq p)))
definition graded_hom.mk_out_in [constructor] {M₁ M₂ : graded_module R I} (d₁ : I ≃ I) (d₂ : I ≃ I)
(fn : Πi, M₁ (d₁ i) →lm M₂ (d₂ i)) : M₁ →gm M₂ :=
graded_hom.mk' (d₁⁻¹ᵉ ⬝e d₂) (λi j p, homomorphism_of_eq (ap M₂ p) ∘lm fn (d₁⁻¹ᵉ i) ∘lm
homomorphism_of_eq (ap M₁ (to_right_inv d₁ i)⁻¹))
definition graded_hom_eq_transport (f : M₁ →gm M₂) {i j : I} (p : deg f i = j) (m : M₁ i) :
f ↘ p m = transport M₂ p (f i m) :=
by induction p; reflexivity
definition graded_hom_eq_zero {f : M₁ →gm M₂} {i j k : I} {q : deg f i = j} {p : deg f i = k}
(m : M₁ i) (r : f ↘ q m = 0) : f ↘ p m = 0 :=
have f ↘ p m = transport M₂ (q⁻¹ ⬝ p) (f ↘ q m), begin induction p, induction q, reflexivity end,
this ⬝ ap (transport M₂ (q⁻¹ ⬝ p)) r ⬝ tr_eq_of_pathover (apd (λi, 0) (q⁻¹ ⬝ p))
variables {f' : M₂ →gm M₃} {f g : M₁ →gm M₂}
definition graded_hom_compose [constructor] (f' : M₂ →gm M₃) (f : M₁ →gm M₂) : M₁ →gm M₃ :=
graded_hom.mk (deg f' ∘ deg f) (λi, f' (deg f i) ∘lm f i)
graded_hom.mk (deg f ⬝e deg f') (λi, f' (deg f i) ∘lm f i)
infixr ` ∘gm `:75 := graded_hom_compose
definition graded_hom_compose_fn (f' : M₂ →gm M₃) (f : M₁ →gm M₂) (i : I) (m : M₁ i) :
(f' ∘gm f) i m = f' (deg f i) (f i m) :=
proof idp qed
variable (M)
definition graded_hom_id [constructor] [refl] : M →gm M :=
graded_hom.mk id (λi, lmid)
graded_hom.mk erfl (λi, lmid)
variable {M}
abbreviation gmid [constructor] := graded_hom_id M
infixr ` ∘gm `:75 := graded_hom_compose
definition gm_constant [constructor] (M₁ M₂ : graded_module R I) (d : I ≃ I) : M₁ →gm M₂ :=
graded_hom.mk' d (gmd_constant d M₁ M₂)
structure graded_iso (M₁ M₂ : graded_module R I) : Type :=
(to_hom : M₁ →gm M₂)
(is_equiv_deg : is_equiv (deg to_hom))
(is_equiv_to_hom : Π⦃i j⦄ (p : deg to_hom i = j), is_equiv (↘to_hom p))
mk' :: (to_hom : M₁ →gm M₂)
(is_equiv_to_hom : Π⦃i j⦄ (p : deg to_hom i = j), is_equiv (to_hom ↘ p))
infix ` ≃gm `:25 := graded_iso
attribute graded_iso.to_hom [coercion]
attribute graded_iso.is_equiv_deg [instance] [priority 1010]
attribute graded_iso._trans_of_to_hom [unfold 5]
definition is_equiv_graded_iso [instance] [priority 1010] (φ : M₁ ≃gm M₂) (i : I) :
@ -77,59 +115,55 @@ graded_iso.is_equiv_to_hom φ idp
definition isomorphism_of_graded_iso' [constructor] (φ : M₁ ≃gm M₂) {i j : I} (p : deg φ i = j) :
M₁ i ≃lm M₂ j :=
isomorphism.mk (↘φ p) !graded_iso.is_equiv_to_hom
isomorphism.mk (φ ↘ p) !graded_iso.is_equiv_to_hom
definition isomorphism_of_graded_iso [constructor] (φ : M₁ ≃gm M₂) (i : I) :
M₁ i ≃lm M₂ (deg φ i) :=
isomorphism.mk (φ i) _
definition graded_iso_of_isomorphism [constructor] (d : I ≃ I) (φ : Πi, M₁ i ≃lm M₂ (d i)) :
definition isomorphism_of_graded_iso_in [constructor] (φ : M₁ ≃gm M₂) (i : I) :
M₁ ((deg φ)⁻¹ i) ≃lm M₂ i :=
isomorphism_of_graded_iso' φ !to_right_inv
protected definition graded_iso.mk [constructor] (d : I ≃ I) (φ : Πi, M₁ i ≃lm M₂ (d i)) :
M₁ ≃gm M₂ :=
begin
apply graded_iso.mk (graded_hom.mk d φ), apply to_is_equiv, intro i j p, induction p,
apply graded_iso.mk' (graded_hom.mk d φ),
intro i j p, induction p,
exact to_is_equiv (equiv_of_isomorphism (φ i)),
end
protected definition graded_iso.mk_in [constructor] (d : I ≃ I) (φ : Πi, M₁ (d⁻¹ i) ≃lm M₂ i) :
M₁ ≃gm M₂ :=
begin
apply graded_iso.mk' (graded_hom.mk_in d φ),
intro i j p, esimp,
exact @is_equiv_compose _ _ _ _ _ !is_equiv_cast _,
end
definition graded_iso_of_eq [constructor] {M₁ M₂ : graded_module R I} (p : M₁ ~ M₂)
: M₁ ≃gm M₂ :=
graded_iso_of_isomorphism erfl (λi, isomorphism_of_eq (p i))
graded_iso.mk erfl (λi, isomorphism_of_eq (p i))
-- definition graded_iso.MK [constructor] (d : I ≃ I) (fn : Πi, M₁ i →lm M₂ (d i))
-- : M₁ ≃gm M₂ :=
-- graded_iso.mk _ _ _ --d (λi j p, homomorphism_of_eq (ap M₂ p) ∘lm fn i)
definition isodeg [unfold 5] (φ : M₁ ≃gm M₂) : I ≃ I :=
equiv.mk (deg φ) _
definition graded_iso_to_lminv [constructor] (φ : M₁ ≃gm M₂) : M₂ →gm M₁ :=
graded_hom.mk (deg φ)⁻¹
abstract begin
intro i, apply to_lminv,
apply isomorphism_of_graded_iso' φ,
apply to_right_inv (isodeg φ) i
end end
definition to_gminv [constructor] (φ : M₁ ≃gm M₂) : M₂ →gm M₁ :=
graded_hom.mk (deg φ)⁻¹
abstract begin
intro i, apply isomorphism.to_hom, symmetry,
apply isomorphism_of_graded_iso' φ,
apply to_right_inv (isodeg φ) i
end end
-- definition to_gminv [constructor] (φ : M₁ ≃gm M₂) : M₂ →gm M₁ :=
-- graded_hom.mk_in (deg φ)⁻¹ᵉ
-- abstract begin
-- intro i, apply isomorphism.to_hom, symmetry,
-- apply isomorphism_of_graded_iso φ
-- end end
variable (M)
definition graded_iso.refl [refl] [constructor] : M ≃gm M :=
graded_iso_of_isomorphism equiv.rfl (λi, isomorphism.rfl)
graded_iso.mk equiv.rfl (λi, isomorphism.rfl)
variable {M}
definition graded_iso.rfl [refl] [constructor] : M ≃gm M := graded_iso.refl M
definition graded_iso.symm [symm] [constructor] (φ : M₁ ≃gm M₂) : M₂ ≃gm M₁ :=
graded_iso.mk (to_gminv φ) !is_equiv_inv
(λi j p, @is_equiv_compose _ _ _ _ _ !isomorphism.is_equiv_to_hom !is_equiv_cast)
graded_iso.mk_in (deg φ)⁻¹ᵉ (λi, (isomorphism_of_graded_iso φ i)⁻¹ˡᵐ)
definition graded_iso.trans [trans] [constructor] (φ : M₁ ≃gm M₂) (ψ : M₂ ≃gm M₃) : M₁ ≃gm M₃ :=
graded_iso_of_isomorphism (isodeg φ ⬝e isodeg ψ)
graded_iso.mk (deg φ ⬝e deg ψ)
(λi, isomorphism_of_graded_iso φ i ⬝lm isomorphism_of_graded_iso ψ (deg φ i))
definition graded_iso.eq_trans [trans] [constructor]
@ -149,6 +183,63 @@ definition graded_hom_of_eq [constructor] {M₁ M₂ : graded_module R I} (p : M
: M₁ →gm M₂ :=
graded_iso_of_eq p
structure graded_homotopy (f g : M₁ →gm M₂) : Type :=
mk' :: (hd : deg f ~ deg g)
(hfn : Π⦃i j : I⦄ (pf : deg f i = j) (pg : deg g i = j) (r : hd i ⬝ pg = pf), f ↘ pf ~ g ↘ pg)
infix ` ~gm `:50 := graded_homotopy
definition graded_homotopy.mk (hd : deg f ~ deg g) (hfn : Πi m, f i m =[hd i] g i m) : f ~gm g :=
graded_homotopy.mk' hd
begin
intro i j pf pg r m, induction r, induction pg, esimp,
exact graded_hom_eq_transport f (hd i) m ⬝ tr_eq_of_pathover (hfn i m),
end
definition graded_homotopy_of_deg (d : I ≃ I) (f g : graded_hom_of_deg d M₁ M₂) : Type :=
Π⦃i j : I⦄ (p : d i = j), f p ~ g p
notation f ` ~[`:50 d:0 `] `:0 g:50 := graded_homotopy_of_deg d f g
variables {d : I ≃ I} {f₁ f₂ : graded_hom_of_deg d M₁ M₂}
-- definition graded_homotopy_elim' [unfold 8] := @graded_homotopy_of_deg.elim'
-- definition graded_homotopy_elim [reducible] [unfold 8] [coercion] (h : f₁ ~[d] f₂) (i : I) :
-- f₁ (refl i) ~ f₂ (refl i) :=
-- graded_homotopy_elim' _ _
-- definition graded_homotopy_elim_in [reducible] [unfold 8] (f : M₁ →gm M₂) (i : I) : M₁ ((deg f)⁻¹ i) →lm M₂ i :=
-- f ↘ (to_right_inv (deg f) i)
definition graded_homotopy_of_deg.mk [constructor] (h : Πi, f₁ (idpath (d i)) ~ f₂ (idpath (d i))) :
f₁ ~[d] f₂ :=
begin
intro i j p, induction p, exact h i
end
-- definition graded_homotopy.mk_in [constructor] {M₁ M₂ : graded_module R I} (d : I ≃ I)
-- (fn : Πi, M₁ (d⁻¹ i) →lm M₂ i) : M₁ →gm M₂ :=
-- graded_hom.mk' d (λi j p, fn j ∘lm homomorphism_of_eq (ap M₁ (eq_inv_of_eq p)))
-- definition is_gconstant (f : M₁ →gm M₂) : Type :=
-- f↘ ~[deg f] gmd0
definition compose_constant (f' : M₂ →gm M₃) (f : M₁ →gm M₂) : Type :=
Π⦃i j k : I⦄ (p : deg f i = j) (q : deg f' j = k) (m : M₁ i), f' ↘ q (f ↘ p m) = 0
definition compose_constant.mk (h : Πi m, f' (deg f i) (f i m) = 0) : compose_constant f' f :=
by intros; induction p; induction q; exact h i m
definition compose_constant.elim (h : compose_constant f' f) (i : I) (m : M₁ i) : f' (deg f i) (f i m) = 0 :=
h idp idp m
definition is_gconstant (f : M₁ →gm M₂) : Type :=
Π⦃i j : I⦄ (p : deg f i = j) (m : M₁ i), f ↘ p m = 0
definition is_gconstant.mk (h : Πi m, f i m = 0) : is_gconstant f :=
by intros; induction p; exact h i m
definition is_gconstant.elim (h : is_gconstant f) (i : I) (m : M₁ i) : f i m = 0 :=
h idp m
/- direct sum of graded R-modules -/
variables {J : Set} (N : graded_module R J)
@ -189,14 +280,59 @@ LeftModule_of_AddAbGroup (dirsum' N) (λr n, dirsum_smul r n)
dirsum_mul_smul
dirsum_one_smul
/- homology of a graded module-homomorphism -/
/- graded variants of left-module constructions -/
definition graded_submodule [constructor] (S : Πi, submodule_rel (M i)) : graded_module R I :=
λi, submodule (S i)
definition graded_submodule_incl [constructor] (S : Πi, submodule_rel (M i)) :
graded_submodule S →gm M :=
graded_hom.mk erfl (λi, submodule_incl (S i))
definition graded_hom_lift [constructor] {S : Πi, submodule_rel (M₂ i)} (φ : M₁ →gm M₂)
(h : Π(i : I) (m : M₁ i), S (deg φ i) (φ i m)) : M₁ →gm graded_submodule S :=
graded_hom.mk (deg φ) (λi, hom_lift (φ i) (h i))
definition graded_image (f : M₁ →gm M₂) : graded_module R I :=
λi, image_module (f ↘ (to_right_inv (deg f) i))
definition graded_image_lift [constructor] (f : M₁ →gm M₂) : M₁ →gm graded_image f :=
graded_hom.mk_in (deg f) (λi, image_lift (f ↘ (to_right_inv (deg f) i)))
definition graded_image_elim [constructor] {f : M₁ →gm M₂} (g : M₁ →gm M₃)
(h : Π⦃i m⦄, f i m = 0 → g i m = 0) :
graded_image f →gm M₃ :=
begin
apply graded_hom.mk_out_in (deg f) (deg g),
intro i,
apply image_elim (g ↘ (ap (deg g) (to_left_inv (deg f) i))),
intro m p,
refine graded_hom_eq_zero m (h _),
exact graded_hom_eq_zero m p
end
definition graded_quotient (S : Πi, submodule_rel (M i)) : graded_module R I :=
λi, quotient_module (S i)
definition graded_quotient_map [constructor] (S : Πi, submodule_rel (M i)) :
M →gm graded_quotient S :=
graded_hom.mk erfl (λi, quotient_map (S i))
definition graded_homology (g : M₂ →gm M₃) (f : M₁ →gm M₂) : graded_module R I :=
λi, homology (g i) (f ↘ (to_right_inv (deg f) i))
definition graded_homology_elim {g : M₂ →gm M₃} {f : M₁ →gm M₂} (h : M₂ →gm M)
(H : compose_constant h f) : graded_homology g f →gm M :=
graded_hom.mk (deg h) (λi, homology_elim (h i) (H _ _))
/- exact couples -/
definition is_exact_gmod (f : M₁ →gm M₂) (f' : M₂ →gm M₃) : Type :=
Π{i j k} (p : deg f i = j) (q : deg f' j = k), is_exact_mod (↘f p) (↘f' q)
Π⦃i j k⦄ (p : deg f i = j) (q : deg f' j = k), is_exact_mod (f ↘ p) (f' ↘ q)
definition gmod_im_in_ker (h : is_exact_gmod f f') : compose_constant f' f :=
λi j k p q, is_exact.im_in_ker (h p q)
structure exact_couple (M₁ M₂ : graded_module R I) : Type :=
(i : M₁ →gm M₁) (j : M₁ →gm M₂) (k : M₂ →gm M₁)
@ -204,14 +340,62 @@ structure exact_couple (M₁ M₂ : graded_module R I) : Type :=
(exact_jk : is_exact_gmod j k)
(exact_ki : is_exact_gmod k i)
variables {i : M₁ →gm M₁} {j : M₁ →gm M₂} {k : M₂ →gm M₁}
end left_module
namespace left_module
namespace derived_couple
section
parameters {R : Ring} {I : Type} {D E : graded_module R I} {i : D →gm D} {j : D →gm E} {k : E →gm D}
(exact_ij : is_exact_gmod i j)
(exact_jk : is_exact_gmod j k)
(exact_ki : is_exact_gmod k i)
namespace derived_couple
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 i' : D' →gm D' := graded_image_lift i ∘gm graded_submodule_incl _
include exact_jk exact_ki
definition d : M₂ →gm M₂ := j ∘gm k
theorem 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 exact_jk) x m
end
theorem 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 : I⦄ ⦃m : D (deg k x)⦄ (p : i (deg k x) m = 0), image (d x) (j (deg k x) m),
begin
intros,
note m_in_im_k := is_exact.ker_in_im (exact_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
exact sorry
end,
intros,
rewrite [graded_hom_compose_fn],
exact quotient_map_eq_zero _ (this p)
end
definition j' : D' →gm E' :=
graded_image_elim (!graded_quotient_map ∘gm graded_hom_lift j j_lemma1) j_lemma2
definition k' : E' →gm D' :=
graded_homology_elim (graded_image_lift i ∘gm k)
abstract 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 exact_jk) (deg k x) (k x m)
end end
end
end derived_couple

12
algebra/left_module.hlean
View file

@ -141,8 +141,17 @@ section module_hom
proposition respect_smul_add_smul (a b : R) (u v : M₁) : f (a • u + b • v) = a • f u + b • f v :=
by rewrite [respect_add f, +respect_smul R f]
end module_hom
section hom_constant
variables {R : Type} {M₁ M₂ : Type}
variables [ring R] [has_scalar R M₁] [add_group M₁] [left_module R M₂]
proposition is_module_hom_constant : is_module_hom R (const M₁ (0 : M₂)) :=
(λm₁ m₂, !add_zero⁻¹, λr m, (smul_zero r)⁻¹ᵖ)
end hom_constant
structure LeftModule (R : Ring) :=
(carrier : Type) (struct : left_module R carrier)
@ -290,6 +299,9 @@ end
abbreviation lmid [constructor] := homomorphism_id M
infixr ` ∘lm `:75 := homomorphism_compose
definition lm_constant [constructor] (M₁ M₂ : LeftModule R) : M₁ →lm M₂ :=
homomorphism.mk (const M₁ 0) !is_module_hom_constant
structure isomorphism (M₁ M₂ : LeftModule R) :=
(to_hom : M₁ →lm M₂)
(is_equiv_to_hom : is_equiv to_hom)

48
algebra/quotient_group.hlean
View file

@ -159,7 +159,7 @@ namespace group
end
namespace quotient
notation `⟦`:max a `⟧`:0 := qg_map a _
notation `⟦`:max a `⟧`:0 := qg_map _ a
end quotient
open quotient
@ -501,4 +501,48 @@ definition codomain_surjection_is_quotient_triangle {A B : AbGroup} (f : A →g
end
end group
end group
namespace group
variables {G H K : Group} {R : normal_subgroup_rel G} {S : normal_subgroup_rel H}
{T : normal_subgroup_rel K}
definition quotient_ab_group_functor [constructor] {G H : AbGroup} {R : subgroup_rel G}
{S : subgroup_rel H} (φ : G →g H)
(h : Πg, R g → S (φ g)) : quotient_ab_group R →g quotient_ab_group S :=
quotient_group_functor φ h
theorem quotient_group_functor_compose (ψ : H →g K) (φ : G →g H)
(hψ : Πg, S g → T (ψ g)) (hφ : Πg, R g → S (φ g)) :
quotient_group_functor ψ hψ ∘g quotient_group_functor φ hφ ~
quotient_group_functor (ψ ∘g φ) (λg, proof hψ (φ g) qed ∘ hφ g) :=
begin
intro g, induction g using set_quotient.rec_prop with g hg, reflexivity
end
definition quotient_group_functor_gid :
quotient_group_functor (gid G) (λg, id) ~ gid (quotient_group R) :=
begin
intro g, induction g using set_quotient.rec_prop with g hg, reflexivity
end
definition quotient_group_functor_mul.{u₁ v₁ u₂ v₂}
{G H : AbGroup} {R : subgroup_rel.{u₁ v₁} G} {S : subgroup_rel.{u₂ v₂} H}
(ψ φ : G →g H) (hψ : Πg, R g → S (ψ g)) (hφ : Πg, R g → S (φ g)) :
homomorphism_mul (quotient_ab_group_functor ψ hψ) (quotient_ab_group_functor φ hφ) ~
quotient_ab_group_functor (homomorphism_mul ψ φ)
(λg hg, subgroup_respect_mul S (hψ g hg) (hφ g hg)) :=
begin
intro g, induction g using set_quotient.rec_prop with g hg, reflexivity
end
definition quotient_group_functor_homotopy {ψ φ : G →g H} (hψ : Πg, R g → S (ψ g))
(hφ : Πg, R g → S (φ g)) (p : φ ~ ψ) :
quotient_group_functor φ hφ ~ quotient_group_functor ψ hψ :=
begin
intro g, induction g using set_quotient.rec_prop with g hg,
exact ap set_quotient.class_of (p g)
end
end group

29
algebra/subgroup.hlean
View file

@ -127,7 +127,7 @@ namespace group
abbreviation is_normal_subgroup [unfold 2] := @normal_subgroup_rel.is_normal_subgroup
section
variables {G G' : Group} (H : subgroup_rel G) (N : normal_subgroup_rel G) {g g' h h' k : G}
variables {G G' G₁ G₂ G₃ : Group} (H : subgroup_rel G) (N : normal_subgroup_rel G) {g g' h h' k : G}
{A B : AbGroup}
theorem is_normal_subgroup' (h : G) (r : N g) : N (h⁻¹ * g * h) :=
@ -315,14 +315,14 @@ namespace group
reflexivity
end
definition iso_surjection_ab_image_incl {A B : AbGroup} (f : A →g B) (H : is_surjective f) : ab_image f ≃g B :=
definition iso_surjection_ab_image_incl [constructor] {A B : AbGroup} (f : A →g B) (H : is_surjective f) : ab_image f ≃g B :=
begin
fapply isomorphism.mk,
exact (ab_image_incl f),
exact is_equiv_surjection_ab_image_incl f H
end
definition hom_lift {G H : Group} (f : G →g H) (K : subgroup_rel H) (Hyp : Π (g : G), K (f g)) : G →g subgroup K :=
definition hom_lift [constructor] {G H : Group} (f : G →g H) (K : subgroup_rel H) (Hyp : Π (g : G), K (f g)) : G →g subgroup K :=
begin
fapply homomorphism.mk,
intro g,
@ -332,7 +332,7 @@ namespace group
intro g h, apply subtype_eq, esimp, apply respect_mul
end
definition image_lift {G H : Group} (f : G →g H) : G →g image f :=
definition image_lift [constructor] {G H : Group} (f : G →g H) : G →g image f :=
begin
fapply hom_lift f,
intro g,
@ -368,6 +368,24 @@ namespace group
intro x,
fapply image_incl_injective f x 1,
end
definition image_elim_lemma {f₁ : G₁ →g G₂} {f₂ : G₁ →g G₃} (h : Π⦃g⦄, f₁ g = 1 → f₂ g = 1)
(g g' : G₁) (p : f₁ g = f₁ g') : f₂ g = f₂ g' :=
have f₁ (g * g'⁻¹) = 1, from !to_respect_mul ⬝ ap (mul _) !to_respect_inv ⬝
mul_inv_eq_of_eq_mul (p ⬝ !one_mul⁻¹),
have f₂ (g * g'⁻¹) = 1, from h this,
eq_of_mul_inv_eq_one (ap (mul _) !to_respect_inv⁻¹ ⬝ !to_respect_mul⁻¹ ⬝ this)
open image
definition image_elim {f₁ : G₁ →g G₂} (f₂ : G₁ →g G₃) (h : Π⦃g⦄, f₁ g = 1 → f₂ g = 1) :
image f₁ →g G₃ :=
homomorphism.mk (total_image.elim_set f₂ (image_elim_lemma h))
begin
refine total_image.rec _, intro g,
refine total_image.rec _, intro g',
exact to_respect_mul f₂ g g'
end
end
variables {G H K : Group} {R : subgroup_rel G} {S : subgroup_rel H} {T : subgroup_rel K}
@ -446,3 +464,6 @@ namespace group
end
end group
open group
attribute image_subgroup [constructor]

201
algebra/submodule.hlean
View file

@ -3,69 +3,13 @@ import .left_module .quotient_group
open algebra eq group sigma is_trunc function trunc
/- move to subgroup -/
namespace group
variables {G H K : Group} {R : subgroup_rel G} {S : subgroup_rel H} {T : subgroup_rel K}
definition subgroup_functor_fun [unfold 7] (φ : G →g H) (h : Πg, R g → S (φ g)) (x : subgroup R) :
subgroup S :=
begin
induction x with g hg,
exact ⟨φ g, h g hg⟩
end
definition subgroup_functor [constructor] (φ : G →g H)
(h : Πg, R g → S (φ g)) : subgroup R →g subgroup S :=
begin
fapply homomorphism.mk,
{ exact subgroup_functor_fun φ h },
{ intro x₁ x₂, induction x₁ with g₁ hg₁, induction x₂ with g₂ hg₂,
exact sigma_eq !to_respect_mul !is_prop.elimo }
end
definition ab_subgroup_functor [constructor] {G H : AbGroup} {R : subgroup_rel G}
{S : subgroup_rel H} (φ : G →g H)
(h : Πg, R g → S (φ g)) : ab_subgroup R →g ab_subgroup S :=
subgroup_functor φ h
theorem subgroup_functor_compose (ψ : H →g K) (φ : G →g H)
(hψ : Πg, S g → T (ψ g)) (hφ : Πg, R g → S (φ g)) :
subgroup_functor ψ hψ ∘g subgroup_functor φ hφ ~
subgroup_functor (ψ ∘g φ) (λg, proof hψ (φ g) qed ∘ hφ g) :=
begin
intro g, induction g with g hg, reflexivity
end
definition subgroup_functor_gid : subgroup_functor (gid G) (λg, id) ~ gid (subgroup R) :=
begin
intro g, induction g with g hg, reflexivity
end
definition subgroup_functor_mul {G H : AbGroup} {R : subgroup_rel G} {S : subgroup_rel H}
(ψ φ : G →g H) (hψ : Πg, R g → S (ψ g)) (hφ : Πg, R g → S (φ g)) :
homomorphism_mul (ab_subgroup_functor ψ hψ) (ab_subgroup_functor φ hφ) ~
ab_subgroup_functor (homomorphism_mul ψ φ)
(λg hg, subgroup_respect_mul S (hψ g hg) (hφ g hg)) :=
begin
intro g, induction g with g hg, reflexivity
end
definition subgroup_functor_homotopy {ψ φ : G →g H} (hψ : Πg, R g → S (ψ g))
(hφ : Πg, R g → S (φ g)) (p : φ ~ ψ) :
subgroup_functor φ hφ ~ subgroup_functor ψ hψ :=
begin
intro g, induction g with g hg,
exact subtype_eq (p g)
end
end group open group
-- move to subgroup
attribute normal_subgroup_rel._trans_of_to_subgroup_rel [unfold 2]
attribute normal_subgroup_rel.to_subgroup_rel [constructor]
namespace left_module
/- submodules -/
variables {R : Ring} {M : LeftModule R} {m m₁ m₂ : M}
variables {R : Ring} {M M₁ M₂ : LeftModule R} {m m₁ m₂ : M}
structure submodule_rel (M : LeftModule R) : Type :=
(S : M → Prop)
@ -140,6 +84,13 @@ lm_homomorphism_of_group_homomorphism (incl_of_subgroup _)
intro r m, induction m with m hm, reflexivity
end
definition hom_lift [constructor] {K : submodule_rel M₂} (φ : M₁ →lm M₂)
(h : Π (m : M₁), K (φ m)) : M₁ →lm submodule K :=
lm_homomorphism_of_group_homomorphism (hom_lift (group_homomorphism_of_lm_homomorphism φ) _ h)
begin
intro r g, exact subtype_eq (to_respect_smul φ r g)
end
definition incl_smul (S : submodule_rel M) (r : R) (m : M) (h : S m) :
r • ⟨m, h⟩ = ⟨_, contains_smul S r h⟩ :> submodule S :=
by reflexivity
@ -155,53 +106,6 @@ submodule_rel.mk (λm, S₁ (submodule_incl S₂ m))
end left_module
/- move to quotient_group -/
namespace group
variables {G H K : Group} {R : normal_subgroup_rel G} {S : normal_subgroup_rel H}
{T : normal_subgroup_rel K}
definition quotient_ab_group_functor [constructor] {G H : AbGroup} {R : subgroup_rel G}
{S : subgroup_rel H} (φ : G →g H)
(h : Πg, R g → S (φ g)) : quotient_ab_group R →g quotient_ab_group S :=
quotient_group_functor φ h
theorem quotient_group_functor_compose (ψ : H →g K) (φ : G →g H)
(hψ : Πg, S g → T (ψ g)) (hφ : Πg, R g → S (φ g)) :
quotient_group_functor ψ hψ ∘g quotient_group_functor φ hφ ~
quotient_group_functor (ψ ∘g φ) (λg, proof hψ (φ g) qed ∘ hφ g) :=
begin
intro g, induction g using set_quotient.rec_prop with g hg, reflexivity
end
definition quotient_group_functor_gid :
quotient_group_functor (gid G) (λg, id) ~ gid (quotient_group R) :=
begin
intro g, induction g using set_quotient.rec_prop with g hg, reflexivity
end
set_option pp.universes true
definition quotient_group_functor_mul.{u₁ v₁ u₂ v₂}
{G H : AbGroup} {R : subgroup_rel.{u₁ v₁} G} {S : subgroup_rel.{u₂ v₂} H}
(ψ φ : G →g H) (hψ : Πg, R g → S (ψ g)) (hφ : Πg, R g → S (φ g)) :
homomorphism_mul (quotient_ab_group_functor ψ hψ) (quotient_ab_group_functor φ hφ) ~
quotient_ab_group_functor (homomorphism_mul ψ φ)
(λg hg, subgroup_respect_mul S (hψ g hg) (hφ g hg)) :=
begin
intro g, induction g using set_quotient.rec_prop with g hg, reflexivity
end
definition quotient_group_functor_homotopy {ψ φ : G →g H} (hψ : Πg, R g → S (ψ g))
(hφ : Πg, R g → S (φ g)) (p : φ ~ ψ) :
quotient_group_functor φ hφ ~ quotient_group_functor ψ hψ :=
begin
intro g, induction g using set_quotient.rec_prop with g hg,
exact ap set_quotient.class_of (p g)
end
end group open group
namespace left_module
/- quotient modules -/
@ -247,9 +151,22 @@ LeftModule_of_AddAbGroup (quotient_module' S) (quotient_module_smul S)
quotient_module_mul_smul
quotient_module_one_smul
definition quotient_map (S : submodule_rel M) : M →lm quotient_module S :=
definition quotient_map [constructor] (S : submodule_rel M) : M →lm quotient_module S :=
lm_homomorphism_of_group_homomorphism (ab_qg_map _) (λr g, idp)
definition quotient_map_eq_zero (m : M) (H : S m) : quotient_map S m = 0 :=
qg_map_eq_one _ H
definition quotient_elim [constructor] (φ : M →lm M₂) (H : Π⦃m⦄, S m → φ m = 0) :
quotient_module S →lm M₂ :=
lm_homomorphism_of_group_homomorphism
(quotient_group_elim (group_homomorphism_of_lm_homomorphism φ) H)
begin
intro r m, esimp,
induction m using set_quotient.rec_prop with m,
exact to_respect_smul φ r m
end
/- specific submodules -/
definition has_scalar_image (φ : M₁ →lm M₂) ⦃m : M₂⦄ (r : R)
(h : image φ m) : image φ (r • m) :=
@ -259,26 +176,48 @@ begin
refine to_respect_smul φ r m' ⬝ ap (λx, r • x) p,
end
definition image_rel (φ : M₁ →lm M₂) : submodule_rel M₂ :=
definition image_rel [constructor] (φ : M₁ →lm M₂) : submodule_rel M₂ :=
submodule_rel_of_subgroup_rel
(image_subgroup (group_homomorphism_of_lm_homomorphism φ))
(has_scalar_image φ)
definition image_module [constructor] (φ : M₁ →lm M₂) : LeftModule R := submodule (image_rel φ)
-- unfortunately this is note definitionally equal:
-- definition foo (φ : M₁ →lm M₂) :
-- (image_module φ : AddAbGroup) = image (group_homomorphism_of_lm_homomorphism φ) :=
-- by reflexivity
variables {ψ : M₂ →lm M₃} {φ : M₁ →lm M₂}
definition image_elim [constructor] (ψ : M₁ →lm M₃) (h : Π⦃g⦄, φ g = 0 → ψ g = 0) :
image_module φ →lm M₃ :=
begin
refine homomorphism.mk (image_elim (group_homomorphism_of_lm_homomorphism ψ) h) _,
split,
{ apply homomorphism.addstruct },
{ intro r, refine @total_image.rec _ _ _ _ (λx, !is_trunc_eq) _, intro g,
apply to_respect_smul }
end
definition has_scalar_kernel (φ : M₁ →lm M₂) ⦃m : M₁⦄ (r : R)
(p : φ m = 0) : φ (r • m) = 0 :=
begin
refine to_respect_smul φ r m ⬝ ap (λx, r • x) p ⬝ smul_zero r,
end
definition kernel_rel (φ : M₁ →lm M₂) : submodule_rel M₁ :=
definition kernel_rel [constructor](φ : M₁ →lm M₂) : submodule_rel M₁ :=
submodule_rel_of_subgroup_rel
(kernel_subgroup (group_homomorphism_of_lm_homomorphism φ))
(has_scalar_kernel φ)
definition kernel_module [constructor] (φ : M₁ →lm M₂) : LeftModule R := submodule (kernel_rel φ)
definition image_lift [constructor] (φ : M₁ →lm M₂) : M₁ →lm image_module φ :=
hom_lift φ (λm, image.mk m idp)
definition homology (ψ : M₂ →lm M₃) (φ : M₁ →lm M₂) : LeftModule R :=
@quotient_module R (submodule (kernel_rel ψ)) (submodule_rel_of_submodule _ (image_rel φ))
variables {ψ : M₂ →lm M₃} {φ : M₁ →lm M₂} (h : Πm, ψ (φ m) = 0)
definition homology.mk (m : M₂) (h : ψ m = 0) : homology ψ φ :=
quotient_map _ ⟨m, h⟩
@ -294,6 +233,19 @@ definition homology_eq {m n : M₂} {hm : ψ m = 0} {hn : ψ n = 0} (h : image
homology.mk m hm = homology.mk n hn :> homology ψ φ :=
eq_of_sub_eq_zero (homology_eq0 h)
definition homology_elim [constructor] (θ : M₂ →lm M) (H : Πm, θ (φ m) = 0) :
homology ψ φ →lm M :=
quotient_elim (θ ∘lm submodule_incl _)
begin
intro m x,
induction m with m h,
esimp at *,
induction x with v, induction v with m' p,
exact ap θ p⁻¹ ⬝ H m'
end
-- remove:
-- definition homology.rec (P : homology ψ φ → Type)
-- [H : Πx, is_set (P x)] (h₀ : Π(m : M₂) (h : ψ m = 0), P (homology.mk m h))
-- (h₁ : Π(m : M₂) (h : ψ m = 0) (k : image φ m), h₀ m h =[homology_eq0' k] h₀ 0 (to_respect_zero ψ))
@ -302,10 +254,33 @@ eq_of_sub_eq_zero (homology_eq0 h)
-- refine @set_quotient.rec _ _ _ H _ _,
-- { intro v, induction v with m h, exact h₀ m h },
-- { intro v v', induction v with m hm, induction v' with n hn,
-- esimp, intro h,
-- exact change_path _ _, }
-- intro h,
-- note x := h₁ (m - n) _ h,
-- esimp,
-- exact change_path _ _,
-- }
-- end
-- definition quotient.rec (P : quotient_group N → Type)
-- [H : Πx, is_set (P x)] (h₀ : Π(g : G), P (qg_map N g))
-- -- (h₀_mul : Π(g h : G), h₀ (g * h))
-- (h₁ : Π(g : G) (h : N g), h₀ g =[qg_map_eq_one g h] h₀ 1)
-- : Πx, P x :=
-- begin
-- refine @set_quotient.rec _ _ _ H _ _,
-- { intro g, exact h₀ g },
-- { intro g g' h,
-- note x := h₁ (g * g'⁻¹) h,
-- }
-- -- { intro v, induction },
-- -- { intro v v', induction v with m hm, induction v' with n hn,
-- -- intro h,
-- -- note x := h₁ (m - n) _ h,
-- -- esimp,
-- -- exact change_path _ _,
-- -- }
-- end
end left_module

27
move_to_lib.hlean
View file

@ -1,10 +1,12 @@
-- definitions, theorems and attributes which should be moved to files in the HoTT library
import homotopy.sphere2 homotopy.cofiber homotopy.wedge
import homotopy.sphere2 homotopy.cofiber homotopy.wedge hit.prop_trunc
open eq nat int susp pointed pmap sigma is_equiv equiv fiber algebra trunc trunc_index pi group
is_trunc function sphere unit sum prod bool
attribute is_prop.elim_set [unfold 6]
definition add_comm_right {A : Type} [add_comm_semigroup A] (n m k : A) : n + m + k = n + k + m :=
!add.assoc ⬝ ap (add n) !add.comm ⬝ !add.assoc⁻¹
@ -1106,6 +1108,29 @@ begin
exact h
end
definition total_image {A B : Type} (f : A → B) : Type := sigma (image f)
local attribute is_prop.elim_set [recursor 6]
definition total_image.elim_set [unfold 8]
{A B : Type} {f : A → B} {C : Type} [is_set C]
(g : A → C) (h : Πa a', f a = f a' → g a = g a') (x : total_image f) : C :=
begin
induction x with b v,
induction v using is_prop.elim_set with x x x',
{ induction x with a p, exact g a },
{ induction x with a p, induction x' with a' p', induction p', exact h _ _ p }
end
definition total_image.rec [unfold 7]
{A B : Type} {f : A → B} {C : total_image f → Type} [H : Πx, is_prop (C x)]
(g : Πa, C ⟨f a, image.mk a idp⟩)
(x : total_image f) : C x :=
begin
induction x with b v,
refine @image.rec _ _ _ _ _ (λv, H ⟨b, v⟩) _ v,
intro a p,
induction p, exact g a
end
definition image.equiv_exists {A B : Type} {f : A → B} {b : B} : image f b ≃ ∃ a, f a = b :=
trunc_equiv_trunc _ (fiber.sigma_char _ _)