Spec.lean

/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import algebraic_geometry.locally_ringed_space
import algebraic_geometry.structure_sheaf
import logic.equiv.transfer_instance
import ring_theory.localization.localization_localization
import topology.sheaves.sheaf_condition.sites
import topology.sheaves.functors

/-!
# $Spec$ as a functor to locally ringed spaces.

We define the functor $Spec$ from commutative rings to locally ringed spaces.

## Implementation notes

We define $Spec$ in three consecutive steps, each with more structure than the last:

1. `Spec.to_Top`, valued in the category of topological spaces,
2. `Spec.to_SheafedSpace`, valued in the category of sheafed spaces and
3. `Spec.to_LocallyRingedSpace`, valued in the category of locally ringed spaces.

Additionally, we provide `Spec.to_PresheafedSpace` as a composition of `Spec.to_SheafedSpace` with
a forgetful functor.

## Related results

The adjunction `Γ ⊣ Spec` is constructed in `algebraic_geometry/Gamma_Spec_adjunction.lean`.

-/

noncomputable theory
universes u v

namespace algebraic_geometry
open opposite
open category_theory
open structure_sheaf Spec (structure_sheaf)

/--
The spectrum of a commutative ring, as a topological space.
-/
def Spec.Top_obj (R : CommRing) : Top := Top.of (prime_spectrum R)

/--
The induced map of a ring homomorphism on the ring spectra, as a morphism of topological spaces.
-/
def Spec.Top_map {R S : CommRing} (f : R ⟶ S) :
  Spec.Top_obj S ⟶ Spec.Top_obj R :=
prime_spectrum.comap f

@[simp] lemma Spec.Top_map_id (R : CommRing) :
  Spec.Top_map (𝟙 R) = 𝟙 (Spec.Top_obj R) :=
prime_spectrum.comap_id

lemma Spec.Top_map_comp {R S T : CommRing} (f : R ⟶ S) (g : S ⟶ T) :
  Spec.Top_map (f ≫ g) = Spec.Top_map g ≫ Spec.Top_map f :=
prime_spectrum.comap_comp _ _

/--
The spectrum, as a contravariant functor from commutative rings to topological spaces.
-/
@[simps] def Spec.to_Top : CommRingᵒᵖ ⥤ Top :=
{ obj := λ R, Spec.Top_obj (unop R),
  map := λ R S f, Spec.Top_map f.unop,
  map_id' := λ R, by rw [unop_id, Spec.Top_map_id],
  map_comp' := λ R S T f g, by rw [unop_comp, Spec.Top_map_comp] }

/--
The spectrum of a commutative ring, as a `SheafedSpace`.
-/
@[simps] def Spec.SheafedSpace_obj (R : CommRing) : SheafedSpace CommRing :=
{ carrier := Spec.Top_obj R,
  presheaf := (structure_sheaf R).1,
  is_sheaf := (structure_sheaf R).2 }

/--
The induced map of a ring homomorphism on the ring spectra, as a morphism of sheafed spaces.
-/
@[simps] def Spec.SheafedSpace_map {R S : CommRing.{u}} (f : R ⟶ S) :
  Spec.SheafedSpace_obj S ⟶ Spec.SheafedSpace_obj R :=
{ base := Spec.Top_map f,
  c :=
  { app := λ U, comap f (unop U) ((topological_space.opens.map (Spec.Top_map f)).obj (unop U))
      (λ p, id),
    naturality' := λ U V i, ring_hom.ext $ λ s, subtype.eq $ funext $ λ p, rfl } }

@[simp] lemma Spec.SheafedSpace_map_id {R : CommRing} :
  Spec.SheafedSpace_map (𝟙 R) = 𝟙 (Spec.SheafedSpace_obj R) :=
PresheafedSpace.ext _ _ (Spec.Top_map_id R) $ nat_trans.ext _ _ $ funext $ λ U,
begin
  dsimp,
  erw [PresheafedSpace.id_c_app, comap_id], swap,
  { rw [Spec.Top_map_id, topological_space.opens.map_id_obj_unop] },
  simpa,
end

lemma Spec.SheafedSpace_map_comp {R S T : CommRing} (f : R ⟶ S) (g : S ⟶ T) :
  Spec.SheafedSpace_map (f ≫ g) = Spec.SheafedSpace_map g ≫ Spec.SheafedSpace_map f :=
PresheafedSpace.ext _ _ (Spec.Top_map_comp f g) $ nat_trans.ext _ _ $ funext $ λ U,
by { dsimp, rw category_theory.functor.map_id, rw category.comp_id, erw comap_comp f g, refl }

/--
Spec, as a contravariant functor from commutative rings to sheafed spaces.
-/
@[simps] def Spec.to_SheafedSpace : CommRingᵒᵖ ⥤ SheafedSpace CommRing :=
{ obj := λ R, Spec.SheafedSpace_obj (unop R),
  map := λ R S f, Spec.SheafedSpace_map f.unop,
  map_id' := λ R, by rw [unop_id, Spec.SheafedSpace_map_id],
  map_comp' := λ R S T f g, by rw [unop_comp, Spec.SheafedSpace_map_comp] }

/--
Spec, as a contravariant functor from commutative rings to presheafed spaces.
-/
def Spec.to_PresheafedSpace : CommRingᵒᵖ ⥤ PresheafedSpace CommRing :=
  Spec.to_SheafedSpace ⋙ SheafedSpace.forget_to_PresheafedSpace

@[simp] lemma Spec.to_PresheafedSpace_obj (R : CommRingᵒᵖ) :
  Spec.to_PresheafedSpace.obj R = (Spec.SheafedSpace_obj (unop R)).to_PresheafedSpace := rfl

lemma Spec.to_PresheafedSpace_obj_op (R : CommRing) :
  Spec.to_PresheafedSpace.obj (op R) = (Spec.SheafedSpace_obj R).to_PresheafedSpace := rfl

@[simp] lemma Spec.to_PresheafedSpace_map (R S : CommRingᵒᵖ) (f : R ⟶ S) :
  Spec.to_PresheafedSpace.map f = Spec.SheafedSpace_map f.unop := rfl

lemma Spec.to_PresheafedSpace_map_op (R S : CommRing) (f : R ⟶ S) :
  Spec.to_PresheafedSpace.map f.op = Spec.SheafedSpace_map f := rfl

lemma Spec.basic_open_hom_ext {X : RingedSpace} {R : CommRing} {α β : X ⟶ Spec.SheafedSpace_obj R}
  (w : α.base = β.base) (h : ∀ r : R, let U := prime_spectrum.basic_open r in
    (to_open R U ≫ α.c.app (op U)) ≫ X.presheaf.map (eq_to_hom (by rw w)) =
     to_open R U ≫ β.c.app (op U)) : α = β :=
begin
  ext1,
  { apply ((Top.sheaf.pushforward β.base).obj X.sheaf).hom_ext _
      prime_spectrum.is_basis_basic_opens,
    intro r,
    apply (structure_sheaf.to_basic_open_epi R r).1,
    simpa using h r },
  exact w,
end

/--
The spectrum of a commutative ring, as a `LocallyRingedSpace`.
-/
@[simps] def Spec.LocallyRingedSpace_obj (R : CommRing) : LocallyRingedSpace :=
{ local_ring := λ x, @@ring_equiv.local_ring _
    (show local_ring (localization.at_prime _), by apply_instance) _
    (iso.CommRing_iso_to_ring_equiv $ stalk_iso R x).symm,
  .. Spec.SheafedSpace_obj R }

@[elementwise]
lemma stalk_map_to_stalk {R S : CommRing} (f : R ⟶ S) (p : prime_spectrum S) :
  to_stalk R (prime_spectrum.comap f p) ≫
  PresheafedSpace.stalk_map (Spec.SheafedSpace_map f) p =
  f ≫ to_stalk S p :=
begin
  erw [← to_open_germ S ⊤ ⟨p, trivial⟩, ← to_open_germ R ⊤ ⟨prime_spectrum.comap f p, trivial⟩,
    category.assoc, PresheafedSpace.stalk_map_germ (Spec.SheafedSpace_map f) ⊤ ⟨p, trivial⟩,
    Spec.SheafedSpace_map_c_app, to_open_comp_comap_assoc],
  refl
end

/--
Under the isomorphisms `stalk_iso`, the map `stalk_map (Spec.SheafedSpace_map f) p` corresponds
to the induced local ring homomorphism `localization.local_ring_hom`.
-/
@[elementwise]
lemma local_ring_hom_comp_stalk_iso {R S : CommRing} (f : R ⟶ S) (p : prime_spectrum S) :
  (stalk_iso R (prime_spectrum.comap f p)).hom ≫
    @category_struct.comp _ _
      (CommRing.of (localization.at_prime (prime_spectrum.comap f p).as_ideal))
      (CommRing.of (localization.at_prime p.as_ideal)) _
      (localization.local_ring_hom (prime_spectrum.comap f p).as_ideal p.as_ideal f rfl)
      (stalk_iso S p).inv =
  PresheafedSpace.stalk_map (Spec.SheafedSpace_map f) p :=
(stalk_iso R (prime_spectrum.comap f p)).eq_inv_comp.mp $ (stalk_iso S p).comp_inv_eq.mpr $
localization.local_ring_hom_unique _ _ _ _ $ λ x, by
rw [stalk_iso_hom, stalk_iso_inv, comp_apply, comp_apply, localization_to_stalk_of,
  stalk_map_to_stalk_apply, stalk_to_fiber_ring_hom_to_stalk]

/--
The induced map of a ring homomorphism on the prime spectra, as a morphism of locally ringed spaces.
-/
@[simps] def Spec.LocallyRingedSpace_map {R S : CommRing} (f : R ⟶ S) :
  Spec.LocallyRingedSpace_obj S ⟶ Spec.LocallyRingedSpace_obj R :=
subtype.mk (Spec.SheafedSpace_map f) $ λ p, is_local_ring_hom.mk $ λ a ha,
begin
  -- Here, we are showing that the map on prime spectra induced by `f` is really a morphism of
  -- *locally* ringed spaces, i.e. that the induced map on the stalks is a local ring homomorphism.
  rw ← local_ring_hom_comp_stalk_iso_apply at ha,
  replace ha := (stalk_iso S p).hom.is_unit_map ha,
  rw coe_inv_hom_id at ha,
  replace ha := is_local_ring_hom.map_nonunit _ ha,
  convert ring_hom.is_unit_map (stalk_iso R (prime_spectrum.comap f p)).inv ha,
  rw coe_hom_inv_id,
end

@[simp] lemma Spec.LocallyRingedSpace_map_id (R : CommRing) :
  Spec.LocallyRingedSpace_map (𝟙 R) = 𝟙 (Spec.LocallyRingedSpace_obj R) :=
subtype.ext $ by { rw [Spec.LocallyRingedSpace_map_coe, Spec.SheafedSpace_map_id], refl }

lemma Spec.LocallyRingedSpace_map_comp {R S T : CommRing} (f : R ⟶ S) (g : S ⟶ T) :
  Spec.LocallyRingedSpace_map (f ≫ g) =
  Spec.LocallyRingedSpace_map g ≫ Spec.LocallyRingedSpace_map f :=
subtype.ext $ by { rw [Spec.LocallyRingedSpace_map_coe, Spec.SheafedSpace_map_comp], refl }

/--
Spec, as a contravariant functor from commutative rings to locally ringed spaces.
-/
@[simps] def Spec.to_LocallyRingedSpace : CommRingᵒᵖ ⥤ LocallyRingedSpace :=
{ obj := λ R, Spec.LocallyRingedSpace_obj (unop R),
  map := λ R S f, Spec.LocallyRingedSpace_map f.unop,
  map_id' := λ R, by rw [unop_id, Spec.LocallyRingedSpace_map_id],
  map_comp' := λ R S T f g, by rw [unop_comp, Spec.LocallyRingedSpace_map_comp] }

section Spec_Γ
open algebraic_geometry.LocallyRingedSpace

/-- The counit morphism `R ⟶ Γ(Spec R)` given by `algebraic_geometry.structure_sheaf.to_open`.  -/
@[simps] def to_Spec_Γ (R : CommRing) : R ⟶ Γ.obj (op (Spec.to_LocallyRingedSpace.obj (op R))) :=
structure_sheaf.to_open R ⊤

instance is_iso_to_Spec_Γ (R : CommRing) : is_iso (to_Spec_Γ R) :=
by { cases R, apply structure_sheaf.is_iso_to_global }

@[reassoc]
lemma Spec_Γ_naturality {R S : CommRing} (f : R ⟶ S) :
  f ≫ to_Spec_Γ S = to_Spec_Γ R ≫ Γ.map (Spec.to_LocallyRingedSpace.map f.op).op :=
by { ext, symmetry, apply localization.local_ring_hom_to_map }

/-- The counit (`Spec_Γ_identity.inv.op`) of the adjunction `Γ ⊣ Spec` is an isomorphism. -/
@[simps hom_app inv_app] def Spec_Γ_identity : Spec.to_LocallyRingedSpace.right_op ⋙ Γ ≅ 𝟭 _ :=
iso.symm $ nat_iso.of_components (λ R, as_iso (to_Spec_Γ R) : _) (λ _ _, Spec_Γ_naturality)

end Spec_Γ

/-- The stalk map of `Spec M⁻¹R ⟶ Spec R` is an iso for each `p : Spec M⁻¹R`. -/
lemma Spec_map_localization_is_iso (R : CommRing) (M : submonoid R)
  (x : prime_spectrum (localization M)) :
  is_iso (PresheafedSpace.stalk_map (Spec.to_PresheafedSpace.map
    (CommRing.of_hom (algebra_map R (localization M))).op) x) :=
begin
  erw ← local_ring_hom_comp_stalk_iso,
  apply_with is_iso.comp_is_iso { instances := ff },
  apply_instance,
  apply_with is_iso.comp_is_iso { instances := ff },
  /- I do not know why this is defeq to the goal, but I'm happy to accept that it is. -/
  exact (show is_iso (is_localization.localization_localization_at_prime_iso_localization
    M x.as_ideal).to_ring_equiv.to_CommRing_iso.hom, by apply_instance),
  apply_instance
end


end algebraic_geometry