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work on graded modules

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This commit is contained in:
Floris van Doorn 2017-04-24 13:33:48 -04:00
parent a7ec040f57
commit 1b4c40413e
4 changed files with 238 additions and 72 deletions
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267
algebra/graded.hlean
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@ -2,7 +2,7 @@
-- Author: Floris van Doorn
import .left_module .direct_sum .submodule
import .left_module .direct_sum .submodule ..heq
open algebra eq left_module pointed function equiv is_equiv is_trunc prod group sigma
@ -11,7 +11,7 @@ namespace left_module
definition graded [reducible] (str : Type) (I : Type) : Type := I → str
definition graded_module [reducible] (R : Ring) : Type → Type := graded (LeftModule R)
variables {R : Ring} {I : Type} {M M₁ M₂ M₃ : graded_module R I}
variables {R : Ring} {I : Set} {M M₁ M₂ M₃ : graded_module R I}
/-
morphisms between graded modules.
@ -31,6 +31,7 @@ variables {R : Ring} {I : Type} {M M₁ M₂ M₃ : graded_module R I}
but as a function taking a path as argument. Specifically, for every path
deg f i = j
we get a function M₁ i → M₂ j.
(3) Note: we do assume that I is a set. This is not strictly necessary, but it simplifies things
-/
definition graded_hom_of_deg (d : I ≃ I) (M₁ M₂ : graded_module R I) : Type :=
@ -49,21 +50,21 @@ mk' :: (d : I ≃ I)
notation M₁ ` →gm ` M₂ := graded_hom M₁ M₂
abbreviation deg [unfold 5] := @graded_hom.d
postfix ` ↘`:(max+10) := graded_hom.fn' -- there is probably a better character for this? Maybe ↷?
postfix ` ↘`:max := graded_hom.fn' -- there is probably a better character for this? Maybe ↷?
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_fn_in [reducible] [unfold 5] (f : M₁ →gm M₂) (i : I) : M₁ ((deg f)⁻¹ i) →lm M₂ i :=
definition graded_hom_fn_out [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
infix ` ← `:101 := graded_hom_fn_out -- 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)
definition graded_hom.mk_in [constructor] {M₁ M₂ : graded_module R I} (d : I ≃ I)
definition graded_hom.mk_out [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)))
@ -81,7 +82,7 @@ definition graded_hom_eq_zero {f : M₁ →gm M₂} {i j k : I} {q : deg f i = j
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}
variables {f' : M₂ →gm M₃} {f g h : M₁ →gm M₂}
definition graded_hom_compose [constructor] (f' : M₂ →gm M₃) (f : M₁ →gm M₂) : M₁ →gm M₃ :=
graded_hom.mk (deg f ⬝e deg f') (λi, f' (deg f i) ∘lm f i)
@ -121,7 +122,7 @@ definition isomorphism_of_graded_iso [constructor] (φ : M₁ ≃gm M₂) (i : I
M₁ i ≃lm M₂ (deg φ i) :=
isomorphism.mk (φ i) _
definition isomorphism_of_graded_iso_in [constructor] (φ : M₁ ≃gm M₂) (i : I) :
definition isomorphism_of_graded_iso_out [constructor] (φ : M₁ ≃gm M₂) (i : I) :
M₁ ((deg φ)⁻¹ i) ≃lm M₂ i :=
isomorphism_of_graded_iso' φ !to_right_inv
@ -133,10 +134,10 @@ begin
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) :
protected definition graded_iso.mk_out [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),
apply graded_iso.mk' (graded_hom.mk_out d φ),
intro i j p, esimp,
exact @is_equiv_compose _ _ _ _ _ !is_equiv_cast _,
end
@ -146,7 +147,7 @@ definition graded_iso_of_eq [constructor] {M₁ M₂ : graded_module R I} (p : M
graded_iso.mk erfl (λi, isomorphism_of_eq (p i))
-- definition to_gminv [constructor] (φ : M₁ ≃gm M₂) : M₂ →gm M₁ :=
-- graded_hom.mk_in (deg φ)⁻¹ᵉ
-- graded_hom.mk_out (deg φ)⁻¹ᵉ
-- abstract begin
-- intro i, apply isomorphism.to_hom, symmetry,
-- apply isomorphism_of_graded_iso φ
@ -160,63 +161,91 @@ 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_in (deg φ)⁻¹ᵉ (λi, (isomorphism_of_graded_iso φ i)⁻¹ˡᵐ)
graded_iso.mk_out (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.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]
{M₁ M₂ : graded_module R I} {M₃ : graded_module R I} (φ : M₁ ~ M₂) (ψ : M₂ ≃gm M₃) : M₁ ≃gm M₃ :=
{M₁ M₂ M₃ : graded_module R I} (φ : M₁ ~ M₂) (ψ : M₂ ≃gm M₃) : M₁ ≃gm M₃ :=
proof graded_iso.trans (graded_iso_of_eq φ) ψ qed
definition graded_iso.trans_eq [trans] [constructor]
{M₁ : graded_module R I} {M₂ M₃ : graded_module R I} (φ : M₁ ≃gm M₂) (ψ : M₂ ~ M₃) : M₁ ≃gm M₃ :=
{M₁ M₂ M₃ : graded_module R I} (φ : M₁ ≃gm M₂) (ψ : M₂ ~ M₃) : M₁ ≃gm M₃ :=
graded_iso.trans φ (graded_iso_of_eq ψ)
postfix `⁻¹ᵍᵐ`:(max + 1) := graded_iso.symm
infixl ` ⬝gm `:75 := graded_iso.trans
infixl ` ⬝gmp `:75 := graded_iso.trans_eq
infixl ` ⬝pgm `:75 := graded_iso.eq_trans
postfix `⁻¹ᵉᵍᵐ`:(max + 1) := graded_iso.symm
infixl ` ⬝egm `:75 := graded_iso.trans
infixl ` ⬝egmp `:75 := graded_iso.trans_eq
infixl ` ⬝epgm `:75 := graded_iso.eq_trans
definition graded_hom_of_eq [constructor] {M₁ M₂ : graded_module R I} (p : M₁ ~ M₂)
: M₁ →gm M₂ :=
graded_iso_of_eq p
definition graded_hom_of_eq [constructor] {M₁ M₂ : graded_module R I} (p : M₁ ~ M₂) : M₁ →gm M₂ :=
proof graded_iso_of_eq p qed
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)
definition graded_homotopy (f g : M₁ →gm M₂) : Type :=
Π⦃i j k⦄ (p : deg f i = j) (q : deg g i = k) (m : M₁ i), f ↘ p m ==[λi, M₂ i] g ↘ q m
-- mk' :: (hd : deg f ~ deg g)
-- (hfn : Π⦃i j : I⦄ (pf : deg f i = j) (pg : deg g i = j), 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.mk2 (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 m, induction (is_set.elim (hd i ⬝ pg) pf), 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 :=
definition graded_homotopy.mk (h : Πi m, f i m ==[λi, M₂ i] g i m) : f ~gm g :=
begin
intro i j p, induction p, exact h i
intros i j k p q m, induction q, induction p, exact h i m
end
-- definition graded_homotopy.mk_in [constructor] {M₁ M₂ : graded_module R I} (d : I ≃ I)
-- definition graded_hom_compose_out {d₁ d₂ : I ≃ I} (f₂ : Πi, M₂ i →lm M₃ (d₂ i))
-- (f₁ : Πi, M₁ (d₁⁻¹ i) →lm M₂ i) : graded_hom.mk d₂ f₂ ∘gm graded_hom.mk_out d₁ f₁ ~gm
-- graded_hom.mk_out_in d₁⁻¹ᵉ d₂ _ :=
-- _
definition graded_hom_out_in_compose_out {d₁ d₂ d₃ : I ≃ I} (f₂ : Πi, M₂ (d₂ i) →lm M₃ (d₃ i))
(f₁ : Πi, M₁ (d₁⁻¹ i) →lm M₂ i) : graded_hom.mk_out_in d₂ d₃ f₂ ∘gm graded_hom.mk_out d₁ f₁ ~gm
graded_hom.mk_out_in (d₂ ⬝e d₁⁻¹ᵉ) d₃ (λi, f₂ i ∘lm (f₁ (d₂ i))) :=
begin
apply graded_homotopy.mk, intro i m, exact sorry
end
definition graded_hom_out_in_rfl {d₁ d₂ : I ≃ I} (f : Πi, M₁ i →lm M₂ (d₂ i))
(p : Πi, d₁ i = i) :
graded_hom.mk_out_in d₁ d₂ (λi, sorry) ~gm graded_hom.mk d₂ f :=
begin
apply graded_homotopy.mk, intro i m, exact sorry
end
definition graded_homotopy.trans (h₁ : f ~gm g) (h₂ : g ~gm h) : f ~gm h :=
begin
exact sorry
end
-- postfix `⁻¹ᵍᵐ`:(max + 1) := graded_iso.symm
infixl ` ⬝gm `:75 := graded_homotopy.trans
-- infixl ` ⬝gmp `:75 := graded_iso.trans_eq
-- infixl ` ⬝pgm `:75 := graded_iso.eq_trans
-- 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_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_out [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 :=
@ -294,10 +323,10 @@ definition graded_hom_lift [constructor] {S : Πi, submodule_rel (M₂ i)} (φ :
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))
λi, image_module (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)))
graded_hom.mk_out (deg f) (λi, image_lift (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) :
@ -311,6 +340,77 @@ begin
exact graded_hom_eq_zero m p
end
definition graded_image' (f : M₁ →gm M₂) : graded_module R I :=
λi, image_module (f i)
definition graded_image'_lift [constructor] (f : M₁ →gm M₂) : M₁ →gm graded_image' f :=
graded_hom.mk erfl (λi, image_lift (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 (deg g),
intro i,
apply image_elim (g i),
intro m p, exact h p
end
theorem graded_image'_elim_compute {f : M₁ →gm M₂} {g : M₁ →gm M₃}
(h : Π⦃i m⦄, f i m = 0 → g i m = 0) :
graded_image'_elim g h ∘gm graded_image'_lift f ~gm g :=
begin
apply graded_homotopy.mk,
intro i m, reflexivity
end
theorem graded_image_elim_compute {f : M₁ →gm M₂} {g : M₁ →gm M₃}
(h : Π⦃i m⦄, f i m = 0 → g i m = 0) :
graded_image_elim g h ∘gm graded_image_lift f ~gm g :=
begin
refine _ ⬝gm graded_image'_elim_compute h,
esimp, exact sorry
-- refine graded_hom_out_in_compose_out _ _ ⬝gm _, exact sorry
-- -- apply graded_homotopy.mk,
-- -- intro i m,
end
variables {α β : I ≃ I}
definition gen_image (f : M₁ →gm M₂) (p : Πi, deg f (α i) = β i) : graded_module R I :=
λi, image_module (f ↘ (p i))
definition gen_image_lift [constructor] (f : M₁ →gm M₂) (p : Πi, deg f (α i) = β i) : M₁ →gm gen_image f p :=
graded_hom.mk_out α⁻¹ᵉ (λi, image_lift (f ↘ (p i)))
definition gen_image_elim [constructor] {f : M₁ →gm M₂} (p : Πi, deg f (α i) = β i) (g : M₁ →gm M₃)
(h : Π⦃i m⦄, f i m = 0 → g i m = 0) :
gen_image f p →gm M₃ :=
begin
apply graded_hom.mk_out_in α⁻¹ᵉ (deg g),
intro i,
apply image_elim (g ↘ (ap (deg g) (to_right_inv α i))),
intro m p,
refine graded_hom_eq_zero m (h _),
exact graded_hom_eq_zero m p
end
theorem gen_image_elim_compute {f : M₁ →gm M₂} {p : deg f ∘ α ~ β} {g : M₁ →gm M₃}
(h : Π⦃i m⦄, f i m = 0 → g i m = 0) :
gen_image_elim p g h ∘gm gen_image_lift f p ~gm g :=
begin
-- induction β with β βe, esimp at *, induction p using homotopy.rec_on_idp,
assert q : β ⬝e (deg f)⁻¹ᵉ = α,
{ apply equiv_eq, intro i, apply inv_eq_of_eq, exact (p i)⁻¹ },
induction q,
-- unfold [gen_image_elim, gen_image_lift],
-- induction (is_prop.elim (λi, to_right_inv (deg f) (β i)) p),
-- apply graded_homotopy.mk,
-- intro i m, reflexivity
end
definition graded_kernel (f : M₁ →gm M₂) : graded_module R I :=
λi, kernel_module (f i)
definition graded_quotient (S : Πi, submodule_rel (M i)) : graded_module R I :=
λi, quotient_module (S i)
@ -321,6 +421,10 @@ 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_intro [constructor] (g : M₂ →gm M₃) (f : M₁ →gm M₂) :
graded_kernel g →gm graded_homology g f :=
graded_quotient_map _
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 _ _))
@ -331,6 +435,11 @@ graded_hom.mk (deg h) (λi, homology_elim (h i) (H _ _))
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)
definition is_exact_gmod.mk {f : M₁ →gm M₂} {f' : M₂ →gm M₃}
(h₁ : Π⦃i⦄ (m : M₁ i), f' (deg f i) (f i m) = 0)
(h₂ : Π⦃i⦄ (m : M₂ (deg f i)), f' (deg f i) m = 0 → image (f i) m) : is_exact_gmod f f' :=
begin intro i j k p q; induction p; induction q; split, apply h₁, apply h₂ end
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)
@ -345,16 +454,19 @@ 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)
parameters {R : Ring} {I : Set} {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)
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
include exact_jk exact_ki exact_ij
definition i' : D' →gm D' :=
graded_image_lift i ∘gm graded_submodule_incl _
-- degree i + 0
theorem j_lemma1 ⦃x : I⦄ (m : D x) : d ((deg j) x) (j x m) = 0 :=
begin
@ -380,7 +492,8 @@ namespace left_module
intros,
refine this _ _ _ p,
exact to_right_inv (deg k) _ ⬝ to_left_inv (deg j) x,
rewrite [ap_con, -adj],
apply is_set.elim
-- rewrite [ap_con, -adj],
end,
intros,
rewrite [graded_hom_compose_fn],
@ -388,7 +501,8 @@ namespace left_module
end
definition j' : D' →gm E' :=
graded_image_elim (!graded_quotient_map ∘gm graded_hom_lift j j_lemma1) j_lemma2
graded_image_elim (graded_homology_intro d d ∘gm graded_hom_lift j j_lemma1) j_lemma2
-- degree -(deg i) + deg j
definition k' : E' →gm D' :=
graded_homology_elim (graded_image_lift i ∘gm k)
@ -398,9 +512,46 @@ namespace left_module
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
-- degree deg i + deg k
theorem is_exact_i'j' : is_exact_gmod i' j' :=
begin
apply is_exact_gmod.mk,
{ intro x, refine total_image.rec _, intro m,
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
theorem is_exact_j'k' : is_exact_gmod j' k' :=
begin
apply is_exact_gmod.mk,
{ },
{ exact sorry }
end
theorem is_exact_k'i' : is_exact_gmod k' i' :=
begin
apply is_exact_gmod.mk,
{ intro x m, },
{ exact sorry }
end
end
-- homomorphism_fn (graded_hom_fn j' (to_fun (deg i') i))
-- (homomorphism_fn (graded_hom_fn i' i) m) = 0
end derived_couple

5
algebra/left_module.hlean
View file

@ -288,11 +288,12 @@ end
definition to_respect_zero (φ : M₁ →lm M₂) : φ 0 = 0 :=
respect_zero φ
definition homomorphism_compose [constructor] (f' : M₂ →lm M₃) (f : M₁ →lm M₂) : M₁ →lm M₃ :=
definition homomorphism_compose [reducible] [constructor] (f' : M₂ →lm M₃) (f : M₁ →lm M₂) :
M₁ →lm M₃ :=
homomorphism.mk (f' ∘ f) !is_module_hom_comp
variable (M)
definition homomorphism_id [constructor] [refl] : M →lm M :=
definition homomorphism_id [reducible] [constructor] [refl] : M →lm M :=
homomorphism.mk (@id M) !is_module_hom_id
variable {M}

35
algebra/submodule.hlean
View file

@ -9,7 +9,7 @@ attribute normal_subgroup_rel.to_subgroup_rel [constructor]
namespace left_module
/- submodules -/
variables {R : Ring} {M M₁ M₂ : LeftModule R} {m m₁ m₂ : M}
variables {R : Ring} {M M₁ M₂ M₃ : LeftModule R} {m m₁ m₂ : M}
structure submodule_rel (M : LeftModule R) : Type :=
(S : M → Prop)
@ -91,6 +91,16 @@ lm_homomorphism_of_group_homomorphism (hom_lift (group_homomorphism_of_lm_homomo
intro r g, exact subtype_eq (to_respect_smul φ r g)
end
definition hom_lift_compose {K : submodule_rel M₃}
(φ : M₂ →lm M₃) (h : Π (m : M₂), K (φ m)) (ψ : M₁ →lm M₂) :
hom_lift φ h ∘lm ψ ~ hom_lift (φ ∘lm ψ) proof (λm, h (ψ m)) qed :=
by reflexivity
definition hom_lift_homotopy {K : submodule_rel M₂} {φ : M₁ →lm M₂}
{h : Π (m : M₁), K (φ m)} {φ' : M₁ →lm M₂}
{h' : Π (m : M₁), K (φ' m)} (p : φ ~ φ') : hom_lift φ h ~ hom_lift φ' h' :=
λg, subtype_eq (p g)
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
@ -104,12 +114,7 @@ submodule_rel.mk (λm, S₁ (submodule_incl S₂ m))
intro m r p, induction m with m hm, exact contains_smul S₁ r p
end
end left_module
namespace left_module
/- quotient modules -/
variables {R : Ring} {M M₁ M₂ M₃ : LeftModule R} {m m₁ m₂ : M} {S : submodule_rel M}
definition quotient_module' (S : submodule_rel M) : AddAbGroup :=
quotient_ab_group (subgroup_rel_of_submodule_rel S)
@ -188,17 +193,26 @@ definition image_module [constructor] (φ : M₁ →lm M₂) : LeftModule R := s
-- (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) :
definition image_lift [constructor] (φ : M₁ →lm M₂) : M₁ →lm image_module φ :=
hom_lift φ (λm, image.mk m idp)
variables {ψ : M₂ →lm M₃} {φ : M₁ →lm M₂} {θ : M₁ →lm M₃}
definition image_elim [constructor] (θ : M₁ →lm M₃) (h : Π⦃g⦄, φ g = 0 → θ g = 0) :
image_module φ →lm M₃ :=
begin
refine homomorphism.mk (image_elim (group_homomorphism_of_lm_homomorphism ψ) h) _,
refine homomorphism.mk (image_elim (group_homomorphism_of_lm_homomorphism θ) h) _,
split,
{ apply homomorphism.addstruct },
{ intro r, refine @total_image.rec _ _ _ _ (λx, !is_trunc_eq) _, intro g,
apply to_respect_smul }
end
definition image_elim_compute (h : Π⦃g⦄, φ g = 0 → θ g = 0) :
image_elim θ h ∘lm image_lift φ ~ θ :=
begin
reflexivity
end
definition has_scalar_kernel (φ : M₁ →lm M₂) ⦃m : M₁⦄ (r : R)
(p : φ m = 0) : φ (r • m) = 0 :=
begin
@ -212,9 +226,6 @@ submodule_rel_of_subgroup_rel
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))

3
known_bugs.txt
View file

@ -3,6 +3,7 @@
instead of
`have g : G, from _,`
- coercions are still displayed by the pretty printer
- When using the calc mode for homotopies, you have to give the proofs using a tactic (e.g. `by exact foo` instead of `foo`)
@ -26,3 +27,5 @@ equiv.MK f
abstract (* long proof *) end
abstract (* long proof *) end
```
- unfold [foo] also does various (sometimes unwanted) reductions (as if you called esimp)