statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
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pushforward_diagram_to_colimit (F : J ⥤ PresheafedSpace.{v} C) :
J ⥤ (presheaf C (colimit (F ⋙ PresheafedSpace.forget C)))ᵒᵖ | { obj := λ j, op ((colimit.ι (F ⋙ PresheafedSpace.forget C) j) _* (F.obj j).presheaf),
map := λ j j' f,
(pushforward_map (colimit.ι (F ⋙ PresheafedSpace.forget C) j') (F.map f).c ≫
(pushforward.comp (F.obj j).presheaf ((F ⋙ PresheafedSpace.forget C).map f)
(colimit.ι (F ⋙ PresheafedSpace.forget C) j')).in... | def | algebraic_geometry.PresheafedSpace.pushforward_diagram_to_colimit | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/has_colimits.lean | [
"algebraic_geometry.presheafed_space",
"topology.category.Top.limits.basic",
"topology.sheaves.limits"
] | [
"opposite.rec"
] | Given a diagram of presheafed spaces,
we can push all the presheaves forward to the colimit `X` of the underlying topological spaces,
obtaining a diagram in `(presheaf C X)ᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit (F : J ⥤ PresheafedSpace.{v} C) : PresheafedSpace C | { carrier := colimit (F ⋙ PresheafedSpace.forget C),
presheaf := limit (pushforward_diagram_to_colimit F).left_op, } | def | algebraic_geometry.PresheafedSpace.colimit | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/has_colimits.lean | [
"algebraic_geometry.presheafed_space",
"topology.category.Top.limits.basic",
"topology.sheaves.limits"
] | [] | Auxiliary definition for `PresheafedSpace.has_colimits`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_carrier (F : J ⥤ PresheafedSpace.{v} C) :
(colimit F).carrier = limits.colimit (F ⋙ PresheafedSpace.forget C) | rfl | lemma | algebraic_geometry.PresheafedSpace.colimit_carrier | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/has_colimits.lean | [
"algebraic_geometry.presheafed_space",
"topology.category.Top.limits.basic",
"topology.sheaves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_presheaf (F : J ⥤ PresheafedSpace.{v} C) :
(colimit F).presheaf = limit (pushforward_diagram_to_colimit F).left_op | rfl | lemma | algebraic_geometry.PresheafedSpace.colimit_presheaf | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/has_colimits.lean | [
"algebraic_geometry.presheafed_space",
"topology.category.Top.limits.basic",
"topology.sheaves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_cocone (F : J ⥤ PresheafedSpace.{v} C) : cocone F | { X := colimit F,
ι :=
{ app := λ j,
{ base := colimit.ι (F ⋙ PresheafedSpace.forget C) j,
c := limit.π _ (op j), },
naturality' := λ j j' f,
begin
fapply PresheafedSpace.ext,
{ ext x,
exact colimit.w_apply (F ⋙ PresheafedSpace.forget C) f x, },
{ ext U,
induction... | def | algebraic_geometry.PresheafedSpace.colimit_cocone | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/has_colimits.lean | [
"algebraic_geometry.presheafed_space",
"topology.category.Top.limits.basic",
"topology.sheaves.limits"
] | [
"opposite.rec"
] | Auxiliary definition for `PresheafedSpace.has_colimits`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc_c_app (F : J ⥤ PresheafedSpace.{v} C) (s : cocone F) (U : (opens ↥(s.X.carrier))ᵒᵖ) :
s.X.presheaf.obj U ⟶
(colimit.desc (F ⋙ PresheafedSpace.forget C)
((PresheafedSpace.forget C).map_cocone s) _*
limit (pushforward_diagram_to_colimit F).left_op).obj
U | begin
refine
limit.lift _ { X := s.X.presheaf.obj U, π := { app := λ j, _, naturality' := λ j j' f, _, }} ≫
(limit_obj_iso_limit_comp_evaluation _ _).inv,
-- We still need to construct the `app` and `naturality'` fields omitted above.
{ refine (s.ι.app (unop j)).c.app U ≫ (F.obj (unop j)).presheaf.map (... | def | algebraic_geometry.PresheafedSpace.colimit_cocone_is_colimit.desc_c_app | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/has_colimits.lean | [
"algebraic_geometry.presheafed_space",
"topology.category.Top.limits.basic",
"topology.sheaves.limits"
] | [] | Auxiliary definition for `PresheafedSpace.colimit_cocone_is_colimit`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc_c_naturality (F : J ⥤ PresheafedSpace.{v} C) (s : cocone F)
{U V : (opens ↥(s.X.carrier))ᵒᵖ} (i : U ⟶ V) :
s.X.presheaf.map i ≫ desc_c_app F s V =
desc_c_app F s U ≫ (colimit.desc (F ⋙ forget C)
((forget C).map_cocone s) _* (colimit_cocone F).X.presheaf).map i | begin
dsimp [desc_c_app],
ext,
simp only [limit.lift_π, nat_trans.naturality, limit.lift_π_assoc, eq_to_hom_map, assoc,
pushforward_obj_map, nat_trans.naturality_assoc, op_map,
limit_obj_iso_limit_comp_evaluation_inv_π_app_assoc,
limit_obj_iso_limit_comp_evaluation_inv_π_app],
dsimp,
have w := fun... | lemma | algebraic_geometry.PresheafedSpace.colimit_cocone_is_colimit.desc_c_naturality | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/has_colimits.lean | [
"algebraic_geometry.presheafed_space",
"topology.category.Top.limits.basic",
"topology.sheaves.limits"
] | [
"quiver.hom.op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
desc (F : J ⥤ PresheafedSpace.{v} C) (s : cocone F) : colimit F ⟶ s.X | { base := colimit.desc (F ⋙ PresheafedSpace.forget C) ((PresheafedSpace.forget C).map_cocone s),
c :=
{ app := λ U, desc_c_app F s U,
naturality' := λ U V i, desc_c_naturality F s i } } | def | algebraic_geometry.PresheafedSpace.colimit_cocone_is_colimit.desc | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/has_colimits.lean | [
"algebraic_geometry.presheafed_space",
"topology.category.Top.limits.basic",
"topology.sheaves.limits"
] | [] | Auxiliary definition for `PresheafedSpace.colimit_cocone_is_colimit`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc_fac (F : J ⥤ PresheafedSpace.{v} C) (s : cocone F) (j : J) :
(colimit_cocone F).ι.app j ≫ desc F s = s.ι.app j | begin
fapply PresheafedSpace.ext,
{ simp [desc] },
{ ext,
dsimp [desc, desc_c_app],
simpa }
end | lemma | algebraic_geometry.PresheafedSpace.colimit_cocone_is_colimit.desc_fac | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/has_colimits.lean | [
"algebraic_geometry.presheafed_space",
"topology.category.Top.limits.basic",
"topology.sheaves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_cocone_is_colimit (F : J ⥤ PresheafedSpace.{v} C) : is_colimit (colimit_cocone F) | { desc := λ s, desc F s,
fac' := λ s, desc_fac F s,
uniq' := λ s m w,
begin
-- We need to use the identity on the continuous maps twice, so we prepare that first:
have t : m.base = colimit.desc (F ⋙ PresheafedSpace.forget C)
((PresheafedSpace.forget C).map_cocone s),
{ apply ca... | def | algebraic_geometry.PresheafedSpace.colimit_cocone_is_colimit | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/has_colimits.lean | [
"algebraic_geometry.presheafed_space",
"topology.category.Top.limits.basic",
"topology.sheaves.limits"
] | [
"category_theory.limits.colimit.hom_ext",
"continuous_map.ext"
] | Auxiliary definition for `PresheafedSpace.has_colimits`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_preserves_colimits [has_limits C] : preserves_colimits (PresheafedSpace.forget C) | { preserves_colimits_of_shape := λ J 𝒥, by exactI
{ preserves_colimit := λ F, preserves_colimit_of_preserves_colimit_cocone
(colimit_cocone_is_colimit F)
begin
apply is_colimit.of_iso_colimit (colimit.is_colimit _),
fapply cocones.ext,
{ refl, },
{ intro j, dsimp, simp, }
end } } | instance | algebraic_geometry.PresheafedSpace.forget_preserves_colimits | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/has_colimits.lean | [
"algebraic_geometry.presheafed_space",
"topology.category.Top.limits.basic",
"topology.sheaves.limits"
] | [] | The underlying topological space of a colimit of presheaved spaces is
the colimit of the underlying topological spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_presheaf_obj_iso_componentwise_limit (F : J ⥤ PresheafedSpace.{v} C) [has_colimit F]
(U : opens (limits.colimit F).carrier) :
(limits.colimit F).presheaf.obj (op U) ≅ limit (componentwise_diagram F U) | begin
refine ((sheaf_iso_of_iso (colimit.iso_colimit_cocone
⟨_, colimit_cocone_is_colimit F⟩).symm).app (op U)).trans _,
refine (limit_obj_iso_limit_comp_evaluation _ _).trans (limits.lim.map_iso _),
fapply nat_iso.of_components,
{ intro X,
refine ((F.obj (unop X)).presheaf.map_iso (eq_to_iso _)),
s... | def | algebraic_geometry.PresheafedSpace.colimit_presheaf_obj_iso_componentwise_limit | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/has_colimits.lean | [
"algebraic_geometry.presheafed_space",
"topology.category.Top.limits.basic",
"topology.sheaves.limits"
] | [
"Top.presheaf.pushforward.comp_inv_app",
"set.preimage_preimage",
"set_like.ext'_iff"
] | The components of the colimit of a diagram of `PresheafedSpace C` is obtained
via taking componentwise limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_presheaf_obj_iso_componentwise_limit_inv_ι_app (F : J ⥤ PresheafedSpace.{v} C)
(U : opens (limits.colimit F).carrier) (j : J) :
(colimit_presheaf_obj_iso_componentwise_limit F U).inv ≫ (colimit.ι F j).c.app (op U) =
limit.π _ (op j) | begin
delta colimit_presheaf_obj_iso_componentwise_limit,
rw [iso.trans_inv, iso.trans_inv, iso.app_inv, sheaf_iso_of_iso_inv, pushforward_to_of_iso_app,
congr_app (iso.symm_inv _)],
simp_rw category.assoc,
rw [← functor.map_comp_assoc, nat_trans.naturality],
erw ← comp_c_app_assoc,
rw congr_app (colimi... | lemma | algebraic_geometry.PresheafedSpace.colimit_presheaf_obj_iso_componentwise_limit_inv_ι_app | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/has_colimits.lean | [
"algebraic_geometry.presheafed_space",
"topology.category.Top.limits.basic",
"topology.sheaves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_presheaf_obj_iso_componentwise_limit_hom_π (F : J ⥤ PresheafedSpace.{v} C)
(U : opens (limits.colimit F).carrier) (j : J) :
(colimit_presheaf_obj_iso_componentwise_limit F U).hom ≫ limit.π _ (op j) =
(colimit.ι F j).c.app (op U) | by rw [← iso.eq_inv_comp, colimit_presheaf_obj_iso_componentwise_limit_inv_ι_app] | lemma | algebraic_geometry.PresheafedSpace.colimit_presheaf_obj_iso_componentwise_limit_hom_π | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/has_colimits.lean | [
"algebraic_geometry.presheafed_space",
"topology.category.Top.limits.basic",
"topology.sheaves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime_spectrum | (as_ideal : ideal R)
(is_prime : as_ideal.is_prime) | structure | prime_spectrum | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal"
] | The prime spectrum of a commutative ring `R` is the type of all prime ideals of `R`.
It is naturally endowed with a topology (the Zariski topology),
and a sheaf of commutative rings (see `algebraic_geometry.structure_sheaf`).
It is a fundamental building block in algebraic geometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
punit (x : prime_spectrum punit) : false | x.1.ne_top_iff_one.1 x.2.1 $ subsingleton.elim (0 : punit) 1 ▸ x.1.zero_mem | lemma | prime_spectrum.punit | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum"
] | The prime spectrum of the zero ring is empty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prime_spectrum_prod_of_sum :
prime_spectrum R ⊕ prime_spectrum S → prime_spectrum (R × S) | | (sum.inl ⟨I, hI⟩) := ⟨ideal.prod I ⊤, by exactI ideal.is_prime_ideal_prod_top⟩
| (sum.inr ⟨J, hJ⟩) := ⟨ideal.prod ⊤ J, by exactI ideal.is_prime_ideal_prod_top'⟩ | def | prime_spectrum.prime_spectrum_prod_of_sum | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum"
] | The map from the direct sum of prime spectra to the prime spectrum of a direct product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prime_spectrum_prod :
prime_spectrum (R × S) ≃ prime_spectrum R ⊕ prime_spectrum S | equiv.symm $ equiv.of_bijective (prime_spectrum_prod_of_sum R S)
begin
split,
{ rintro (⟨I, hI⟩|⟨J, hJ⟩) (⟨I', hI'⟩|⟨J', hJ'⟩) h;
simp only [ideal.prod.ext_iff, prime_spectrum_prod_of_sum] at h,
{ simp only [h] },
{ exact false.elim (hI.ne_top h.left) },
{ exact false.elim (hJ.ne_top h.right) },
... | def | prime_spectrum.prime_spectrum_prod | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"equiv.of_bijective",
"equiv.symm",
"ideal.ideal_prod_prime",
"ideal.prod.ext_iff",
"prime_spectrum"
] | The prime spectrum of `R × S` is in bijection with the disjoint unions of the prime spectrum of
`R` and the prime spectrum of `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prime_spectrum_prod_symm_inl_as_ideal (x : prime_spectrum R) :
((prime_spectrum_prod R S).symm $ sum.inl x).as_ideal = ideal.prod x.as_ideal ⊤ | by { cases x, refl } | lemma | prime_spectrum.prime_spectrum_prod_symm_inl_as_ideal | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal.prod",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime_spectrum_prod_symm_inr_as_ideal (x : prime_spectrum S) :
((prime_spectrum_prod R S).symm $ sum.inr x).as_ideal = ideal.prod ⊤ x.as_ideal | by { cases x, refl } | lemma | prime_spectrum.prime_spectrum_prod_symm_inr_as_ideal | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal.prod",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus (s : set R) : set (prime_spectrum R) | {x | s ⊆ x.as_ideal} | def | prime_spectrum.zero_locus | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum"
] | The zero locus of a set `s` of elements of a commutative ring `R` is the set of all prime ideals
of the ring that contain the set `s`.
An element `f` of `R` can be thought of as a dependent function on the prime spectrum of `R`.
At a point `x` (a prime ideal) the function (i.e., element) `f` takes values in the quotie... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_zero_locus (x : prime_spectrum R) (s : set R) :
x ∈ zero_locus s ↔ s ⊆ x.as_ideal | iff.rfl | lemma | prime_spectrum.mem_zero_locus | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_span (s : set R) :
zero_locus (ideal.span s : set R) = zero_locus s | by { ext x, exact (submodule.gi R R).gc s x.as_ideal } | lemma | prime_spectrum.zero_locus_span | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal.span",
"submodule.gi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_ideal (t : set (prime_spectrum R)) : ideal R | ⨅ (x : prime_spectrum R) (h : x ∈ t), x.as_ideal | def | prime_spectrum.vanishing_ideal | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal",
"prime_spectrum"
] | The vanishing ideal of a set `t` of points of the prime spectrum of a commutative ring `R` is
the intersection of all the prime ideals in the set `t`.
An element `f` of `R` can be thought of as a dependent function on the prime spectrum of `R`.
At a point `x` (a prime ideal) the function (i.e., element) `f` takes valu... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_vanishing_ideal (t : set (prime_spectrum R)) :
(vanishing_ideal t : set R) = {f : R | ∀ x : prime_spectrum R, x ∈ t → f ∈ x.as_ideal} | begin
ext f,
rw [vanishing_ideal, set_like.mem_coe, submodule.mem_infi],
apply forall_congr, intro x,
rw [submodule.mem_infi],
end | lemma | prime_spectrum.coe_vanishing_ideal | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum",
"set_like.mem_coe",
"submodule.mem_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_vanishing_ideal (t : set (prime_spectrum R)) (f : R) :
f ∈ vanishing_ideal t ↔ ∀ x : prime_spectrum R, x ∈ t → f ∈ x.as_ideal | by rw [← set_like.mem_coe, coe_vanishing_ideal, set.mem_set_of_eq] | lemma | prime_spectrum.mem_vanishing_ideal | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum",
"set_like.mem_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_ideal_singleton (x : prime_spectrum R) :
vanishing_ideal ({x} : set (prime_spectrum R)) = x.as_ideal | by simp [vanishing_ideal] | lemma | prime_spectrum.vanishing_ideal_singleton | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_zero_locus_iff_le_vanishing_ideal (t : set (prime_spectrum R)) (I : ideal R) :
t ⊆ zero_locus I ↔ I ≤ vanishing_ideal t | ⟨λ h f k, (mem_vanishing_ideal _ _).mpr (λ x j, (mem_zero_locus _ _).mpr (h j) k), λ h,
λ x j, (mem_zero_locus _ _).mpr (le_trans h (λ f h, ((mem_vanishing_ideal _ _).mp h) x j))⟩ | lemma | prime_spectrum.subset_zero_locus_iff_le_vanishing_ideal | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gc : @galois_connection (ideal R) (set (prime_spectrum R))ᵒᵈ _ _
(λ I, zero_locus I) (λ t, vanishing_ideal t) | λ I t, subset_zero_locus_iff_le_vanishing_ideal t I | lemma | prime_spectrum.gc | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"galois_connection",
"ideal",
"prime_spectrum"
] | `zero_locus` and `vanishing_ideal` form a galois connection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gc_set : @galois_connection (set R) (set (prime_spectrum R))ᵒᵈ _ _
(λ s, zero_locus s) (λ t, vanishing_ideal t) | have ideal_gc : galois_connection (ideal.span) coe := (submodule.gi R R).gc,
by simpa [zero_locus_span, function.comp] using ideal_gc.compose (gc R) | lemma | prime_spectrum.gc_set | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"galois_connection",
"ideal.span",
"prime_spectrum",
"submodule.gi"
] | `zero_locus` and `vanishing_ideal` form a galois connection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subset_zero_locus_iff_subset_vanishing_ideal (t : set (prime_spectrum R)) (s : set R) :
t ⊆ zero_locus s ↔ s ⊆ vanishing_ideal t | (gc_set R) s t | lemma | prime_spectrum.subset_zero_locus_iff_subset_vanishing_ideal | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_vanishing_ideal_zero_locus (s : set R) :
s ⊆ vanishing_ideal (zero_locus s) | (gc_set R).le_u_l s | lemma | prime_spectrum.subset_vanishing_ideal_zero_locus | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_vanishing_ideal_zero_locus (I : ideal R) :
I ≤ vanishing_ideal (zero_locus I) | (gc R).le_u_l I | lemma | prime_spectrum.le_vanishing_ideal_zero_locus | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_ideal_zero_locus_eq_radical (I : ideal R) :
vanishing_ideal (zero_locus (I : set R)) = I.radical | ideal.ext $ λ f,
begin
rw [mem_vanishing_ideal, ideal.radical_eq_Inf, submodule.mem_Inf],
exact ⟨(λ h x hx, h ⟨x, hx.2⟩ hx.1), (λ h x hx, h x.1 ⟨hx, x.2⟩)⟩
end | lemma | prime_spectrum.vanishing_ideal_zero_locus_eq_radical | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal",
"ideal.ext",
"ideal.radical_eq_Inf",
"submodule.mem_Inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_radical (I : ideal R) : zero_locus (I.radical : set R) = zero_locus I | vanishing_ideal_zero_locus_eq_radical I ▸ (gc R).l_u_l_eq_l I | lemma | prime_spectrum.zero_locus_radical | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_zero_locus_vanishing_ideal (t : set (prime_spectrum R)) :
t ⊆ zero_locus (vanishing_ideal t) | (gc R).l_u_le t | lemma | prime_spectrum.subset_zero_locus_vanishing_ideal | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_anti_mono {s t : set R} (h : s ⊆ t) : zero_locus t ⊆ zero_locus s | (gc_set R).monotone_l h | lemma | prime_spectrum.zero_locus_anti_mono | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_anti_mono_ideal {s t : ideal R} (h : s ≤ t) :
zero_locus (t : set R) ⊆ zero_locus (s : set R) | (gc R).monotone_l h | lemma | prime_spectrum.zero_locus_anti_mono_ideal | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_ideal_anti_mono {s t : set (prime_spectrum R)} (h : s ⊆ t) :
vanishing_ideal t ≤ vanishing_ideal s | (gc R).monotone_u h | lemma | prime_spectrum.vanishing_ideal_anti_mono | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_subset_zero_locus_iff (I J : ideal R) :
zero_locus (I : set R) ⊆ zero_locus (J : set R) ↔ J ≤ I.radical | ⟨λ h, ideal.radical_le_radical_iff.mp (vanishing_ideal_zero_locus_eq_radical I ▸
vanishing_ideal_zero_locus_eq_radical J ▸ vanishing_ideal_anti_mono h),
λ h, zero_locus_radical I ▸ zero_locus_anti_mono_ideal h⟩ | lemma | prime_spectrum.zero_locus_subset_zero_locus_iff | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_subset_zero_locus_singleton_iff (f g : R) :
zero_locus ({f} : set R) ⊆ zero_locus {g} ↔ g ∈ (ideal.span ({f} : set R)).radical | by rw [← zero_locus_span {f}, ← zero_locus_span {g}, zero_locus_subset_zero_locus_iff,
ideal.span_le, set.singleton_subset_iff, set_like.mem_coe] | lemma | prime_spectrum.zero_locus_subset_zero_locus_singleton_iff | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal.span",
"ideal.span_le",
"set.singleton_subset_iff",
"set_like.mem_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_bot :
zero_locus ((⊥ : ideal R) : set R) = set.univ | (gc R).l_bot | lemma | prime_spectrum.zero_locus_bot | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_singleton_zero :
zero_locus ({0} : set R) = set.univ | zero_locus_bot | lemma | prime_spectrum.zero_locus_singleton_zero | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_empty :
zero_locus (∅ : set R) = set.univ | (gc_set R).l_bot | lemma | prime_spectrum.zero_locus_empty | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_ideal_univ :
vanishing_ideal (∅ : set (prime_spectrum R)) = ⊤ | by simpa using (gc R).u_top | lemma | prime_spectrum.vanishing_ideal_univ | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_empty_of_one_mem {s : set R} (h : (1:R) ∈ s) :
zero_locus s = ∅ | begin
rw set.eq_empty_iff_forall_not_mem,
intros x hx,
rw mem_zero_locus at hx,
have x_prime : x.as_ideal.is_prime := by apply_instance,
have eq_top : x.as_ideal = ⊤, { rw ideal.eq_top_iff_one, exact hx h },
apply x_prime.ne_top eq_top,
end | lemma | prime_spectrum.zero_locus_empty_of_one_mem | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal.eq_top_iff_one",
"set.eq_empty_iff_forall_not_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_singleton_one :
zero_locus ({1} : set R) = ∅ | zero_locus_empty_of_one_mem (set.mem_singleton (1 : R)) | lemma | prime_spectrum.zero_locus_singleton_one | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"set.mem_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_empty_iff_eq_top {I : ideal R} :
zero_locus (I : set R) = ∅ ↔ I = ⊤ | begin
split,
{ contrapose!,
intro h,
rcases ideal.exists_le_maximal I h with ⟨M, hM, hIM⟩,
exact set.nonempty.ne_empty ⟨⟨M, hM.is_prime⟩, hIM⟩ },
{ rintro rfl, apply zero_locus_empty_of_one_mem, trivial }
end | lemma | prime_spectrum.zero_locus_empty_iff_eq_top | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal",
"ideal.exists_le_maximal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_univ :
zero_locus (set.univ : set R) = ∅ | zero_locus_empty_of_one_mem (set.mem_univ 1) | lemma | prime_spectrum.zero_locus_univ | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"set.mem_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_ideal_eq_top_iff {s : set (prime_spectrum R)} : vanishing_ideal s = ⊤ ↔ s = ∅ | by rw [← top_le_iff, ← subset_zero_locus_iff_le_vanishing_ideal,
submodule.top_coe, zero_locus_univ, set.subset_empty_iff] | lemma | prime_spectrum.vanishing_ideal_eq_top_iff | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum",
"set.subset_empty_iff",
"submodule.top_coe",
"top_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_sup (I J : ideal R) :
zero_locus ((I ⊔ J : ideal R) : set R) = zero_locus I ∩ zero_locus J | (gc R).l_sup | lemma | prime_spectrum.zero_locus_sup | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_union (s s' : set R) :
zero_locus (s ∪ s') = zero_locus s ∩ zero_locus s' | (gc_set R).l_sup | lemma | prime_spectrum.zero_locus_union | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_ideal_union (t t' : set (prime_spectrum R)) :
vanishing_ideal (t ∪ t') = vanishing_ideal t ⊓ vanishing_ideal t' | (gc R).u_inf | lemma | prime_spectrum.vanishing_ideal_union | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_supr {ι : Sort*} (I : ι → ideal R) :
zero_locus ((⨆ i, I i : ideal R) : set R) = (⋂ i, zero_locus (I i)) | (gc R).l_supr | lemma | prime_spectrum.zero_locus_supr | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_Union {ι : Sort*} (s : ι → set R) :
zero_locus (⋃ i, s i) = (⋂ i, zero_locus (s i)) | (gc_set R).l_supr | lemma | prime_spectrum.zero_locus_Union | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_bUnion (s : set (set R)) :
zero_locus (⋃ s' ∈ s, s' : set R) = ⋂ s' ∈ s, zero_locus s' | by simp only [zero_locus_Union] | lemma | prime_spectrum.zero_locus_bUnion | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_ideal_Union {ι : Sort*} (t : ι → set (prime_spectrum R)) :
vanishing_ideal (⋃ i, t i) = (⨅ i, vanishing_ideal (t i)) | (gc R).u_infi | lemma | prime_spectrum.vanishing_ideal_Union | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_inf (I J : ideal R) :
zero_locus ((I ⊓ J : ideal R) : set R) = zero_locus I ∪ zero_locus J | set.ext $ λ x, x.2.inf_le | lemma | prime_spectrum.zero_locus_inf | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
union_zero_locus (s s' : set R) :
zero_locus s ∪ zero_locus s' = zero_locus ((ideal.span s) ⊓ (ideal.span s') : ideal R) | by { rw zero_locus_inf, simp } | lemma | prime_spectrum.union_zero_locus | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal",
"ideal.span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_mul (I J : ideal R) :
zero_locus ((I * J : ideal R) : set R) = zero_locus I ∪ zero_locus J | set.ext $ λ x, x.2.mul_le | lemma | prime_spectrum.zero_locus_mul | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_singleton_mul (f g : R) :
zero_locus ({f * g} : set R) = zero_locus {f} ∪ zero_locus {g} | set.ext $ λ x, by simpa using x.2.mul_mem_iff_mem_or_mem | lemma | prime_spectrum.zero_locus_singleton_mul | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_pow (I : ideal R) {n : ℕ} (hn : 0 < n) :
zero_locus ((I ^ n : ideal R) : set R) = zero_locus I | zero_locus_radical (I ^ n) ▸ (I.radical_pow n hn).symm ▸ zero_locus_radical I | lemma | prime_spectrum.zero_locus_pow | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_singleton_pow (f : R) (n : ℕ) (hn : 0 < n) :
zero_locus ({f ^ n} : set R) = zero_locus {f} | set.ext $ λ x, by simpa using x.2.pow_mem_iff_mem n hn | lemma | prime_spectrum.zero_locus_singleton_pow | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_vanishing_ideal_le (t t' : set (prime_spectrum R)) :
vanishing_ideal t ⊔ vanishing_ideal t' ≤ vanishing_ideal (t ∩ t') | begin
intros r,
rw [submodule.mem_sup, mem_vanishing_ideal],
rintro ⟨f, hf, g, hg, rfl⟩ x ⟨hxt, hxt'⟩,
rw mem_vanishing_ideal at hf hg,
apply submodule.add_mem; solve_by_elim
end | lemma | prime_spectrum.sup_vanishing_ideal_le | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum",
"submodule.add_mem",
"submodule.mem_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_compl_zero_locus_iff_not_mem {f : R} {I : prime_spectrum R} :
I ∈ (zero_locus {f} : set (prime_spectrum R))ᶜ ↔ f ∉ I.as_ideal | by rw [set.mem_compl_iff, mem_zero_locus, set.singleton_subset_iff]; refl | lemma | prime_spectrum.mem_compl_zero_locus_iff_not_mem | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum",
"set.mem_compl_iff",
"set.singleton_subset_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zariski_topology : topological_space (prime_spectrum R) | topological_space.of_closed (set.range prime_spectrum.zero_locus)
(⟨set.univ, by simp⟩)
begin
intros Zs h,
rw set.sInter_eq_Inter,
choose f hf using λ i : Zs, h i.prop,
simp only [← hf],
exact ⟨_, zero_locus_Union _⟩
end
(by { rintro _ ⟨s, rfl⟩ _ ⟨t, rfl⟩, exact ⟨_, (union_zero_locus s t).sy... | instance | prime_spectrum.zariski_topology | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum",
"prime_spectrum.zero_locus",
"set.range",
"set.sInter_eq_Inter",
"topological_space",
"topological_space.of_closed"
] | The Zariski topology on the prime spectrum of a commutative ring is defined via the closed sets
of the topology: they are exactly those sets that are the zero locus of a subset of the ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_iff (U : set (prime_spectrum R)) :
is_open U ↔ ∃ s, Uᶜ = zero_locus s | by simp only [@eq_comm _ Uᶜ]; refl | lemma | prime_spectrum.is_open_iff | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"is_open",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_iff_zero_locus (Z : set (prime_spectrum R)) :
is_closed Z ↔ ∃ s, Z = zero_locus s | by rw [← is_open_compl_iff, is_open_iff, compl_compl] | lemma | prime_spectrum.is_closed_iff_zero_locus | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"compl_compl",
"is_closed",
"is_open_compl_iff",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_iff_zero_locus_ideal (Z : set (prime_spectrum R)) :
is_closed Z ↔ ∃ (I : ideal R), Z = zero_locus I | (is_closed_iff_zero_locus _).trans
⟨λ ⟨s, hs⟩, ⟨_, (zero_locus_span s).substr hs⟩, λ ⟨I, hI⟩, ⟨I, hI⟩⟩ | lemma | prime_spectrum.is_closed_iff_zero_locus_ideal | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal",
"is_closed",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_iff_zero_locus_radical_ideal (Z : set (prime_spectrum R)) :
is_closed Z ↔ ∃ (I : ideal R), I.is_radical ∧ Z = zero_locus I | (is_closed_iff_zero_locus_ideal _).trans
⟨λ ⟨I, hI⟩, ⟨_, I.radical_is_radical, (zero_locus_radical I).substr hI⟩, λ ⟨I, _, hI⟩, ⟨I, hI⟩⟩ | lemma | prime_spectrum.is_closed_iff_zero_locus_radical_ideal | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal",
"is_closed",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_zero_locus (s : set R) :
is_closed (zero_locus s) | by { rw [is_closed_iff_zero_locus], exact ⟨s, rfl⟩ } | lemma | prime_spectrum.is_closed_zero_locus | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_singleton_iff_is_maximal (x : prime_spectrum R) :
is_closed ({x} : set (prime_spectrum R)) ↔ x.as_ideal.is_maximal | begin
refine (is_closed_iff_zero_locus _).trans ⟨λ h, _, λ h, _⟩,
{ obtain ⟨s, hs⟩ := h,
rw [eq_comm, set.eq_singleton_iff_unique_mem] at hs,
refine ⟨⟨x.2.1, λ I hI, not_not.1 (mt (ideal.exists_le_maximal I) $
not_exists.2 (λ J, not_and.2 $ λ hJ hIJ,_))⟩⟩,
exact ne_of_lt (lt_of_lt_of_le hI hIJ) (s... | lemma | prime_spectrum.is_closed_singleton_iff_is_maximal | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal.exists_le_maximal",
"is_closed",
"prime_spectrum",
"set.eq_singleton_iff_unique_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_locus_vanishing_ideal_eq_closure (t : set (prime_spectrum R)) :
zero_locus (vanishing_ideal t : set R) = closure t | begin
apply set.subset.antisymm,
{ rintro x hx t' ⟨ht', ht⟩,
obtain ⟨fs, rfl⟩ : ∃ s, t' = zero_locus s,
by rwa [is_closed_iff_zero_locus] at ht',
rw [subset_zero_locus_iff_subset_vanishing_ideal] at ht,
exact set.subset.trans ht hx },
{ rw (is_closed_zero_locus _).closure_subset_iff,
exact sub... | lemma | prime_spectrum.zero_locus_vanishing_ideal_eq_closure | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"closure",
"prime_spectrum",
"set.subset.antisymm",
"set.subset.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_ideal_closure (t : set (prime_spectrum R)) :
vanishing_ideal (closure t) = vanishing_ideal t | zero_locus_vanishing_ideal_eq_closure t ▸ (gc R).u_l_u_eq_u t | lemma | prime_spectrum.vanishing_ideal_closure | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"closure",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_singleton (x) : closure ({x} : set (prime_spectrum R)) = zero_locus x.as_ideal | by rw [← zero_locus_vanishing_ideal_eq_closure, vanishing_ideal_singleton] | lemma | prime_spectrum.closure_singleton | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"closure",
"closure_singleton",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_radical_vanishing_ideal (s : set (prime_spectrum R)) :
(vanishing_ideal s).is_radical | by { rw [← vanishing_ideal_closure, ← zero_locus_vanishing_ideal_eq_closure,
vanishing_ideal_zero_locus_eq_radical], apply ideal.radical_is_radical } | lemma | prime_spectrum.is_radical_vanishing_ideal | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal.radical_is_radical",
"is_radical",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_ideal_anti_mono_iff {s t : set (prime_spectrum R)}
(ht : is_closed t) : s ⊆ t ↔ vanishing_ideal t ≤ vanishing_ideal s | ⟨vanishing_ideal_anti_mono, λ h,
begin
rw [← ht.closure_subset_iff, ← ht.closure_eq],
convert ← zero_locus_anti_mono_ideal h;
apply zero_locus_vanishing_ideal_eq_closure,
end⟩ | lemma | prime_spectrum.vanishing_ideal_anti_mono_iff | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"is_closed",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vanishing_ideal_strict_anti_mono_iff {s t : set (prime_spectrum R)}
(hs : is_closed s) (ht : is_closed t) :
s ⊂ t ↔ vanishing_ideal t < vanishing_ideal s | by rw [set.ssubset_def, vanishing_ideal_anti_mono_iff hs,
vanishing_ideal_anti_mono_iff ht, lt_iff_le_not_le] | lemma | prime_spectrum.vanishing_ideal_strict_anti_mono_iff | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"is_closed",
"prime_spectrum",
"set.ssubset_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closeds_embedding (R : Type*) [comm_ring R] :
(topological_space.closeds $ prime_spectrum R)ᵒᵈ ↪o ideal R | order_embedding.of_map_le_iff (λ s, vanishing_ideal s.of_dual)
(λ s t, (vanishing_ideal_anti_mono_iff s.2).symm) | def | prime_spectrum.closeds_embedding | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"comm_ring",
"ideal",
"order_embedding.of_map_le_iff",
"prime_spectrum",
"topological_space.closeds"
] | The antitone order embedding of closed subsets of `Spec R` into ideals of `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
t1_space_iff_is_field [is_domain R] :
t1_space (prime_spectrum R) ↔ is_field R | begin
refine ⟨_, λ h, _⟩,
{ introI h,
have hbot : ideal.is_prime (⊥ : ideal R) := ideal.bot_prime,
exact not_not.1 (mt (ring.ne_bot_of_is_maximal_of_not_is_field $
(is_closed_singleton_iff_is_maximal _).1 (t1_space.t1 ⟨⊥, hbot⟩)) (not_not.2 rfl)) },
{ refine ⟨λ x, (is_closed_singleton_iff_is_maximal... | lemma | prime_spectrum.t1_space_iff_is_field | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal",
"ideal.bot_is_maximal",
"ideal.bot_prime",
"ideal.is_prime",
"is_domain",
"is_field",
"prime_spectrum",
"ring.ne_bot_of_is_maximal_of_not_is_field",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_irreducible_zero_locus_iff_of_radical (I : ideal R) (hI : I.is_radical) :
is_irreducible (zero_locus (I : set R)) ↔ I.is_prime | begin
rw [ideal.is_prime_iff, is_irreducible],
apply and_congr,
{ rw [set.nonempty_iff_ne_empty, ne.def, zero_locus_empty_iff_eq_top] },
{ transitivity ∀ (x y : ideal R), Z(I) ⊆ Z(x) ∪ Z(y) → Z(I) ⊆ Z(x) ∨ Z(I) ⊆ Z(y),
{ simp_rw [is_preirreducible_iff_closed_union_closed, is_closed_iff_zero_locus_ideal],
... | lemma | prime_spectrum.is_irreducible_zero_locus_iff_of_radical | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal",
"ideal.is_prime_iff",
"ideal.mul_mem_left",
"ideal.radical_inf",
"ideal.radical_mul",
"ideal.span_le",
"ideal.span_singleton_mul_span_singleton",
"is_irreducible",
"is_preirreducible_iff_closed_union_closed",
"or_iff_not_imp_left",
"set.nonempty_iff_ne_empty",
"set.singleton_subset_if... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_irreducible_zero_locus_iff (I : ideal R) :
is_irreducible (zero_locus (I : set R)) ↔ I.radical.is_prime | zero_locus_radical I ▸ is_irreducible_zero_locus_iff_of_radical _ I.radical_is_radical | lemma | prime_spectrum.is_irreducible_zero_locus_iff | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal",
"is_irreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_irreducible_iff_vanishing_ideal_is_prime {s : set (prime_spectrum R)} :
is_irreducible s ↔ (vanishing_ideal s).is_prime | by rw [← is_irreducible_iff_closure, ← zero_locus_vanishing_ideal_eq_closure,
is_irreducible_zero_locus_iff_of_radical _ (is_radical_vanishing_ideal s)] | lemma | prime_spectrum.is_irreducible_iff_vanishing_ideal_is_prime | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"is_irreducible",
"is_irreducible_iff_closure",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_comap_zero_locus_aux (f : R →+* S) (s : set R) :
(λ y, ⟨ideal.comap f y.as_ideal, infer_instance⟩ :
prime_spectrum S → prime_spectrum R) ⁻¹' (zero_locus s) = zero_locus (f '' s) | begin
ext x,
simp only [mem_zero_locus, set.image_subset_iff],
refl
end | lemma | prime_spectrum.preimage_comap_zero_locus_aux | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum",
"set.image_subset_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap (f : R →+* S) : C(prime_spectrum S, prime_spectrum R) | { to_fun := λ y, ⟨ideal.comap f y.as_ideal, infer_instance⟩,
continuous_to_fun :=
begin
simp only [continuous_iff_is_closed, is_closed_iff_zero_locus],
rintro _ ⟨s, rfl⟩,
exact ⟨_, preimage_comap_zero_locus_aux f s⟩
end } | def | prime_spectrum.comap | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"continuous_iff_is_closed",
"prime_spectrum"
] | The function between prime spectra of commutative rings induced by a ring homomorphism.
This function is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_as_ideal (y : prime_spectrum S) :
(comap f y).as_ideal = ideal.comap f y.as_ideal | rfl | lemma | prime_spectrum.comap_as_ideal | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal.comap",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_id : comap (ring_hom.id R) = continuous_map.id _ | by { ext, refl } | lemma | prime_spectrum.comap_id | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"continuous_map.id",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_comp (f : R →+* S) (g : S →+* S') :
comap (g.comp f) = (comap f).comp (comap g) | rfl | lemma | prime_spectrum.comap_comp | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_comp_apply (f : R →+* S) (g : S →+* S') (x : prime_spectrum S') :
prime_spectrum.comap (g.comp f) x = (prime_spectrum.comap f) (prime_spectrum.comap g x) | rfl | lemma | prime_spectrum.comap_comp_apply | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"prime_spectrum",
"prime_spectrum.comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_comap_zero_locus (s : set R) :
(comap f) ⁻¹' (zero_locus s) = zero_locus (f '' s) | preimage_comap_zero_locus_aux f s | lemma | prime_spectrum.preimage_comap_zero_locus | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_injective_of_surjective (f : R →+* S) (hf : function.surjective f) :
function.injective (comap f) | λ x y h, prime_spectrum.ext _ _ (ideal.comap_injective_of_surjective f hf
(congr_arg prime_spectrum.as_ideal h : (comap f x).as_ideal = (comap f y).as_ideal)) | lemma | prime_spectrum.comap_injective_of_surjective | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal.comap_injective_of_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_singleton_is_closed_of_surjective (f : R →+* S) (hf : function.surjective f)
(x : prime_spectrum S) (hx : is_closed ({x} : set (prime_spectrum S))) :
is_closed ({comap f x} : set (prime_spectrum R)) | begin
haveI : x.as_ideal.is_maximal := (is_closed_singleton_iff_is_maximal x).1 hx,
exact (is_closed_singleton_iff_is_maximal _).2 (ideal.comap_is_maximal_of_surjective f hf)
end | lemma | prime_spectrum.comap_singleton_is_closed_of_surjective | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal.comap_is_maximal_of_surjective",
"is_closed",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_singleton_is_closed_of_is_integral (f : R →+* S) (hf : f.is_integral)
(x : prime_spectrum S) (hx : is_closed ({x} : set (prime_spectrum S))) :
is_closed ({comap f x} : set (prime_spectrum R)) | (is_closed_singleton_iff_is_maximal _).2 (ideal.is_maximal_comap_of_is_integral_of_is_maximal'
f hf x.as_ideal $ (is_closed_singleton_iff_is_maximal x).1 hx) | lemma | prime_spectrum.comap_singleton_is_closed_of_is_integral | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"ideal.is_maximal_comap_of_is_integral_of_is_maximal'",
"is_closed",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_comap_inducing [algebra R S] (M : submonoid R)
[is_localization M S] : inducing (comap (algebra_map R S)) | begin
constructor,
rw topological_space_eq_iff,
intro U,
simp_rw ← is_closed_compl_iff,
generalize : Uᶜ = Z,
simp_rw [is_closed_induced_iff, is_closed_iff_zero_locus],
split,
{ rintro ⟨s, rfl⟩,
refine ⟨_,⟨(algebra_map R S) ⁻¹' (ideal.span s),rfl⟩,_⟩,
rw [preimage_comap_zero_locus, ← zero_locus_s... | lemma | prime_spectrum.localization_comap_inducing | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"algebra",
"algebra_map",
"ideal.span",
"inducing",
"is_closed_compl_iff",
"is_closed_induced_iff",
"is_localization",
"is_localization.map_comap",
"submonoid",
"topological_space_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_comap_injective [algebra R S] (M : submonoid R)
[is_localization M S] : function.injective (comap (algebra_map R S)) | begin
intros p q h,
replace h := congr_arg (λ (x : prime_spectrum R), ideal.map (algebra_map R S) x.as_ideal) h,
dsimp only at h,
erw [is_localization.map_comap M S, is_localization.map_comap M S] at h,
ext1,
exact h
end | lemma | prime_spectrum.localization_comap_injective | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"algebra",
"algebra_map",
"ideal.map",
"is_localization",
"is_localization.map_comap",
"prime_spectrum",
"submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_comap_embedding [algebra R S] (M : submonoid R)
[is_localization M S] : embedding (comap (algebra_map R S)) | ⟨localization_comap_inducing S M, localization_comap_injective S M⟩ | lemma | prime_spectrum.localization_comap_embedding | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"algebra",
"algebra_map",
"embedding",
"is_localization",
"submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_comap_range [algebra R S] (M : submonoid R)
[is_localization M S] :
set.range (comap (algebra_map R S)) = { p | disjoint (M : set R) p.as_ideal } | begin
ext x,
split,
{ simp_rw disjoint_iff_inf_le,
rintro ⟨p, rfl⟩ x ⟨hx₁, hx₂⟩,
exact (p.2.1 : ¬ _)
(p.as_ideal.eq_top_of_is_unit_mem hx₂ (is_localization.map_units S ⟨x, hx₁⟩)) },
{ intro h,
use ⟨x.as_ideal.map (algebra_map R S),
is_localization.is_prime_of_is_prime_disjoint M S _ x.2 ... | lemma | prime_spectrum.localization_comap_range | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"algebra",
"algebra_map",
"disjoint",
"disjoint_iff_inf_le",
"is_localization",
"is_localization.comap_map_of_is_prime_disjoint",
"is_localization.is_prime_of_is_prime_disjoint",
"set.range",
"submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_inducing_of_surjective (hf : surjective f) : inducing (comap f) | { induced := begin
simp_rw [topological_space_eq_iff, ←is_closed_compl_iff, is_closed_induced_iff,
is_closed_iff_zero_locus],
refine λ s, ⟨λ ⟨F, hF⟩, ⟨zero_locus (f ⁻¹' F), ⟨f ⁻¹' F, rfl⟩,
by rw [preimage_comap_zero_locus, surjective.image_preimage hf, hF]⟩, _⟩,
rintros ⟨-, ⟨F, rfl⟩, hF⟩,
ex... | lemma | prime_spectrum.comap_inducing_of_surjective | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"inducing",
"is_closed_induced_iff",
"topological_space_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_comap_zero_locus_eq_zero_locus_comap (hf : surjective f) (I : ideal S) :
comap f '' zero_locus I = zero_locus (I.comap f) | begin
simp only [set.ext_iff, set.mem_image, mem_zero_locus, set_like.coe_subset_coe],
refine λ p, ⟨_, λ h_I_p, _⟩,
{ rintro ⟨p, hp, rfl⟩ a ha,
exact hp ha },
{ have hp : ker f ≤ p.as_ideal := (ideal.comap_mono bot_le).trans h_I_p,
refine ⟨⟨p.as_ideal.map f, ideal.map_is_prime_of_surjective hf hp⟩, λ x ... | lemma | prime_spectrum.image_comap_zero_locus_eq_zero_locus_comap | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"bot_le",
"ideal",
"ideal.comap_mono",
"ideal.map_is_prime_of_surjective",
"ideal.mem_map_iff_of_surjective",
"ideal.mem_map_of_mem",
"set.ext_iff",
"set.mem_image",
"set_like.coe_subset_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_comap_of_surjective (hf : surjective f) :
set.range (comap f) = zero_locus (ker f) | begin
rw ← set.image_univ,
convert image_comap_zero_locus_eq_zero_locus_comap _ _ hf _,
rw zero_locus_bot,
end | lemma | prime_spectrum.range_comap_of_surjective | algebraic_geometry.prime_spectrum | src/algebraic_geometry/prime_spectrum/basic.lean | [
"algebra.punit_instances",
"linear_algebra.finsupp",
"ring_theory.ideal.over",
"ring_theory.ideal.prod",
"ring_theory.localization.away.basic",
"ring_theory.nilpotent",
"topology.sets.closeds",
"topology.sober"
] | [
"set.image_univ",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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