algebraic_geometry.morphisms.universally_closed
⟷
Mathlib.AlgebraicGeometry.Morphisms.UniversallyClosed
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
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mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -111,7 +111,7 @@ theorem universallyClosed_is_local_at_target : PropertyIsLocalAtTarget @Universa
stable_under_composition.respects_iso (fun X Y Z f g hf hg => IsClosedMap.comp hg hf)
fun X Y f => (TopCat.homeoOfIso (Scheme.forget_to_Top.map_iso f)).IsClosedMap
· intro X Y f ι U hU H
- simp_rw [topologically, morphism_restrict_base] at H
+ simp_rw [topologically, morphism_restrict_base] at H
exact (isClosedMap_iff_isClosedMap_of_iSup_eq_top hU).mpr H
#align algebraic_geometry.universally_closed_is_local_at_target AlgebraicGeometry.universallyClosed_is_local_at_target
-/
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,8 +3,8 @@ Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
-import Mathbin.AlgebraicGeometry.Morphisms.Basic
-import Mathbin.Topology.LocalAtTarget
+import AlgebraicGeometry.Morphisms.Basic
+import Topology.LocalAtTarget
#align_import algebraic_geometry.morphisms.universally_closed from "leanprover-community/mathlib"@"728ef9dbb281241906f25cbeb30f90d83e0bb451"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,15 +2,12 @@
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-
-! This file was ported from Lean 3 source module algebraic_geometry.morphisms.universally_closed
-! leanprover-community/mathlib commit 728ef9dbb281241906f25cbeb30f90d83e0bb451
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.AlgebraicGeometry.Morphisms.Basic
import Mathbin.Topology.LocalAtTarget
+#align_import algebraic_geometry.morphisms.universally_closed from "leanprover-community/mathlib"@"728ef9dbb281241906f25cbeb30f90d83e0bb451"
+
/-!
# Universally closed morphism
mathlib commit https://github.com/leanprover-community/mathlib/commit/728ef9dbb281241906f25cbeb30f90d83e0bb451
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
! This file was ported from Lean 3 source module algebraic_geometry.morphisms.universally_closed
-! leanprover-community/mathlib commit a8ae1b3f7979249a0af6bc7cf20c1f6bf656ca73
+! leanprover-community/mathlib commit 728ef9dbb281241906f25cbeb30f90d83e0bb451
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -14,6 +14,9 @@ import Mathbin.Topology.LocalAtTarget
/-!
# Universally closed morphism
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
A morphism of schemes `f : X ⟶ Y` is universally closed if `X ×[Y] Y' ⟶ Y'` is a closed map
for all base change `Y' ⟶ Y`.
mathlib commit https://github.com/leanprover-community/mathlib/commit/9240e8be927a0955b9a82c6c85ef499ee3a626b8
@@ -37,6 +37,7 @@ open CategoryTheory.MorphismProperty
open AlgebraicGeometry.MorphismProperty (topologically)
+#print AlgebraicGeometry.UniversallyClosed /-
/-- A morphism of schemes `f : X ⟶ Y` is universally closed if the base change `X ×[Y] Y' ⟶ Y'`
along any morphism `Y' ⟶ Y` is (topologically) a closed map.
-/
@@ -44,45 +45,63 @@ along any morphism `Y' ⟶ Y` is (topologically) a closed map.
class UniversallyClosed (f : X ⟶ Y) : Prop where
out : universally (topologically @IsClosedMap) f
#align algebraic_geometry.universally_closed AlgebraicGeometry.UniversallyClosed
+-/
+#print AlgebraicGeometry.universallyClosed_eq /-
theorem universallyClosed_eq : @UniversallyClosed = universally (topologically @IsClosedMap) := by
ext X Y f; rw [universally_closed_iff]
#align algebraic_geometry.universally_closed_eq AlgebraicGeometry.universallyClosed_eq
+-/
+#print AlgebraicGeometry.universallyClosed_respectsIso /-
theorem universallyClosed_respectsIso : RespectsIso @UniversallyClosed :=
universallyClosed_eq.symm ▸ universally_respectsIso (topologically @IsClosedMap)
#align algebraic_geometry.universally_closed_respects_iso AlgebraicGeometry.universallyClosed_respectsIso
+-/
+#print AlgebraicGeometry.universallyClosed_stableUnderBaseChange /-
theorem universallyClosed_stableUnderBaseChange : StableUnderBaseChange @UniversallyClosed :=
universallyClosed_eq.symm ▸ universally_stableUnderBaseChange (topologically @IsClosedMap)
#align algebraic_geometry.universally_closed_stable_under_base_change AlgebraicGeometry.universallyClosed_stableUnderBaseChange
+-/
+#print AlgebraicGeometry.universallyClosed_stableUnderComposition /-
theorem universallyClosed_stableUnderComposition : StableUnderComposition @UniversallyClosed :=
by
rw [universally_closed_eq]
exact stable_under_composition.universally fun X Y Z f g hf hg => IsClosedMap.comp hg hf
#align algebraic_geometry.universally_closed_stable_under_composition AlgebraicGeometry.universallyClosed_stableUnderComposition
+-/
+#print AlgebraicGeometry.universallyClosedTypeComp /-
instance universallyClosedTypeComp {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z)
[hf : UniversallyClosed f] [hg : UniversallyClosed g] : UniversallyClosed (f ≫ g) :=
universallyClosed_stableUnderComposition f g hf hg
#align algebraic_geometry.universally_closed_type_comp AlgebraicGeometry.universallyClosedTypeComp
+-/
+#print AlgebraicGeometry.universallyClosedFst /-
instance universallyClosedFst {X Y Z : Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [hg : UniversallyClosed g] :
UniversallyClosed (pullback.fst : pullback f g ⟶ _) :=
universallyClosed_stableUnderBaseChange.fst f g hg
#align algebraic_geometry.universally_closed_fst AlgebraicGeometry.universallyClosedFst
+-/
+#print AlgebraicGeometry.universallyClosedSnd /-
instance universallyClosedSnd {X Y Z : Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [hf : UniversallyClosed f] :
UniversallyClosed (pullback.snd : pullback f g ⟶ _) :=
universallyClosed_stableUnderBaseChange.snd f g hf
#align algebraic_geometry.universally_closed_snd AlgebraicGeometry.universallyClosedSnd
+-/
+#print AlgebraicGeometry.morphismRestrict_base /-
theorem morphismRestrict_base {X Y : Scheme} (f : X ⟶ Y) (U : Opens Y.carrier) :
⇑(f ∣_ U).1.base = U.1.restrictPreimage f.1 :=
funext fun x => Subtype.ext <| morphismRestrict_base_coe f U x
#align algebraic_geometry.morphism_restrict_base AlgebraicGeometry.morphismRestrict_base
+-/
+#print AlgebraicGeometry.universallyClosed_is_local_at_target /-
theorem universallyClosed_is_local_at_target : PropertyIsLocalAtTarget @UniversallyClosed :=
by
rw [universally_closed_eq]
@@ -95,12 +114,15 @@ theorem universallyClosed_is_local_at_target : PropertyIsLocalAtTarget @Universa
simp_rw [topologically, morphism_restrict_base] at H
exact (isClosedMap_iff_isClosedMap_of_iSup_eq_top hU).mpr H
#align algebraic_geometry.universally_closed_is_local_at_target AlgebraicGeometry.universallyClosed_is_local_at_target
+-/
+#print AlgebraicGeometry.UniversallyClosed.openCover_iff /-
theorem UniversallyClosed.openCover_iff {X Y : Scheme.{u}} (f : X ⟶ Y)
(𝒰 : Scheme.OpenCover.{u} Y) :
UniversallyClosed f ↔ ∀ i, UniversallyClosed (pullback.snd : pullback f (𝒰.map i) ⟶ _) :=
universallyClosed_is_local_at_target.openCover_iff f 𝒰
#align algebraic_geometry.universally_closed.open_cover_iff AlgebraicGeometry.UniversallyClosed.openCover_iff
+-/
end AlgebraicGeometry
mathlib commit https://github.com/leanprover-community/mathlib/commit/9240e8be927a0955b9a82c6c85ef499ee3a626b8
@@ -63,27 +63,27 @@ theorem universallyClosed_stableUnderComposition : StableUnderComposition @Unive
exact stable_under_composition.universally fun X Y Z f g hf hg => IsClosedMap.comp hg hf
#align algebraic_geometry.universally_closed_stable_under_composition AlgebraicGeometry.universallyClosed_stableUnderComposition
-instance universallyClosed_type_comp {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z)
+instance universallyClosedTypeComp {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z)
[hf : UniversallyClosed f] [hg : UniversallyClosed g] : UniversallyClosed (f ≫ g) :=
universallyClosed_stableUnderComposition f g hf hg
-#align algebraic_geometry.universally_closed_type_comp AlgebraicGeometry.universallyClosed_type_comp
+#align algebraic_geometry.universally_closed_type_comp AlgebraicGeometry.universallyClosedTypeComp
-instance universallyClosed_fst {X Y Z : Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [hg : UniversallyClosed g] :
+instance universallyClosedFst {X Y Z : Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [hg : UniversallyClosed g] :
UniversallyClosed (pullback.fst : pullback f g ⟶ _) :=
universallyClosed_stableUnderBaseChange.fst f g hg
-#align algebraic_geometry.universally_closed_fst AlgebraicGeometry.universallyClosed_fst
+#align algebraic_geometry.universally_closed_fst AlgebraicGeometry.universallyClosedFst
-instance universallyClosed_snd {X Y Z : Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [hf : UniversallyClosed f] :
+instance universallyClosedSnd {X Y Z : Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [hf : UniversallyClosed f] :
UniversallyClosed (pullback.snd : pullback f g ⟶ _) :=
universallyClosed_stableUnderBaseChange.snd f g hf
-#align algebraic_geometry.universally_closed_snd AlgebraicGeometry.universallyClosed_snd
+#align algebraic_geometry.universally_closed_snd AlgebraicGeometry.universallyClosedSnd
theorem morphismRestrict_base {X Y : Scheme} (f : X ⟶ Y) (U : Opens Y.carrier) :
⇑(f ∣_ U).1.base = U.1.restrictPreimage f.1 :=
funext fun x => Subtype.ext <| morphismRestrict_base_coe f U x
#align algebraic_geometry.morphism_restrict_base AlgebraicGeometry.morphismRestrict_base
-theorem universallyClosedIsLocalAtTarget : PropertyIsLocalAtTarget @UniversallyClosed :=
+theorem universallyClosed_is_local_at_target : PropertyIsLocalAtTarget @UniversallyClosed :=
by
rw [universally_closed_eq]
apply universally_is_local_at_target_of_morphism_restrict
@@ -94,12 +94,12 @@ theorem universallyClosedIsLocalAtTarget : PropertyIsLocalAtTarget @UniversallyC
· intro X Y f ι U hU H
simp_rw [topologically, morphism_restrict_base] at H
exact (isClosedMap_iff_isClosedMap_of_iSup_eq_top hU).mpr H
-#align algebraic_geometry.universally_closed_is_local_at_target AlgebraicGeometry.universallyClosedIsLocalAtTarget
+#align algebraic_geometry.universally_closed_is_local_at_target AlgebraicGeometry.universallyClosed_is_local_at_target
theorem UniversallyClosed.openCover_iff {X Y : Scheme.{u}} (f : X ⟶ Y)
(𝒰 : Scheme.OpenCover.{u} Y) :
UniversallyClosed f ↔ ∀ i, UniversallyClosed (pullback.snd : pullback f (𝒰.map i) ⟶ _) :=
- universallyClosedIsLocalAtTarget.openCover_iff f 𝒰
+ universallyClosed_is_local_at_target.openCover_iff f 𝒰
#align algebraic_geometry.universally_closed.open_cover_iff AlgebraicGeometry.UniversallyClosed.openCover_iff
end AlgebraicGeometry
mathlib commit https://github.com/leanprover-community/mathlib/commit/2a0ce625dbb0ffbc7d1316597de0b25c1ec75303
@@ -46,7 +46,7 @@ class UniversallyClosed (f : X ⟶ Y) : Prop where
#align algebraic_geometry.universally_closed AlgebraicGeometry.UniversallyClosed
theorem universallyClosed_eq : @UniversallyClosed = universally (topologically @IsClosedMap) := by
- ext (X Y f); rw [universally_closed_iff]
+ ext X Y f; rw [universally_closed_iff]
#align algebraic_geometry.universally_closed_eq AlgebraicGeometry.universallyClosed_eq
theorem universallyClosed_respectsIso : RespectsIso @UniversallyClosed :=
mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb
@@ -92,7 +92,7 @@ theorem universallyClosedIsLocalAtTarget : PropertyIsLocalAtTarget @UniversallyC
stable_under_composition.respects_iso (fun X Y Z f g hf hg => IsClosedMap.comp hg hf)
fun X Y f => (TopCat.homeoOfIso (Scheme.forget_to_Top.map_iso f)).IsClosedMap
· intro X Y f ι U hU H
- simp_rw [topologically, morphism_restrict_base] at H
+ simp_rw [topologically, morphism_restrict_base] at H
exact (isClosedMap_iff_isClosedMap_of_iSup_eq_top hU).mpr H
#align algebraic_geometry.universally_closed_is_local_at_target AlgebraicGeometry.universallyClosedIsLocalAtTarget
mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee
@@ -93,7 +93,7 @@ theorem universallyClosedIsLocalAtTarget : PropertyIsLocalAtTarget @UniversallyC
fun X Y f => (TopCat.homeoOfIso (Scheme.forget_to_Top.map_iso f)).IsClosedMap
· intro X Y f ι U hU H
simp_rw [topologically, morphism_restrict_base] at H
- exact (isClosedMap_iff_isClosedMap_of_supᵢ_eq_top hU).mpr H
+ exact (isClosedMap_iff_isClosedMap_of_iSup_eq_top hU).mpr H
#align algebraic_geometry.universally_closed_is_local_at_target AlgebraicGeometry.universallyClosedIsLocalAtTarget
theorem UniversallyClosed.openCover_iff {X Y : Scheme.{u}} (f : X ⟶ Y)
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
The type class MorphismProperty.HasTwoOutOfThreeProperty
is introduced. The structure StableUnderComposition
is also changed into a type class IsStableUnderComposition
.
Co-authored-by: Joël Riou <37772949+joelriou@users.noreply.github.com>
@@ -54,17 +54,27 @@ theorem universallyClosed_stableUnderBaseChange : StableUnderBaseChange @Univers
universallyClosed_eq.symm ▸ universally_stableUnderBaseChange (topologically @IsClosedMap)
#align algebraic_geometry.universally_closed_stable_under_base_change AlgebraicGeometry.universallyClosed_stableUnderBaseChange
-theorem universallyClosed_stableUnderComposition : StableUnderComposition @UniversallyClosed := by
+instance isClosedMap_isStableUnderComposition :
+ IsStableUnderComposition (topologically @IsClosedMap) where
+ comp_mem f g hf hg := IsClosedMap.comp (f := f.1.base) (g := g.1.base) hg hf
+
+instance universallyClosed_isStableUnderComposition :
+ IsStableUnderComposition @UniversallyClosed := by
rw [universallyClosed_eq]
- exact StableUnderComposition.universally (fun X Y Z f g hf hg =>
- IsClosedMap.comp (f := f.1.base) (g := g.1.base) hg hf)
-#align algebraic_geometry.universally_closed_stable_under_composition AlgebraicGeometry.universallyClosed_stableUnderComposition
+ infer_instance
+#align algebraic_geometry.universally_closed_stable_under_composition AlgebraicGeometry.universallyClosed_isStableUnderComposition
instance universallyClosedTypeComp {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z)
[hf : UniversallyClosed f] [hg : UniversallyClosed g] : UniversallyClosed (f ≫ g) :=
- universallyClosed_stableUnderComposition f g hf hg
+ comp_mem _ _ _ hf hg
#align algebraic_geometry.universally_closed_type_comp AlgebraicGeometry.universallyClosedTypeComp
+theorem topologically_isClosedMap_respectsIso : RespectsIso (topologically @IsClosedMap) := by
+ apply MorphismProperty.respectsIso_of_isStableUnderComposition
+ intro _ _ f hf
+ have : IsIso f := hf
+ exact (TopCat.homeoOfIso (Scheme.forgetToTop.mapIso (asIso f))).isClosedMap
+
instance universallyClosedFst {X Y Z : Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [hg : UniversallyClosed g] :
UniversallyClosed (pullback.fst : pullback f g ⟶ _) :=
universallyClosed_stableUnderBaseChange.fst f g hg
@@ -83,9 +93,7 @@ theorem morphismRestrict_base {X Y : Scheme} (f : X ⟶ Y) (U : Opens Y.carrier)
theorem universallyClosed_is_local_at_target : PropertyIsLocalAtTarget @UniversallyClosed := by
rw [universallyClosed_eq]
apply universallyIsLocalAtTargetOfMorphismRestrict
- · exact StableUnderComposition.respectsIso (fun X Y Z f g hf hg =>
- IsClosedMap.comp (f := f.1.base) (g := g.1.base) hg hf)
- (fun f => (TopCat.homeoOfIso (Scheme.forgetToTop.mapIso f)).isClosedMap)
+ · exact topologically_isClosedMap_respectsIso
· intro X Y f ι U hU H
simp_rw [topologically, morphismRestrict_base] at H
exact (isClosedMap_iff_isClosedMap_of_iSup_eq_top hU).mpr H
TopologicalSpace
assumptions.Co-authored-by: grunweg <grunweg@posteo.de>
@@ -57,7 +57,7 @@ theorem universallyClosed_stableUnderBaseChange : StableUnderBaseChange @Univers
theorem universallyClosed_stableUnderComposition : StableUnderComposition @UniversallyClosed := by
rw [universallyClosed_eq]
exact StableUnderComposition.universally (fun X Y Z f g hf hg =>
- @IsClosedMap.comp _ _ _ _ _ _ g.1.base f.1.base hg hf)
+ IsClosedMap.comp (f := f.1.base) (g := g.1.base) hg hf)
#align algebraic_geometry.universally_closed_stable_under_composition AlgebraicGeometry.universallyClosed_stableUnderComposition
instance universallyClosedTypeComp {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z)
@@ -84,7 +84,7 @@ theorem universallyClosed_is_local_at_target : PropertyIsLocalAtTarget @Universa
rw [universallyClosed_eq]
apply universallyIsLocalAtTargetOfMorphismRestrict
· exact StableUnderComposition.respectsIso (fun X Y Z f g hf hg =>
- @IsClosedMap.comp _ _ _ _ _ _ g.1.base f.1.base hg hf)
+ IsClosedMap.comp (f := f.1.base) (g := g.1.base) hg hf)
(fun f => (TopCat.homeoOfIso (Scheme.forgetToTop.mapIso f)).isClosedMap)
· intro X Y f ι U hU H
simp_rw [topologically, morphismRestrict_base] at H
@@ -43,7 +43,7 @@ class UniversallyClosed (f : X ⟶ Y) : Prop where
#align algebraic_geometry.universally_closed AlgebraicGeometry.UniversallyClosed
theorem universallyClosed_eq : @UniversallyClosed = universally (topologically @IsClosedMap) := by
- ext X Y f; rw [UniversallyClosed_iff]
+ ext X Y f; rw [universallyClosed_iff]
#align algebraic_geometry.universally_closed_eq AlgebraicGeometry.universallyClosed_eq
theorem universallyClosed_respectsIso : RespectsIso @UniversallyClosed :=
@@ -2,15 +2,12 @@
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-
-! This file was ported from Lean 3 source module algebraic_geometry.morphisms.universally_closed
-! leanprover-community/mathlib commit a8ae1b3f7979249a0af6bc7cf20c1f6bf656ca73
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.Topology.LocalAtTarget
+#align_import algebraic_geometry.morphisms.universally_closed from "leanprover-community/mathlib"@"a8ae1b3f7979249a0af6bc7cf20c1f6bf656ca73"
+
/-!
# Universally closed morphism
The unported dependencies are