[Merged by Bors] - feat: port AlgebraicGeometry.Limits

Mathlib.lean

@@ -417,6 +417,7 @@ import Mathlib.AlgebraicGeometry.AffineScheme
 import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
 import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
 import Mathlib.AlgebraicGeometry.Gluing
+import Mathlib.AlgebraicGeometry.Limits
 import Mathlib.AlgebraicGeometry.LocallyRingedSpace
 import Mathlib.AlgebraicGeometry.LocallyRingedSpace.HasColimits
 import Mathlib.AlgebraicGeometry.OpenImmersion.Basic

Mathlib/AlgebraicGeometry/Limits.lean

@@ -0,0 +1,154 @@
+/-
+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.limits
+! leanprover-community/mathlib commit 70fd9563a21e7b963887c9360bd29b2393e6225a
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.AlgebraicGeometry.Pullbacks
+import Mathlib.AlgebraicGeometry.AffineScheme
+
+/-!
+# (Co)Limits of Schemes
+
+We construct various limits and colimits in the category of schemes.
+
+* The existence of fibred products was shown in `algebraic_geometry/pullbacks.lean`.
+* `Spec ℤ` is the terminal object.
+* The preceding two results imply that `Scheme` has all finite limits.
+* The empty scheme is the (strict) initial object.
+
+## Todo
+
+* Coproducts exists (and the forgetful functors preserve them).
+
+-/
+
+set_option linter.uppercaseLean3 false
+
+universe u
+
+open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
+
+namespace AlgebraicGeometry
+
+/-- `Spec ℤ` is the terminal object in the category of schemes. -/
+noncomputable def specZIsTerminal : IsTerminal (Scheme.Spec.obj (op <| CommRingCat.of ℤ)) :=
+  @IsTerminal.isTerminalObj _ _ _ _ Scheme.Spec _ inferInstance
+    (terminalOpOfInitial CommRingCat.zIsInitial)
+#align algebraic_geometry.Spec_Z_is_terminal AlgebraicGeometry.specZIsTerminal
+
+instance : HasTerminal Scheme :=
+  hasTerminal_of_hasTerminal_of_preservesLimit Scheme.Spec
+
+instance : IsAffine (⊤_ Scheme.{u}) :=
+  isAffineOfIso (PreservesTerminal.iso Scheme.Spec).inv
+
+instance : HasFiniteLimits Scheme :=
+  hasFiniteLimits_of_hasTerminal_and_pullbacks
+
+section Initial
+
+/-- The map from the empty scheme. -/
+@[simps]
+def Scheme.emptyTo (X : Scheme.{u}) : ∅ ⟶ X :=
+  ⟨{  base := ⟨fun x => PEmpty.elim x, by continuity⟩
+      c := { app := fun U => CommRingCat.punitIsTerminal.from _ } }, fun x => PEmpty.elim x⟩
+#align algebraic_geometry.Scheme.empty_to AlgebraicGeometry.Scheme.emptyTo
+
+@[ext]
+theorem Scheme.empty_ext {X : Scheme.{u}} (f g : ∅ ⟶ X) : f = g :=
+  -- Porting note : `ext` regression
+  -- see https://github.com/leanprover-community/mathlib4/issues/5229
+  LocallyRingedSpace.Hom.ext _ _ <| PresheafedSpace.ext _ _ (by ext a; exact PEmpty.elim a) <|
+    NatTrans.ext _ _ <| funext fun a => by aesop_cat
+#align algebraic_geometry.Scheme.empty_ext AlgebraicGeometry.Scheme.empty_ext
+
+theorem Scheme.eq_emptyTo {X : Scheme.{u}} (f : ∅ ⟶ X) : f = Scheme.emptyTo X :=
+  Scheme.empty_ext f (Scheme.emptyTo X)
+#align algebraic_geometry.Scheme.eq_empty_to AlgebraicGeometry.Scheme.eq_emptyTo
+
+instance Scheme.hom_unique_of_empty_source (X : Scheme.{u}) : Unique (∅ ⟶ X) :=
+  ⟨⟨Scheme.emptyTo _⟩, fun _ => Scheme.empty_ext _ _⟩
+
+/-- The empty scheme is the initial object in the category of schemes. -/
+def emptyIsInitial : IsInitial (∅ : Scheme.{u}) :=
+  IsInitial.ofUnique _
+#align algebraic_geometry.empty_is_initial AlgebraicGeometry.emptyIsInitial
+
+@[simp]
+theorem emptyIsInitial_to : emptyIsInitial.to = Scheme.emptyTo :=
+  rfl
+#align algebraic_geometry.empty_is_initial_to AlgebraicGeometry.emptyIsInitial_to
+
+instance : IsEmpty Scheme.empty.carrier :=
+  show IsEmpty PEmpty by infer_instance
+
+instance spec_pUnit_isEmpty : IsEmpty (Scheme.Spec.obj (op <| CommRingCat.of PUnit)).carrier :=
+  ⟨PrimeSpectrum.pUnit⟩
+#align algebraic_geometry.Spec_punit_is_empty AlgebraicGeometry.spec_pUnit_isEmpty
+
+instance (priority := 100) isOpenImmersionCat_of_isEmpty {X Y : Scheme} (f : X ⟶ Y)
+    [IsEmpty X.carrier] : IsOpenImmersion f := by
+  apply (config := { allowSynthFailures := true }) IsOpenImmersion.of_stalk_iso
+  · apply openEmbedding_of_continuous_injective_open
+    -- Porting note : `continuity` failed
+    -- see https://github.com/leanprover-community/mathlib4/issues/5030
+    · exact f.1.base.2
+    · rintro (i : X.carrier); exact isEmptyElim i
+    · intro U _; convert isOpen_empty (α := Y); ext; rw [Set.mem_empty_iff_false, iff_false_iff]
+      exact fun x => isEmptyElim (show X.carrier from x.choose)
+  · rintro (i : X.carrier); exact isEmptyElim i
+#align algebraic_geometry.is_open_immersion_of_is_empty AlgebraicGeometry.isOpenImmersionCat_of_isEmpty
+
+instance (priority := 100) isIso_of_isEmpty {X Y : Scheme} (f : X ⟶ Y) [IsEmpty Y.carrier] :
+    IsIso f := by
+  haveI : IsEmpty X.carrier := ⟨fun x => isEmptyElim (show Y.carrier from f.1.base x)⟩
+  have : Epi f.1.base
+  . rw [TopCat.epi_iff_surjective]; rintro (x : Y.carrier)
+    exact isEmptyElim x
+  apply IsOpenImmersion.to_iso
+#align algebraic_geometry.is_iso_of_is_empty AlgebraicGeometry.isIso_of_isEmpty
+
+/-- A scheme is initial if its underlying space is empty . -/
+noncomputable def isInitialOfIsEmpty {X : Scheme} [IsEmpty X.carrier] : IsInitial X :=
+  emptyIsInitial.ofIso (asIso <| emptyIsInitial.to _)
+#align algebraic_geometry.is_initial_of_is_empty AlgebraicGeometry.isInitialOfIsEmpty
+
+/-- `Spec 0` is the initial object in the category of schemes. -/
+noncomputable def specPunitIsInitial : IsInitial (Scheme.Spec.obj (op <| CommRingCat.of PUnit)) :=
+  emptyIsInitial.ofIso (asIso <| emptyIsInitial.to _)
+#align algebraic_geometry.Spec_punit_is_initial AlgebraicGeometry.specPunitIsInitial
+
+instance (priority := 100) isAffine_of_isEmpty {X : Scheme} [IsEmpty X.carrier] : IsAffine X :=
+  isAffineOfIso
+    (inv (emptyIsInitial.to X) ≫ emptyIsInitial.to (Scheme.Spec.obj (op <| CommRingCat.of PUnit)))
+#align algebraic_geometry.is_affine_of_is_empty AlgebraicGeometry.isAffine_of_isEmpty
+
+instance : HasInitial Scheme :=
+  -- Porting note : this instance was not needed
+  haveI : (Y : Scheme) → Unique (Scheme.empty ⟶ Y) := Scheme.hom_unique_of_empty_source
+  hasInitial_of_unique Scheme.empty
+
+instance initial_isEmpty : IsEmpty (⊥_ Scheme).carrier :=
+  ⟨fun x => ((initial.to Scheme.empty : _).1.base x).elim⟩
+#align algebraic_geometry.initial_is_empty AlgebraicGeometry.initial_isEmpty
+
+theorem bot_isAffineOpen (X : Scheme) : IsAffineOpen (⊥ : Opens X.carrier) := by
+  convert rangeIsAffineOpenOfOpenImmersion (initial.to X)
+  ext
+  -- Porting note : added this `erw` to turn LHS to `False`
+  erw [Set.mem_empty_iff_false]
+  rw [false_iff_iff]
+  exact fun x => isEmptyElim (show (⊥_ Scheme).carrier from x.choose)
+#align algebraic_geometry.bot_is_affine_open AlgebraicGeometry.bot_isAffineOpen
+
+instance : HasStrictInitialObjects Scheme :=
+  hasStrictInitialObjects_of_initial_is_strict fun A f => by infer_instance
+
+end Initial
+
+end AlgebraicGeometry