statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
image_to_kernel_epi_of_epi_of_zero [has_images V] [epi f] :
epi (image_to_kernel f (0 : B ⟶ C) (by simp)) | begin
simp only [image_to_kernel_zero_right],
haveI := epi_image_of_epi f,
rw ←image_subobject_arrow,
refine @epi_comp _ _ _ _ _ _ (epi_comp _ _) _ _,
end | instance | image_to_kernel_epi_of_epi_of_zero | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"image_to_kernel",
"image_to_kernel_zero_right"
] | `image_to_kernel` for `A --f--> B --0--> C`, where `g` is an epi is itself an epi
(i.e. the sequence is exact at `B`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology {A B C : V} (f : A ⟶ B) [has_image f] (g : B ⟶ C) [has_kernel g]
(w : f ≫ g = 0) [has_cokernel (image_to_kernel f g w)] : V | cokernel (image_to_kernel f g w) | def | homology | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"image_to_kernel"
] | The homology of a pair of morphisms `f : A ⟶ B` and `g : B ⟶ C` satisfying `f ≫ g = 0`
is the cokernel of the `image_to_kernel` morphism for `f` and `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology.π : (kernel_subobject g : V) ⟶ homology f g w | cokernel.π _ | def | homology.π | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"homology"
] | The morphism from cycles to homology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology.condition : image_to_kernel f g w ≫ homology.π f g w = 0 | cokernel.condition _ | lemma | homology.condition | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"homology.π",
"image_to_kernel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homology.desc {D : V} (k : (kernel_subobject g : V) ⟶ D) (p : image_to_kernel f g w ≫ k = 0) :
homology f g w ⟶ D | cokernel.desc _ k p | def | homology.desc | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"homology",
"image_to_kernel"
] | To construct a map out of homology, it suffices to construct a map out of the cycles
which vanishes on boundaries. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology.π_desc
{D : V} (k : (kernel_subobject g : V) ⟶ D) (p : image_to_kernel f g w ≫ k = 0) :
homology.π f g w ≫ homology.desc f g w k p = k | by { simp [homology.π, homology.desc], } | lemma | homology.π_desc | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"homology.desc",
"homology.π",
"image_to_kernel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homology.ext {D : V} {k k' : homology f g w ⟶ D}
(p : homology.π f g w ≫ k = homology.π f g w ≫ k') : k = k' | by { ext, exact p, } | lemma | homology.ext | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"homology",
"homology.π"
] | To check two morphisms out of `homology f g w` are equal, it suffices to check on cycles. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology_of_zero_right [has_cokernel (image_to_kernel f (0 : B ⟶ C) comp_zero)]
[has_cokernel f] [has_cokernel (image.ι f)] [epi (factor_thru_image f)] :
homology f (0 : B ⟶ C) comp_zero ≅ cokernel f | (cokernel.map_iso _ _ (image_subobject_iso _) ((kernel_subobject_iso 0).trans
kernel_zero_iso_source) (by simp)).trans (cokernel_image_ι _) | def | homology_of_zero_right | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"homology",
"image_to_kernel"
] | The cokernel of the map `Im f ⟶ Ker 0` is isomorphic to the cokernel of `f.` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology_of_zero_left [has_zero_object V] [has_kernels V] [has_image (0 : A ⟶ B)]
[has_cokernel (image_to_kernel (0 : A ⟶ B) g zero_comp)] :
homology (0 : A ⟶ B) g zero_comp ≅ kernel g | ((cokernel_iso_of_eq $ image_to_kernel_zero_left _).trans cokernel_zero_iso_target).trans
(kernel_subobject_iso _) | def | homology_of_zero_left | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"homology",
"image_to_kernel",
"image_to_kernel_zero_left"
] | The kernel of the map `Im 0 ⟶ Ker f` is isomorphic to the kernel of `f.` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology_zero_zero [has_zero_object V]
[has_image (0 : A ⟶ B)] [has_cokernel (image_to_kernel (0 : A ⟶ B) (0 : B ⟶ C) (by simp))] :
homology (0 : A ⟶ B) (0 : B ⟶ C) (by simp) ≅ B | { hom := homology.desc (0 : A ⟶ B) (0 : B ⟶ C) (by simp) (kernel_subobject 0).arrow (by simp),
inv := inv (kernel_subobject 0).arrow ≫ homology.π _ _ _, } | def | homology_zero_zero | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"homology",
"homology.desc",
"homology.π",
"image_to_kernel"
] | `homology 0 0 _` is just the middle object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_subobject_map_comp_image_to_kernel (p : α.right = β.left) :
image_to_kernel f g w ≫ kernel_subobject_map β =
image_subobject_map α ≫ image_to_kernel f' g' w' | by { ext, simp [p], } | lemma | image_subobject_map_comp_image_to_kernel | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"image_to_kernel"
] | Given compatible commutative squares between
a pair `f g` and a pair `f' g'` satisfying `f ≫ g = 0` and `f' ≫ g' = 0`,
the `image_to_kernel` morphisms intertwine the induced map on kernels and the induced map on images. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology.map (p : α.right = β.left) :
homology f g w ⟶ homology f' g' w' | cokernel.desc _ (kernel_subobject_map β ≫ cokernel.π _)
begin
rw [image_subobject_map_comp_image_to_kernel_assoc w w' α β p],
simp only [cokernel.condition, comp_zero],
end | def | homology.map | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"homology"
] | Given compatible commutative squares between
a pair `f g` and a pair `f' g'` satisfying `f ≫ g = 0` and `f' ≫ g' = 0`,
we get a morphism on homology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology.π_map (p : α.right = β.left) :
homology.π f g w ≫ homology.map w w' α β p = kernel_subobject_map β ≫ homology.π f' g' w' | by simp only [homology.π, homology.map, cokernel.π_desc] | lemma | homology.π_map | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"homology.map",
"homology.π"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homology.map_desc (p : α.right = β.left)
{D : V} (k : (kernel_subobject g' : V) ⟶ D) (z : image_to_kernel f' g' w' ≫ k = 0) :
homology.map w w' α β p ≫ homology.desc f' g' w' k z =
homology.desc f g w (kernel_subobject_map β ≫ k)
(by simp only [image_subobject_map_comp_image_to_kernel_assoc w w' α β p, z,... | by ext; simp only [homology.π_desc, homology.π_map_assoc] | lemma | homology.map_desc | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"homology.desc",
"homology.map",
"homology.π_desc",
"image_to_kernel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homology.map_id : homology.map w w (𝟙 _) (𝟙 _) rfl = 𝟙 _ | by ext; simp only [homology.π_map, kernel_subobject_map_id, category.id_comp, category.comp_id] | lemma | homology.map_id | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"homology.map",
"homology.π_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homology.comp_right_eq_comp_left
{V : Type*} [category V] {A₁ B₁ C₁ A₂ B₂ C₂ A₃ B₃ C₃ : V}
{f₁ : A₁ ⟶ B₁} {g₁ : B₁ ⟶ C₁} {f₂ : A₂ ⟶ B₂} {g₂ : B₂ ⟶ C₂} {f₃ : A₃ ⟶ B₃} {g₃ : B₃ ⟶ C₃}
{α₁ : arrow.mk f₁ ⟶ arrow.mk f₂} {β₁ : arrow.mk g₁ ⟶ arrow.mk g₂}
{α₂ : arrow.mk f₂ ⟶ arrow.mk f₃} {β₂ : arrow.mk g₂ ⟶ arrow.mk g₃}... | by simp only [comma.comp_left, comma.comp_right, p₁, p₂] | lemma | homology.comp_right_eq_comp_left | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [] | Auxiliary lemma for homology computations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology.map_comp (p₁ : α₁.right = β₁.left) (p₂ : α₂.right = β₂.left) :
homology.map w₁ w₂ α₁ β₁ p₁ ≫ homology.map w₂ w₃ α₂ β₂ p₂ =
homology.map w₁ w₃ (α₁ ≫ α₂) (β₁ ≫ β₂) (homology.comp_right_eq_comp_left p₁ p₂) | by ext; simp only [kernel_subobject_map_comp, homology.π_map_assoc, homology.π_map, category.assoc] | lemma | homology.map_comp | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"homology.comp_right_eq_comp_left",
"homology.map",
"homology.π_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homology.map_iso (α : arrow.mk f₁ ≅ arrow.mk f₂) (β : arrow.mk g₁ ≅ arrow.mk g₂)
(p : α.hom.right = β.hom.left) :
homology f₁ g₁ w₁ ≅ homology f₂ g₂ w₂ | { hom := homology.map w₁ w₂ α.hom β.hom p,
inv := homology.map w₂ w₁ α.inv β.inv
(by { rw [← cancel_mono (α.hom.right), ← comma.comp_right, α.inv_hom_id, comma.id_right, p,
← comma.comp_left, β.inv_hom_id, comma.id_left], refl }),
hom_inv_id' := by { rw [homology.map_comp], convert homology.map_id _; rw [is... | def | homology.map_iso | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"homology",
"homology.map",
"homology.map_comp",
"homology.map_id"
] | An isomorphism between two three-term complexes induces an isomorphism on homology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aux_tac : tactic unit | `[ dsimp only [auto_param_eq], erw [category.id_comp, category.comp_id], cases pf, cases pg, refl ] | def | aux_tac | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [] | Custom tactic to golf and speedup boring proofs in `homology.congr`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology.congr (pf : f = f') (pg : g = g') : homology f g w ≅ homology f' g' w' | { hom := homology.map w w' ⟨𝟙 _, 𝟙 _, by aux_tac⟩ ⟨𝟙 _, 𝟙 _, by aux_tac⟩ rfl,
inv := homology.map w' w ⟨𝟙 _, 𝟙 _, by aux_tac⟩ ⟨𝟙 _, 𝟙 _, by aux_tac⟩ rfl,
hom_inv_id' := begin
cases pf, cases pg, rw [homology.map_comp, ← homology.map_id],
congr' 1; exact category.comp_id _,
end,
inv_hom_id' := be... | def | homology.congr | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"homology",
"homology.map",
"homology.map_comp",
"homology.map_id"
] | `homology f g w ≅ homology f' g' w'` if `f = f'` and `g = g'`.
(Note the objects are not changing here.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_to_kernel' (w : f ≫ g = 0) : image f ⟶ kernel g | kernel.lift g (image.ι f) (by { ext, simpa using w, }) | def | image_to_kernel' | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [] | While `image_to_kernel f g w` provides a morphism
`image_subobject f ⟶ kernel_subobject g`
in terms of the subobject API,
this variant provides a morphism
`image f ⟶ kernel g`,
which is sometimes more convenient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_subobject_iso_image_to_kernel' (w : f ≫ g = 0) :
(image_subobject_iso f).hom ≫ image_to_kernel' f g w =
image_to_kernel f g w ≫ (kernel_subobject_iso g).hom | by { ext, simp [image_to_kernel'], } | lemma | image_subobject_iso_image_to_kernel' | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"image_to_kernel",
"image_to_kernel'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_to_kernel'_kernel_subobject_iso (w : f ≫ g = 0) :
image_to_kernel' f g w ≫ (kernel_subobject_iso g).inv =
(image_subobject_iso f).inv ≫ image_to_kernel f g w | by { ext, simp [image_to_kernel'], } | lemma | image_to_kernel'_kernel_subobject_iso | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"image_to_kernel",
"image_to_kernel'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homology_iso_cokernel_image_to_kernel' (w : f ≫ g = 0) :
homology f g w ≅ cokernel (image_to_kernel' f g w) | { hom := cokernel.map _ _ (image_subobject_iso f).hom (kernel_subobject_iso g).hom
(by simp only [image_subobject_iso_image_to_kernel']),
inv := cokernel.map _ _ (image_subobject_iso f).inv (kernel_subobject_iso g).inv
(by simp only [image_to_kernel'_kernel_subobject_iso]),
hom_inv_id' := begin
apply co... | def | homology_iso_cokernel_image_to_kernel' | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"homology",
"image_subobject_iso_image_to_kernel'",
"image_to_kernel'",
"image_to_kernel'_kernel_subobject_iso"
] | `homology f g w` can be computed as the cokernel of `image_to_kernel' f g w`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology_iso_cokernel_lift (w : f ≫ g = 0) :
homology f g w ≅ cokernel (kernel.lift g f w) | begin
refine homology_iso_cokernel_image_to_kernel' f g w ≪≫ _,
have p : factor_thru_image f ≫ image_to_kernel' f g w = kernel.lift g f w,
{ ext, simp [image_to_kernel'], },
exact (cokernel_epi_comp _ _).symm ≪≫ cokernel_iso_of_eq p,
end | def | homology_iso_cokernel_lift | algebra.homology | src/algebra/homology/image_to_kernel.lean | [
"category_theory.subobject.limits"
] | [
"homology",
"homology_iso_cokernel_image_to_kernel'",
"image_to_kernel'"
] | `homology f g w` can be computed as the cokernel of `kernel.lift g f w`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_mod_ideals (I : D ⥤ ideal R) : D ⥤ Module.{u} R | { obj := λ t, Module.of R $ R ⧸ (I.obj t),
map := λ s t w, submodule.mapq _ _ (linear_map.id) (I.map w).down.down } | def | local_cohomology.ring_mod_ideals | algebra.homology | src/algebra/homology/local_cohomology.lean | [
"ring_theory.ideal.basic",
"algebra.category.Module.colimits",
"algebra.category.Module.projective",
"category_theory.abelian.ext",
"category_theory.limits.final",
"ring_theory.noetherian"
] | [
"Module.of",
"ideal",
"linear_map.id",
"submodule.mapq"
] | The directed system of `R`-modules of the form `R/J`, where `J` is an ideal of `R`,
determined by the functor `I` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Module_enough_projectives' : enough_projectives (Module.{u} R) | Module.Module_enough_projectives.{u} | instance | local_cohomology.Module_enough_projectives' | algebra.homology | src/algebra/homology/local_cohomology.lean | [
"ring_theory.ideal.basic",
"algebra.category.Module.colimits",
"algebra.category.Module.projective",
"category_theory.abelian.ext",
"category_theory.limits.final",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram (I : D ⥤ ideal R) (i : ℕ) : Dᵒᵖ ⥤ Module.{u} R ⥤ Module.{u} R | (ring_mod_ideals I).op ⋙ Ext R (Module.{u} R) i | def | local_cohomology.diagram | algebra.homology | src/algebra/homology/local_cohomology.lean | [
"ring_theory.ideal.basic",
"algebra.category.Module.colimits",
"algebra.category.Module.projective",
"category_theory.abelian.ext",
"category_theory.limits.final",
"ring_theory.noetherian"
] | [
"Ext",
"ideal"
] | The diagram we will take the colimit of to define local cohomology, corresponding to the
directed system determined by the functor `I` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_diagram (I : D ⥤ ideal R) (i : ℕ) :
Module.{max u v} R ⥤ Module.{max u v} R | colimit (diagram.{(max u v) v} I i) | def | local_cohomology.of_diagram | algebra.homology | src/algebra/homology/local_cohomology.lean | [
"ring_theory.ideal.basic",
"algebra.category.Module.colimits",
"algebra.category.Module.projective",
"category_theory.abelian.ext",
"category_theory.limits.final",
"ring_theory.noetherian"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram_comp (i : ℕ) : diagram (I' ⋙ I) i ≅ I'.op ⋙ (diagram I i) | iso.refl _ | def | local_cohomology.diagram_comp | algebra.homology | src/algebra/homology/local_cohomology.lean | [
"ring_theory.ideal.basic",
"algebra.category.Module.colimits",
"algebra.category.Module.projective",
"category_theory.abelian.ext",
"category_theory.limits.final",
"ring_theory.noetherian"
] | [] | Local cohomology along a composition of diagrams. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_of_final [functor.initial I'] (i : ℕ) :
of_diagram.{(max u v) v'} (I' ⋙ I) i ≅ of_diagram.{(max u v') v} I i | (has_colimit.iso_of_nat_iso (diagram_comp _ _ _))
≪≫ (functor.final.colimit_iso _ _) | def | local_cohomology.iso_of_final | algebra.homology | src/algebra/homology/local_cohomology.lean | [
"ring_theory.ideal.basic",
"algebra.category.Module.colimits",
"algebra.category.Module.projective",
"category_theory.abelian.ext",
"category_theory.limits.final",
"ring_theory.noetherian"
] | [] | Local cohomology agrees along precomposition with a cofinal diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal_powers_diagram (J : ideal R) : ℕᵒᵖ ⥤ ideal R | { obj := λ t, J^(unop t),
map := λ s t w, ⟨⟨ideal.pow_le_pow w.unop.down.down⟩⟩, } | def | local_cohomology.ideal_powers_diagram | algebra.homology | src/algebra/homology/local_cohomology.lean | [
"ring_theory.ideal.basic",
"algebra.category.Module.colimits",
"algebra.category.Module.projective",
"category_theory.abelian.ext",
"category_theory.limits.final",
"ring_theory.noetherian"
] | [
"ideal"
] | The functor sending a natural number `i` to the `i`-th power of the ideal `J` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
self_le_radical (J : ideal R) : Type u | full_subcategory (λ J' : ideal R, J ≤ J'.radical) | def | local_cohomology.self_le_radical | algebra.homology | src/algebra/homology/local_cohomology.lean | [
"ring_theory.ideal.basic",
"algebra.category.Module.colimits",
"algebra.category.Module.projective",
"category_theory.abelian.ext",
"category_theory.limits.final",
"ring_theory.noetherian"
] | [
"ideal"
] | The full subcategory of all ideals with radical containing `J` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
self_le_radical.inhabited (J : ideal R) : inhabited (self_le_radical J) | { default := ⟨J, ideal.le_radical⟩ } | instance | local_cohomology.self_le_radical.inhabited | algebra.homology | src/algebra/homology/local_cohomology.lean | [
"ring_theory.ideal.basic",
"algebra.category.Module.colimits",
"algebra.category.Module.projective",
"category_theory.abelian.ext",
"category_theory.limits.final",
"ring_theory.noetherian"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_le_radical_diagram (J : ideal R) : (self_le_radical J) ⥤ ideal R | full_subcategory_inclusion _ | def | local_cohomology.self_le_radical_diagram | algebra.homology | src/algebra/homology/local_cohomology.lean | [
"ring_theory.ideal.basic",
"algebra.category.Module.colimits",
"algebra.category.Module.projective",
"category_theory.abelian.ext",
"category_theory.limits.final",
"ring_theory.noetherian"
] | [
"ideal"
] | The diagram of all ideals with radical containing `J`, represented as a functor.
This is the "largest" diagram that computes local cohomology with support in `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
local_cohomology (J : ideal R) (i : ℕ) : Module.{u} R ⥤ Module.{u} R | of_diagram (ideal_powers_diagram J) i | def | local_cohomology | algebra.homology | src/algebra/homology/local_cohomology.lean | [
"ring_theory.ideal.basic",
"algebra.category.Module.colimits",
"algebra.category.Module.projective",
"category_theory.abelian.ext",
"category_theory.limits.final",
"ring_theory.noetherian"
] | [
"ideal"
] | `local_cohomology J i` is `i`-th the local cohomology module of a module `M` over
a commutative ring `R` with support in the ideal `J` of `R`, defined as the direct limit
of `Ext^i(R/J^t, M)` over all powers `t : ℕ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
local_cohomology.of_self_le_radical (J : ideal R) (i : ℕ) : Module.{u} R ⥤ Module.{u} R | of_diagram.{u} (self_le_radical_diagram.{u} J) i | def | local_cohomology.of_self_le_radical | algebra.homology | src/algebra/homology/local_cohomology.lean | [
"ring_theory.ideal.basic",
"algebra.category.Module.colimits",
"algebra.category.Module.projective",
"category_theory.abelian.ext",
"category_theory.limits.final",
"ring_theory.noetherian"
] | [
"ideal"
] | Local cohomology as the direct limit of `Ext^i(R/J', M)` over *all* ideals `J'` with radical
containing `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal_powers_to_self_le_radical (J : ideal R) : ℕᵒᵖ ⥤ self_le_radical J | full_subcategory.lift _ (ideal_powers_diagram J)
(λ k, begin
change _ ≤ (J^(unop k)).radical,
cases (unop k),
{ simp only [ideal.radical_top, pow_zero, ideal.one_eq_top, le_top] },
{ simp only [J.radical_pow _ n.succ_pos, ideal.le_radical] },
end) | def | local_cohomology.ideal_powers_to_self_le_radical | algebra.homology | src/algebra/homology/local_cohomology.lean | [
"ring_theory.ideal.basic",
"algebra.category.Module.colimits",
"algebra.category.Module.projective",
"category_theory.abelian.ext",
"category_theory.limits.final",
"ring_theory.noetherian"
] | [
"ideal",
"ideal.le_radical",
"ideal.one_eq_top",
"ideal.radical_top",
"le_top",
"pow_zero"
] | Lifting `ideal_powers_diagram J` from a diagram valued in `ideals R` to a diagram
valued in `self_le_radical J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.exists_pow_le_of_le_radical_of_fg (hIJ : I ≤ J.radical) (hJ : J.radical.fg) :
∃ (k : ℕ), I^k ≤ J | begin
obtain ⟨k, hk⟩ := J.exists_radical_pow_le_of_fg hJ,
use k,
calc I^k ≤ J.radical^k : ideal.pow_mono hIJ _
... ≤ J : hk,
end | lemma | local_cohomology.ideal.exists_pow_le_of_le_radical_of_fg | algebra.homology | src/algebra/homology/local_cohomology.lean | [
"ring_theory.ideal.basic",
"algebra.category.Module.colimits",
"algebra.category.Module.projective",
"category_theory.abelian.ext",
"category_theory.limits.final",
"ring_theory.noetherian"
] | [
"ideal.pow_mono"
] | PORTING NOTE: This lemma should probably be moved to `ring_theory/finiteness.lean`
to be near `ideal.exists_radical_pow_le_of_fg`, which it generalizes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal_powers_initial [hR : is_noetherian R R] :
functor.initial (ideal_powers_to_self_le_radical J) | { out := λ J', begin
apply @zigzag_is_connected _ _ _,
{ intros j1 j2,
apply relation.refl_trans_gen.single,
-- The inclusions `J^n1 ≤ J'` and `J^n2 ≤ J'` always form a triangle, based on
-- which exponent is larger.
cases le_total (unop j1.left) (unop j2.left) with h,
right, exact ⟨costructur... | instance | local_cohomology.ideal_powers_initial | algebra.homology | src/algebra/homology/local_cohomology.lean | [
"ring_theory.ideal.basic",
"algebra.category.Module.colimits",
"algebra.category.Module.projective",
"category_theory.abelian.ext",
"category_theory.limits.final",
"ring_theory.noetherian"
] | [
"is_noetherian",
"relation.refl_trans_gen.single"
] | The diagram of powers of `J` is initial in the diagram of all ideals with
radical containing `J`. This uses noetherianness. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_self_le_radical (J : ideal R) [is_noetherian R R] (i : ℕ) :
local_cohomology.of_self_le_radical J i ≅ local_cohomology J i | (local_cohomology.iso_of_final.{u u 0}
(ideal_powers_to_self_le_radical J) (self_le_radical_diagram J) i).symm
≪≫ has_colimit.iso_of_nat_iso (iso.refl _) | def | local_cohomology.iso_self_le_radical | algebra.homology | src/algebra/homology/local_cohomology.lean | [
"ring_theory.ideal.basic",
"algebra.category.Module.colimits",
"algebra.category.Module.projective",
"category_theory.abelian.ext",
"category_theory.limits.final",
"ring_theory.noetherian"
] | [
"ideal",
"is_noetherian",
"local_cohomology",
"local_cohomology.of_self_le_radical"
] | Local cohomology (defined in terms of powers of `J`) agrees with local
cohomology computed over all ideals with radical containing `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
self_le_radical.cast (hJK : J.radical = K.radical) :
self_le_radical J ⥤ self_le_radical K | full_subcategory.map (λ L hL, begin
rw ← ideal.radical_le_radical_iff at ⊢ hL,
exact hJK.symm.trans_le hL,
end) | def | local_cohomology.self_le_radical.cast | algebra.homology | src/algebra/homology/local_cohomology.lean | [
"ring_theory.ideal.basic",
"algebra.category.Module.colimits",
"algebra.category.Module.projective",
"category_theory.abelian.ext",
"category_theory.limits.final",
"ring_theory.noetherian"
] | [
"ideal.radical_le_radical_iff"
] | Casting from the full subcategory of ideals with radical containing `J` to the full
subcategory of ideals with radical containing `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
self_le_radical.cast_is_equivalence (hJK : J.radical = K.radical) :
is_equivalence (self_le_radical.cast hJK) | { inverse := self_le_radical.cast hJK.symm,
unit_iso := by tidy,
counit_iso := by tidy } | instance | local_cohomology.self_le_radical.cast_is_equivalence | algebra.homology | src/algebra/homology/local_cohomology.lean | [
"ring_theory.ideal.basic",
"algebra.category.Module.colimits",
"algebra.category.Module.projective",
"category_theory.abelian.ext",
"category_theory.limits.final",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_le_radical.iso_of_same_radical (hJK : J.radical = K.radical) (i : ℕ) :
of_self_le_radical J i ≅ of_self_le_radical K i | (iso_of_final.{u u u} (self_le_radical.cast hJK.symm) _ _).symm | def | local_cohomology.self_le_radical.iso_of_same_radical | algebra.homology | src/algebra/homology/local_cohomology.lean | [
"ring_theory.ideal.basic",
"algebra.category.Module.colimits",
"algebra.category.Module.projective",
"category_theory.abelian.ext",
"category_theory.limits.final",
"ring_theory.noetherian"
] | [] | The natural isomorphism between local cohomology defined using the `of_self_le_radical`
diagram, assuming `J.radical = K.radical`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_of_same_radical [is_noetherian R R] (hJK : J.radical = K.radical) (i : ℕ) :
local_cohomology J i ≅ local_cohomology K i | (iso_self_le_radical J i).symm
≪≫ self_le_radical.iso_of_same_radical hJK i
≪≫ iso_self_le_radical K i | def | local_cohomology.iso_of_same_radical | algebra.homology | src/algebra/homology/local_cohomology.lean | [
"ring_theory.ideal.basic",
"algebra.category.Module.colimits",
"algebra.category.Module.projective",
"category_theory.abelian.ext",
"category_theory.limits.final",
"ring_theory.noetherian"
] | [
"is_noetherian",
"local_cohomology"
] | Local cohomology agrees on ideals with the same radical. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology_ext {L M N K : Module R} {f : L ⟶ M} {g : M ⟶ N} (w : f ≫ g = 0)
{h k : homology f g w ⟶ K}
(w : ∀ (x : linear_map.ker g),
h (cokernel.π (image_to_kernel _ _ w) (to_kernel_subobject x)) =
k (cokernel.π (image_to_kernel _ _ w) (to_kernel_subobject x))) : h = k | begin
refine cokernel_funext (λ n, _),
-- Gosh it would be nice if `equiv_rw` could directly use an isomorphism, or an enriched `≃`.
equiv_rw (kernel_subobject_iso g ≪≫ Module.kernel_iso_ker g).to_linear_equiv.to_equiv at n,
convert w n; simp [to_kernel_subobject],
end | lemma | Module.homology_ext | algebra.homology | src/algebra/homology/Module.lean | [
"algebra.homology.homotopy",
"algebra.category.Module.abelian",
"algebra.category.Module.subobject",
"category_theory.limits.concrete_category"
] | [
"Module",
"Module.kernel_iso_ker",
"homology",
"image_to_kernel",
"linear_map.ker"
] | To prove that two maps out of a homology group are equal,
it suffices to check they are equal on the images of cycles. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_cycles {C : homological_complex (Module.{u} R) c}
{i : ι} (x : linear_map.ker (C.d_from i)) : C.cycles i | to_kernel_subobject x | abbreviation | Module.to_cycles | algebra.homology | src/algebra/homology/Module.lean | [
"algebra.homology.homotopy",
"algebra.category.Module.abelian",
"algebra.category.Module.subobject",
"category_theory.limits.concrete_category"
] | [
"homological_complex",
"linear_map.ker"
] | Bundle an element `C.X i` such that `C.d_from i x = 0` as a term of `C.cycles i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cycles_ext {C : homological_complex (Module.{u} R) c} {i : ι}
{x y : C.cycles i} (w : (C.cycles i).arrow x = (C.cycles i).arrow y) : x = y | begin
apply_fun (C.cycles i).arrow using (Module.mono_iff_injective _).mp (cycles C i).arrow_mono,
exact w,
end | lemma | Module.cycles_ext | algebra.homology | src/algebra/homology/Module.lean | [
"algebra.homology.homotopy",
"algebra.category.Module.abelian",
"algebra.category.Module.subobject",
"category_theory.limits.concrete_category"
] | [
"Module.mono_iff_injective",
"homological_complex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cycles_map_to_cycles (f : C ⟶ D) {i : ι} (x : linear_map.ker (C.d_from i)) :
(cycles_map f i) (to_cycles x) = to_cycles ⟨f.f i x.1, by simp [x.2]⟩ | by { ext, simp, } | lemma | Module.cycles_map_to_cycles | algebra.homology | src/algebra/homology/Module.lean | [
"algebra.homology.homotopy",
"algebra.category.Module.abelian",
"algebra.category.Module.subobject",
"category_theory.limits.concrete_category"
] | [
"cycles_map",
"linear_map.ker"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_homology
{C : homological_complex (Module.{u} R) c} {i : ι} (x : linear_map.ker (C.d_from i)) :
C.homology i | homology.π (C.d_to i) (C.d_from i) _ (to_cycles x) | abbreviation | Module.to_homology | algebra.homology | src/algebra/homology/Module.lean | [
"algebra.homology.homotopy",
"algebra.category.Module.abelian",
"algebra.category.Module.subobject",
"category_theory.limits.concrete_category"
] | [
"homological_complex",
"homology.π",
"linear_map.ker"
] | Build a term of `C.homology i` from an element `C.X i` such that `C.d_from i x = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology_ext' {M : Module R} (i : ι) {h k : C.homology i ⟶ M}
(w : ∀ (x : linear_map.ker (C.d_from i)), h (to_homology x) = k (to_homology x)) :
h = k | homology_ext _ w | lemma | Module.homology_ext' | algebra.homology | src/algebra/homology/Module.lean | [
"algebra.homology.homotopy",
"algebra.category.Module.abelian",
"algebra.category.Module.subobject",
"category_theory.limits.concrete_category"
] | [
"Module",
"linear_map.ker"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_to_kernel_op {X Y Z : V} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
image_to_kernel g.op f.op (by rw [←op_comp, w, op_zero]) = ((image_subobject_iso _)
≪≫ (image_op_op _).symm).hom ≫ (cokernel.desc f (factor_thru_image g)
(by rw [←cancel_mono (image.ι g), category.assoc, image.fac, w, zero_comp])).op
≫ (... | begin
ext,
simpa only [iso.trans_hom, iso.symm_hom, iso.trans_inv, kernel_op_op_inv, category.assoc,
image_to_kernel_arrow, kernel_subobject_arrow', kernel.lift_ι, ←op_comp, cokernel.π_desc,
←image_subobject_arrow, ←image_unop_op_inv_comp_op_factor_thru_image g.op],
end | lemma | image_to_kernel_op | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"image_to_kernel",
"image_to_kernel_arrow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_to_kernel_unop {X Y Z : Vᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
image_to_kernel g.unop f.unop (by rw [←unop_comp, w, unop_zero]) = ((image_subobject_iso _)
≪≫ (image_unop_unop _).symm).hom ≫ (cokernel.desc f (factor_thru_image g)
(by rw [←cancel_mono (image.ι g), category.assoc, image.fac, w, zero_... | begin
ext,
dunfold image_unop_unop,
simp only [iso.trans_hom, iso.symm_hom, iso.trans_inv, kernel_unop_unop_inv, category.assoc,
image_to_kernel_arrow, kernel_subobject_arrow', kernel.lift_ι, cokernel.π_desc,
iso.unop_inv, ←unop_comp, factor_thru_image_comp_image_unop_op_inv, quiver.hom.unop_op,
image... | lemma | image_to_kernel_unop | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"image_to_kernel",
"image_to_kernel_arrow",
"quiver.hom.unop_op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homology_op {X Y Z : V} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
homology g.op f.op (by rw [←op_comp, w, op_zero]) ≅ opposite.op (homology f g w) | cokernel_iso_of_eq (image_to_kernel_op _ _ w) ≪≫ (cokernel_epi_comp _ _)
≪≫ cokernel_comp_is_iso _ _ ≪≫ cokernel_op_op _ ≪≫ ((homology_iso_kernel_desc _ _ _)
≪≫ (kernel_iso_of_eq (by ext; simp only [image.fac, cokernel.π_desc, cokernel.π_desc_assoc]))
≪≫ (kernel_comp_mono _ (image.ι g))).op | def | homology_op | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"homology",
"homology_iso_kernel_desc",
"image_to_kernel_op",
"opposite.op"
] | Given `f, g` with `f ≫ g = 0`, the homology of `g.op, f.op` is the opposite of the homology of
`f, g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology_unop {X Y Z : Vᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
homology g.unop f.unop (by rw [←unop_comp, w, unop_zero]) ≅ opposite.unop (homology f g w) | cokernel_iso_of_eq (image_to_kernel_unop _ _ w) ≪≫ (cokernel_epi_comp _ _)
≪≫ cokernel_comp_is_iso _ _ ≪≫ cokernel_unop_unop _
≪≫ ((homology_iso_kernel_desc _ _ _)
≪≫ (kernel_iso_of_eq (by ext; simp only [image.fac, cokernel.π_desc, cokernel.π_desc_assoc]))
≪≫ (kernel_comp_mono _ (image.ι g))).unop | def | homology_unop | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"homology",
"homology_iso_kernel_desc",
"image_to_kernel_unop",
"opposite.unop"
] | Given morphisms `f, g` in `Vᵒᵖ` with `f ≫ g = 0`, the homology of `g.unop, f.unop` is the
opposite of the homology of `f, g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op (X : homological_complex V c) : homological_complex Vᵒᵖ c.symm | { X := λ i, op (X.X i),
d := λ i j, (X.d j i).op,
shape' := λ i j hij, by { rw [X.shape j i hij, op_zero], },
d_comp_d' := by { intros, rw [← op_comp, X.d_comp_d, op_zero], } } | def | homological_complex.op | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"homological_complex"
] | Sends a complex `X` with objects in `V` to the corresponding complex with objects in `Vᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_symm (X : homological_complex V c.symm) : homological_complex Vᵒᵖ c | { X := λ i, op (X.X i),
d := λ i j, (X.d j i).op,
shape' := λ i j hij, by { rw [X.shape j i hij, op_zero], },
d_comp_d' := by { intros, rw [← op_comp, X.d_comp_d, op_zero], } } | def | homological_complex.op_symm | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"homological_complex"
] | Sends a complex `X` with objects in `V` to the corresponding complex with objects in `Vᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop (X : homological_complex Vᵒᵖ c) : homological_complex V c.symm | { X := λ i, unop (X.X i),
d := λ i j, (X.d j i).unop,
shape' := λ i j hij, by { rw [X.shape j i hij, unop_zero], },
d_comp_d' := by { intros, rw [← unop_comp, X.d_comp_d, unop_zero], } } | def | homological_complex.unop | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"homological_complex"
] | Sends a complex `X` with objects in `Vᵒᵖ` to the corresponding complex with objects in `V`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_symm (X : homological_complex Vᵒᵖ c.symm) : homological_complex V c | { X := λ i, unop (X.X i),
d := λ i j, (X.d j i).unop,
shape' := λ i j hij, by { rw [X.shape j i hij, unop_zero], },
d_comp_d' := by { intros, rw [← unop_comp, X.d_comp_d, unop_zero], } } | def | homological_complex.unop_symm | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"homological_complex"
] | Sends a complex `X` with objects in `Vᵒᵖ` to the corresponding complex with objects in `V`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_functor : (homological_complex V c)ᵒᵖ ⥤ homological_complex Vᵒᵖ c.symm | { obj := λ X, (unop X).op,
map := λ X Y f,
{ f := λ i, (f.unop.f i).op,
comm' := λ i j hij, by simp only [op_d, ← op_comp, f.unop.comm] }, } | def | homological_complex.op_functor | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"homological_complex"
] | Auxilliary definition for `op_equivalence`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_inverse : homological_complex Vᵒᵖ c.symm ⥤ (homological_complex V c)ᵒᵖ | { obj := λ X, op X.unop_symm,
map := λ X Y f, quiver.hom.op
{ f := λ i, (f.f i).unop,
comm' := λ i j hij, by simp only [unop_symm_d, ←unop_comp, f.comm], }} | def | homological_complex.op_inverse | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"homological_complex",
"quiver.hom.op"
] | Auxilliary definition for `op_equivalence`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_unit_iso : 𝟭 (homological_complex V c)ᵒᵖ ≅ op_functor V c ⋙ op_inverse V c | nat_iso.of_components (λ X, (homological_complex.hom.iso_of_components (λ i, iso.refl _)
(λ i j hij, by simp only [iso.refl_hom, category.id_comp, unop_symm_d, op_d, quiver.hom.unop_op,
category.comp_id]) : (opposite.unop X).op.unop_symm ≅ unop X).op)
begin
intros X Y f,
refine quiver.hom.unop_inj _... | def | homological_complex.op_unit_iso | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"homological_complex",
"homological_complex.hom.iso_of_components",
"opposite.unop",
"quiver.hom.unop_inj",
"quiver.hom.unop_op"
] | Auxilliary definition for `op_equivalence`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_counit_iso : op_inverse V c ⋙ op_functor V c ≅ 𝟭 (homological_complex Vᵒᵖ c.symm) | nat_iso.of_components (λ X, homological_complex.hom.iso_of_components (λ i, iso.refl _)
(λ i j hij, by simpa only [iso.refl_hom, category.id_comp, category.comp_id]))
begin
intros X Y f,
ext,
simpa only [quiver.hom.unop_op, quiver.hom.op_unop, functor.comp_map, functor.id_map,
iso.refl_hom, catego... | def | homological_complex.op_counit_iso | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"homological_complex",
"homological_complex.hom.iso_of_components",
"quiver.hom.op_unop",
"quiver.hom.unop_op"
] | Auxilliary definition for `op_equivalence`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_equivalence : (homological_complex V c)ᵒᵖ ≌ homological_complex Vᵒᵖ c.symm | { functor := op_functor V c,
inverse := op_inverse V c,
unit_iso := op_unit_iso V c,
counit_iso := op_counit_iso V c,
functor_unit_iso_comp' :=
begin
intro X,
ext,
simp only [op_unit_iso, op_counit_iso, nat_iso.of_components_hom_app, iso.op_hom,
comp_f, op_functor_map_f, quiver.hom.unop_op, ... | def | homological_complex.op_equivalence | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"homological_complex",
"quiver.hom.unop_op"
] | Given a category of complexes with objects in `V`, there is a natural equivalence between its
opposite category and a category of complexes with objects in `Vᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_functor : (homological_complex Vᵒᵖ c)ᵒᵖ ⥤ homological_complex V c.symm | { obj := λ X, (unop X).unop,
map := λ X Y f,
{ f := λ i, (f.unop.f i).unop,
comm' := λ i j hij, by simp only [unop_d, ← unop_comp, f.unop.comm] }, } | def | homological_complex.unop_functor | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"homological_complex"
] | Auxilliary definition for `unop_equivalence`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_inverse : homological_complex V c.symm ⥤ (homological_complex Vᵒᵖ c)ᵒᵖ | { obj := λ X, op X.op_symm,
map := λ X Y f, quiver.hom.op
{ f := λ i, (f.f i).op,
comm' := λ i j hij, by simp only [op_symm_d, ←op_comp, f.comm], }} | def | homological_complex.unop_inverse | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"homological_complex",
"quiver.hom.op"
] | Auxilliary definition for `unop_equivalence`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_unit_iso : 𝟭 (homological_complex Vᵒᵖ c)ᵒᵖ ≅ unop_functor V c ⋙ unop_inverse V c | nat_iso.of_components (λ X, (homological_complex.hom.iso_of_components (λ i, iso.refl _)
(λ i j hij, by simp only [iso.refl_hom, category.id_comp, unop_symm_d, op_d, quiver.hom.unop_op,
category.comp_id]) : (opposite.unop X).op.unop_symm ≅ unop X).op)
begin
intros X Y f,
refine quiver.hom.unop_inj _... | def | homological_complex.unop_unit_iso | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"homological_complex",
"homological_complex.hom.iso_of_components",
"opposite.unop",
"quiver.hom.unop_inj",
"quiver.hom.unop_op"
] | Auxilliary definition for `unop_equivalence`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_counit_iso : unop_inverse V c ⋙ unop_functor V c ≅ 𝟭 (homological_complex V c.symm) | nat_iso.of_components (λ X, homological_complex.hom.iso_of_components (λ i, iso.refl _)
(λ i j hij, by simpa only [iso.refl_hom, category.id_comp, category.comp_id]))
begin
intros X Y f,
ext,
simpa only [quiver.hom.unop_op, quiver.hom.op_unop, functor.comp_map, functor.id_map,
iso.refl_hom, catego... | def | homological_complex.unop_counit_iso | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"homological_complex",
"homological_complex.hom.iso_of_components",
"quiver.hom.op_unop",
"quiver.hom.unop_op"
] | Auxilliary definition for `unop_equivalence`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_equivalence : (homological_complex Vᵒᵖ c)ᵒᵖ ≌ homological_complex V c.symm | { functor := unop_functor V c,
inverse := unop_inverse V c,
unit_iso := unop_unit_iso V c,
counit_iso := unop_counit_iso V c,
functor_unit_iso_comp' :=
begin
intro X,
ext,
simp only [op_unit_iso, op_counit_iso, nat_iso.of_components_hom_app, iso.op_hom,
comp_f, op_functor_map_f, quiver.hom.u... | def | homological_complex.unop_equivalence | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"homological_complex",
"quiver.hom.unop_op"
] | Given a category of complexes with objects in `Vᵒᵖ`, there is a natural equivalence between its
opposite category and a category of complexes with objects in `V`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_functor_additive : (@op_functor ι V _ c _).additive | {} | instance | homological_complex.op_functor_additive | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"additive"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_functor_additive : (@unop_functor ι V _ c _).additive | {} | instance | homological_complex.unop_functor_additive | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"additive"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homology_op_def :
C.op.homology i ≅ _root_.homology (C.d_from i).op (C.d_to i).op
(by rw [←op_comp, C.d_to_comp_d_from i, op_zero]) | iso.refl _ | def | homological_complex.homology_op_def | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [] | Auxilliary tautological definition for `homology_op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology_op : C.op.homology i ≅ opposite.op (C.homology i) | homology_op_def _ _ ≪≫ homology_op _ _ _ | def | homological_complex.homology_op | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"homology_op",
"opposite.op"
] | Given a complex `C` of objects in `V`, the `i`th homology of its 'opposite' complex (with
objects in `Vᵒᵖ`) is the opposite of the `i`th homology of `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology_unop_def (C : homological_complex Vᵒᵖ c) :
C.unop.homology i ≅ _root_.homology (C.d_from i).unop (C.d_to i).unop
(by rw [←unop_comp, C.d_to_comp_d_from i, unop_zero]) | iso.refl _ | def | homological_complex.homology_unop_def | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"homological_complex"
] | Auxilliary tautological definition for `homology_unop`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology_unop (C : homological_complex Vᵒᵖ c) :
C.unop.homology i ≅ opposite.unop (C.homology i) | homology_unop_def _ _ ≪≫ homology_unop _ _ _ | def | homological_complex.homology_unop | algebra.homology | src/algebra/homology/opposite.lean | [
"category_theory.abelian.opposite",
"category_theory.abelian.homology",
"algebra.homology.additive"
] | [
"homological_complex",
"homology_unop",
"opposite.unop"
] | Given a complex `C` of objects in `Vᵒᵖ`, the `i`th homology of its 'opposite' complex (with
objects in `V`) is the opposite of the `i`th homology of `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quasi_iso (f : C ⟶ D) : Prop | (is_iso : ∀ i, is_iso ((homology_functor V c i).map f)) | class | quasi_iso | algebra.homology | src/algebra/homology/quasi_iso.lean | [
"algebra.homology.homotopy",
"category_theory.abelian.homology"
] | [
"homology_functor"
] | A chain map is a quasi-isomorphism if it induces isomorphisms on homology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quasi_iso_of_iso (f : C ⟶ D) [is_iso f] : quasi_iso f | { is_iso := λ i, begin
change is_iso (((homology_functor V c i).map_iso (as_iso f)).hom),
apply_instance,
end } | instance | quasi_iso_of_iso | algebra.homology | src/algebra/homology/quasi_iso.lean | [
"algebra.homology.homotopy",
"category_theory.abelian.homology"
] | [
"homology_functor",
"quasi_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_iso_comp (f : C ⟶ D) [quasi_iso f] (g : D ⟶ E) [quasi_iso g] : quasi_iso (f ≫ g) | { is_iso := λ i, begin
rw functor.map_comp,
apply_instance,
end } | instance | quasi_iso_comp | algebra.homology | src/algebra/homology/quasi_iso.lean | [
"algebra.homology.homotopy",
"category_theory.abelian.homology"
] | [
"quasi_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_iso_of_comp_left (f : C ⟶ D) [quasi_iso f] (g : D ⟶ E) [quasi_iso (f ≫ g)] :
quasi_iso g | { is_iso := λ i, is_iso.of_is_iso_fac_left ((homology_functor V c i).map_comp f g).symm } | lemma | quasi_iso_of_comp_left | algebra.homology | src/algebra/homology/quasi_iso.lean | [
"algebra.homology.homotopy",
"category_theory.abelian.homology"
] | [
"homology_functor",
"map_comp",
"quasi_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_iso_of_comp_right (f : C ⟶ D) (g : D ⟶ E) [quasi_iso g] [quasi_iso (f ≫ g)] :
quasi_iso f | { is_iso := λ i, is_iso.of_is_iso_fac_right ((homology_functor V c i).map_comp f g).symm } | lemma | quasi_iso_of_comp_right | algebra.homology | src/algebra/homology/quasi_iso.lean | [
"algebra.homology.homotopy",
"category_theory.abelian.homology"
] | [
"homology_functor",
"map_comp",
"quasi_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_quasi_iso {C D : homological_complex W c} (e : homotopy_equiv C D) :
quasi_iso e.hom | ⟨λ i, begin
refine ⟨⟨(homology_functor W c i).map e.inv, _⟩⟩,
simp only [← functor.map_comp, ← (homology_functor W c i).map_id],
split; apply homology_map_eq_of_homotopy,
exacts [e.homotopy_hom_inv_id, e.homotopy_inv_hom_id],
end⟩ | lemma | homotopy_equiv.to_quasi_iso | algebra.homology | src/algebra/homology/quasi_iso.lean | [
"algebra.homology.homotopy",
"category_theory.abelian.homology"
] | [
"homological_complex",
"homology_functor",
"homology_map_eq_of_homotopy",
"homotopy_equiv",
"map_id",
"quasi_iso"
] | An homotopy equivalence is a quasi-isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_quasi_iso_inv {C D : homological_complex W c} (e : homotopy_equiv C D) (i : ι) :
(@as_iso _ _ _ _ _ (e.to_quasi_iso.1 i)).inv = (homology_functor W c i).map e.inv | begin
symmetry,
simp only [←iso.hom_comp_eq_id, as_iso_hom, ←functor.map_comp, ←(homology_functor W c i).map_id,
homology_map_eq_of_homotopy e.homotopy_hom_inv_id _],
end | lemma | homotopy_equiv.to_quasi_iso_inv | algebra.homology | src/algebra/homology/quasi_iso.lean | [
"algebra.homology.homotopy",
"category_theory.abelian.homology"
] | [
"homological_complex",
"homology_functor",
"homology_map_eq_of_homotopy",
"homotopy_equiv",
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_single₀_cokernel_at_zero_iso : cokernel (X.d 1 0) ≅ Y | (X.homology_zero_iso.symm.trans ((@as_iso _ _ _ _ _ (hf.1 0)).trans
((chain_complex.homology_functor_0_single₀ W).app Y))) | def | homological_complex.hom.to_single₀_cokernel_at_zero_iso | algebra.homology | src/algebra/homology/quasi_iso.lean | [
"algebra.homology.homotopy",
"category_theory.abelian.homology"
] | [
"chain_complex.homology_functor_0_single₀"
] | If a chain map `f : X ⟶ Y[0]` is a quasi-isomorphism, then the cokernel of the differential
`d : X₁ → X₀` is isomorphic to `Y.` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_single₀_cokernel_at_zero_iso_hom_eq [hf : quasi_iso f] :
f.to_single₀_cokernel_at_zero_iso.hom = cokernel.desc (X.d 1 0) (f.f 0)
(by rw ←f.2 1 0 rfl; exact comp_zero) | begin
ext,
dunfold to_single₀_cokernel_at_zero_iso chain_complex.homology_zero_iso homology_of_zero_right
homology.map_iso chain_complex.homology_functor_0_single₀ cokernel.map,
dsimp,
simp only [cokernel.π_desc, category.assoc, homology.map_desc, cokernel.π_desc_assoc],
simp [homology.desc, iso.refl_inv ... | lemma | homological_complex.hom.to_single₀_cokernel_at_zero_iso_hom_eq | algebra.homology | src/algebra/homology/quasi_iso.lean | [
"algebra.homology.homotopy",
"category_theory.abelian.homology"
] | [
"chain_complex.homology_functor_0_single₀",
"chain_complex.homology_zero_iso",
"homology.desc",
"homology.map_desc",
"homology.map_iso",
"homology_of_zero_right",
"quasi_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_single₀_epi_at_zero [hf : quasi_iso f] :
epi (f.f 0) | begin
constructor,
intros Z g h Hgh,
rw [←cokernel.π_desc (X.d 1 0) (f.f 0) (by rw ←f.2 1 0 rfl; exact comp_zero),
←to_single₀_cokernel_at_zero_iso_hom_eq] at Hgh,
rw (@cancel_epi _ _ _ _ _ _ (epi_comp _ _) _ _).1 Hgh,
end | lemma | homological_complex.hom.to_single₀_epi_at_zero | algebra.homology | src/algebra/homology/quasi_iso.lean | [
"algebra.homology.homotopy",
"category_theory.abelian.homology"
] | [
"quasi_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_single₀_exact_d_f_at_zero [hf : quasi_iso f] :
exact (X.d 1 0) (f.f 0) | begin
rw preadditive.exact_iff_homology_zero,
have h : X.d 1 0 ≫ f.f 0 = 0,
{ simp only [← f.2 1 0 rfl, chain_complex.single₀_obj_X_d, comp_zero], },
refine ⟨h, nonempty.intro (homology_iso_kernel_desc _ _ _ ≪≫ _)⟩,
{ suffices : is_iso (cokernel.desc _ _ h),
{ haveI := this, apply kernel.of_mono, },
... | lemma | homological_complex.hom.to_single₀_exact_d_f_at_zero | algebra.homology | src/algebra/homology/quasi_iso.lean | [
"algebra.homology.homotopy",
"category_theory.abelian.homology"
] | [
"chain_complex.single₀_obj_X_d",
"homology_iso_kernel_desc",
"quasi_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_single₀_exact_at_succ [hf : quasi_iso f] (n : ℕ) :
exact (X.d (n + 2) (n + 1)) (X.d (n + 1) n) | (preadditive.exact_iff_homology_zero _ _).2 ⟨X.d_comp_d _ _ _,
⟨(chain_complex.homology_succ_iso _ _).symm.trans
((@as_iso _ _ _ _ _ (hf.1 (n + 1))).trans homology_zero_zero)⟩⟩ | lemma | homological_complex.hom.to_single₀_exact_at_succ | algebra.homology | src/algebra/homology/quasi_iso.lean | [
"algebra.homology.homotopy",
"category_theory.abelian.homology"
] | [
"chain_complex.homology_succ_iso",
"homology_zero_zero",
"quasi_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_single₀_kernel_at_zero_iso [hf : quasi_iso f] : kernel (X.d 0 1) ≅ Y | (X.homology_zero_iso.symm.trans ((@as_iso _ _ _ _ _ (hf.1 0)).symm.trans
((cochain_complex.homology_functor_0_single₀ W).app Y))) | def | homological_complex.hom.from_single₀_kernel_at_zero_iso | algebra.homology | src/algebra/homology/quasi_iso.lean | [
"algebra.homology.homotopy",
"category_theory.abelian.homology"
] | [
"cochain_complex.homology_functor_0_single₀",
"quasi_iso"
] | If a cochain map `f : Y[0] ⟶ X` is a quasi-isomorphism, then the kernel of the differential
`d : X₀ → X₁` is isomorphic to `Y.` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
from_single₀_kernel_at_zero_iso_inv_eq [hf : quasi_iso f] :
f.from_single₀_kernel_at_zero_iso.inv = kernel.lift (X.d 0 1) (f.f 0)
(by rw f.2 0 1 rfl; exact zero_comp) | begin
ext,
dunfold from_single₀_kernel_at_zero_iso cochain_complex.homology_zero_iso homology_of_zero_left
homology.map_iso cochain_complex.homology_functor_0_single₀ kernel.map,
simp only [iso.trans_inv, iso.app_inv, iso.symm_inv, category.assoc,
equalizer_as_kernel, kernel.lift_ι],
dsimp,
simp only ... | lemma | homological_complex.hom.from_single₀_kernel_at_zero_iso_inv_eq | algebra.homology | src/algebra/homology/quasi_iso.lean | [
"algebra.homology.homotopy",
"category_theory.abelian.homology"
] | [
"cochain_complex.homology_functor_0_single₀",
"cochain_complex.homology_zero_iso",
"homology.map_iso",
"homology.π",
"homology.π_map",
"homology_of_zero_left",
"quasi_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_single₀_mono_at_zero [hf : quasi_iso f] :
mono (f.f 0) | begin
constructor,
intros Z g h Hgh,
rw [←kernel.lift_ι (X.d 0 1) (f.f 0) (by rw f.2 0 1 rfl; exact zero_comp),
←from_single₀_kernel_at_zero_iso_inv_eq] at Hgh,
rw (@cancel_mono _ _ _ _ _ _ (mono_comp _ _) _ _).1 Hgh,
end | lemma | homological_complex.hom.from_single₀_mono_at_zero | algebra.homology | src/algebra/homology/quasi_iso.lean | [
"algebra.homology.homotopy",
"category_theory.abelian.homology"
] | [
"quasi_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_single₀_exact_f_d_at_zero [hf : quasi_iso f] :
exact (f.f 0) (X.d 0 1) | begin
rw preadditive.exact_iff_homology_zero,
have h : f.f 0 ≫ X.d 0 1 = 0,
{ simp only [homological_complex.hom.comm, cochain_complex.single₀_obj_X_d, zero_comp] },
refine ⟨h, nonempty.intro (homology_iso_cokernel_lift _ _ _ ≪≫ _)⟩,
{ suffices : is_iso (kernel.lift (X.d 0 1) (f.f 0) h),
{ haveI := this, ... | lemma | homological_complex.hom.from_single₀_exact_f_d_at_zero | algebra.homology | src/algebra/homology/quasi_iso.lean | [
"algebra.homology.homotopy",
"category_theory.abelian.homology"
] | [
"cochain_complex.single₀_obj_X_d",
"homological_complex.hom.comm",
"homology_iso_cokernel_lift",
"quasi_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_single₀_exact_at_succ [hf : quasi_iso f] (n : ℕ) :
exact (X.d n (n + 1)) (X.d (n + 1) (n + 2)) | (preadditive.exact_iff_homology_zero _ _).2
⟨X.d_comp_d _ _ _, ⟨(cochain_complex.homology_succ_iso _ _).symm.trans
((@as_iso _ _ _ _ _ (hf.1 (n + 1))).symm.trans homology_zero_zero)⟩⟩ | lemma | homological_complex.hom.from_single₀_exact_at_succ | algebra.homology | src/algebra/homology/quasi_iso.lean | [
"algebra.homology.homotopy",
"category_theory.abelian.homology"
] | [
"cochain_complex.homology_succ_iso",
"homology_zero_zero",
"quasi_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
category_theory.functor.quasi_iso_of_map_quasi_iso
{C D : homological_complex A c} (f : C ⟶ D)
(hf : quasi_iso ((F.map_homological_complex _).map f)) : quasi_iso f | ⟨λ i, begin
haveI : is_iso (F.map ((homology_functor A c i).map f)),
{ rw [← functor.comp_map, ← nat_iso.naturality_2 (F.homology_functor_iso i) f,
functor.comp_map],
apply_instance, },
exact is_iso_of_reflects_iso _ F,
end⟩ | lemma | category_theory.functor.quasi_iso_of_map_quasi_iso | algebra.homology | src/algebra/homology/quasi_iso.lean | [
"algebra.homology.homotopy",
"category_theory.abelian.homology"
] | [
"homological_complex",
"homology_functor",
"quasi_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single (j : ι) : V ⥤ homological_complex V c | { obj := λ A,
{ X := λ i, if i = j then A else 0,
d := λ i j, 0, },
map := λ A B f,
{ f := λ i, if h : i = j then
eq_to_hom (by { dsimp, rw if_pos h, }) ≫ f ≫ eq_to_hom (by { dsimp, rw if_pos h, })
else
0, },
map_id' := λ A, begin
ext,
dsimp,
split_ifs with h,
{ subst h, simp... | def | homological_complex.single | algebra.homology | src/algebra/homology/single.lean | [
"algebra.homology.homology"
] | [
"homological_complex"
] | The functor `V ⥤ homological_complex V c` creating a chain complex supported in a single degree.
See also `chain_complex.single₀ : V ⥤ chain_complex V ℕ`,
which has better definitional properties,
if you are working with `ℕ`-indexed complexes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
single_obj_X_self (j : ι) (A : V) : ((single V c j).obj A).X j ≅ A | eq_to_iso (by simp) | def | homological_complex.single_obj_X_self | algebra.homology | src/algebra/homology/single.lean | [
"algebra.homology.homology"
] | [] | The object in degree `j` of `(single V c h).obj A` is just `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
single_map_f_self (j : ι) {A B : V} (f : A ⟶ B) :
((single V c j).map f).f j =
(single_obj_X_self V c j A).hom ≫ f ≫ (single_obj_X_self V c j B).inv | by { simp, refl, } | lemma | homological_complex.single_map_f_self | algebra.homology | src/algebra/homology/single.lean | [
"algebra.homology.homology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single₀ : V ⥤ chain_complex V ℕ | { obj := λ X,
{ X := λ n, match n with
| 0 := X
| (n+1) := 0
end,
d := λ i j, 0, },
map := λ X Y f,
{ f := λ n, match n with
| 0 := f
| (n+1) := 0
end, },
map_id' := λ X, by { ext n, cases n, refl, dsimp, unfold_aux, simp, },
map_comp' := λ X Y Z f g, by { ext n, cases n, refl, dsi... | def | chain_complex.single₀ | algebra.homology | src/algebra/homology/single.lean | [
"algebra.homology.homology"
] | [
"chain_complex"
] | `chain_complex.single₀ V` is the embedding of `V` into `chain_complex V ℕ`
as chain complexes supported in degree 0.
This is naturally isomorphic to `single V _ 0`, but has better definitional properties. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
single₀_obj_X_0 (X : V) : ((single₀ V).obj X).X 0 = X | rfl | lemma | chain_complex.single₀_obj_X_0 | algebra.homology | src/algebra/homology/single.lean | [
"algebra.homology.homology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single₀_obj_X_succ (X : V) (n : ℕ) : ((single₀ V).obj X).X (n+1) = 0 | rfl | lemma | chain_complex.single₀_obj_X_succ | algebra.homology | src/algebra/homology/single.lean | [
"algebra.homology.homology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single₀_obj_X_d (X : V) (i j : ℕ) : ((single₀ V).obj X).d i j = 0 | rfl | lemma | chain_complex.single₀_obj_X_d | algebra.homology | src/algebra/homology/single.lean | [
"algebra.homology.homology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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