prime_spectrum.lean

/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import topology.opens
import ring_theory.ideal.prod
import linear_algebra.finsupp
import algebra.punit_instances

/-!
# Prime spectrum of a commutative ring

The prime spectrum of a commutative ring is the type of all prime ideals.
It is naturally endowed with a topology: the Zariski topology.

(It is also naturally endowed with a sheaf of rings,
which is constructed in `algebraic_geometry.structure_sheaf`.)

## Main definitions

* `prime_spectrum R`: The prime spectrum of a commutative ring `R`,
  i.e., the set of all prime ideals of `R`.
* `zero_locus s`: The zero locus of a subset `s` of `R`
  is the subset of `prime_spectrum R` consisting of all prime ideals that contain `s`.
* `vanishing_ideal t`: The vanishing ideal of a subset `t` of `prime_spectrum R`
  is the intersection of points in `t` (viewed as prime ideals).

## Conventions

We denote subsets of rings with `s`, `s'`, etc...
whereas we denote subsets of prime spectra with `t`, `t'`, etc...

## Inspiration/contributors

The contents of this file draw inspiration from
<https://github.com/ramonfmir/lean-scheme>
which has contributions from Ramon Fernandez Mir, Kevin Buzzard, Kenny Lau,
and Chris Hughes (on an earlier repository).

-/

noncomputable theory
open_locale classical

universe variables u v

variables (R : Type u) [comm_ring R]

/-- 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. -/
@[nolint has_inhabited_instance]
def prime_spectrum := {I : ideal R // I.is_prime}

variable {R}

namespace prime_spectrum

/-- A method to view a point in the prime spectrum of a commutative ring
as an ideal of that ring. -/
abbreviation as_ideal (x : prime_spectrum R) : ideal R := x.val

instance is_prime (x : prime_spectrum R) :
  x.as_ideal.is_prime := x.2

/--
The prime spectrum of the zero ring is empty.
-/
lemma 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

section
variables (R) (S : Type v) [comm_ring S]

/-- 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`. -/
noncomputable def prime_spectrum_prod :
  prime_spectrum (R × S) ≃ prime_spectrum R ⊕ prime_spectrum S :=
ideal.prime_ideals_equiv R S

variables {R S}

@[simp] lemma 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 }
@[simp] lemma 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 }

end

@[ext] lemma ext {x y : prime_spectrum R} :
  x = y ↔ x.as_ideal = y.as_ideal :=
subtype.ext_iff_val

/-- 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 quotient ring `R` modulo the prime ideal `x`.
In this manner, `zero_locus s` is exactly the subset of `prime_spectrum R`
where all "functions" in `s` vanish simultaneously.
-/
def zero_locus (s : set R) : set (prime_spectrum R) :=
{x | s ⊆ x.as_ideal}

@[simp] lemma mem_zero_locus (x : prime_spectrum R) (s : set R) :
  x ∈ zero_locus s ↔ s ⊆ x.as_ideal := iff.rfl

@[simp] lemma 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 }

/-- 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 values in the quotient ring `R` modulo the prime ideal `x`.
In this manner, `vanishing_ideal t` is exactly the ideal of `R`
consisting of all "functions" that vanish on all of `t`.
-/
def vanishing_ideal (t : set (prime_spectrum R)) : ideal R :=
⨅ (x : prime_spectrum R) (h : x ∈ t), x.as_ideal

lemma 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, submodule.mem_coe, submodule.mem_infi],
  apply forall_congr, intro x,
  rw [submodule.mem_infi],
end

lemma 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 [← submodule.mem_coe, coe_vanishing_ideal, set.mem_set_of_eq]

lemma subset_zero_locus_iff_le_vanishing_ideal (t : set (prime_spectrum R)) (I : ideal R) :
  t ⊆ zero_locus I ↔ I ≤ vanishing_ideal t :=
begin
  split; intro h,
  { intros f hf,
    rw [mem_vanishing_ideal],
    intros x hx,
    have hxI := h hx,
    rw mem_zero_locus at hxI,
    exact hxI hf },
  { intros x hx,
    rw mem_zero_locus,
    refine le_trans h _,
    intros f hf,
    rw [mem_vanishing_ideal] at hf,
    exact hf x hx }
end

section gc
variable (R)

/-- `zero_locus` and `vanishing_ideal` form a galois connection. -/
lemma gc : @galois_connection
  (ideal R) (order_dual (set (prime_spectrum R))) _ _
  (λ I, zero_locus I) (λ t, vanishing_ideal t) :=
λ I t, subset_zero_locus_iff_le_vanishing_ideal t I

/-- `zero_locus` and `vanishing_ideal` form a galois connection. -/
lemma gc_set : @galois_connection
  (set R) (order_dual (set (prime_spectrum R))) _ _
  (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 galois_connection.compose _ _ _ _ ideal_gc (gc R)

lemma 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

end gc

-- TODO: we actually get the radical ideal,
-- but I think that isn't in mathlib yet.
lemma subset_vanishing_ideal_zero_locus (s : set R) :
  s ⊆ vanishing_ideal (zero_locus s) :=
(gc_set R).le_u_l s

lemma le_vanishing_ideal_zero_locus (I : ideal R) :
  I ≤ vanishing_ideal (zero_locus I) :=
(gc R).le_u_l I

lemma subset_zero_locus_vanishing_ideal (t : set (prime_spectrum R)) :
  t ⊆ zero_locus (vanishing_ideal t) :=
(gc R).l_u_le t

lemma zero_locus_bot :
  zero_locus ((⊥ : ideal R) : set R) = set.univ :=
(gc R).l_bot

@[simp] lemma zero_locus_singleton_zero :
  zero_locus (0 : set R) = set.univ :=
zero_locus_bot

@[simp] lemma zero_locus_empty :
  zero_locus (∅ : set R) = set.univ :=
(gc_set R).l_bot

@[simp] lemma vanishing_ideal_univ :
  vanishing_ideal (∅ : set (prime_spectrum R)) = ⊤ :=
by simpa using (gc R).u_top

lemma 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.1 eq_top,
end

lemma zero_locus_empty_iff_eq_top {I : ideal R} :
  zero_locus (I : set R) = ∅ ↔ I = ⊤ :=
begin
  split,
  { contrapose!,
    intro h,
    apply set.ne_empty_iff_nonempty.mpr,
    rcases ideal.exists_le_maximal I h with ⟨M, hM, hIM⟩,
    exact ⟨⟨M, hM.is_prime⟩, hIM⟩ },
  { rintro rfl, apply zero_locus_empty_of_one_mem, trivial }
end

@[simp] lemma zero_locus_univ :
  zero_locus (set.univ : set R) = ∅ :=
zero_locus_empty_of_one_mem (set.mem_univ 1)

lemma 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 zero_locus_union (s s' : set R) :
  zero_locus (s ∪ s') = zero_locus s ∩ zero_locus s' :=
(gc_set R).l_sup

lemma vanishing_ideal_union (t t' : set (prime_spectrum R)) :
  vanishing_ideal (t ∪ t') = vanishing_ideal t ⊓ vanishing_ideal t' :=
(gc R).u_inf

lemma 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 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 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 zero_locus_inf (I J : ideal R) :
  zero_locus ((I ⊓ J : ideal R) : set R) = zero_locus I ∪ zero_locus J :=
begin
  ext x,
  split,
  { rintro h,
    rw set.mem_union,
    simp only [mem_zero_locus] at h ⊢,
    -- TODO: The rest of this proof should be factored out.
    rw or_iff_not_imp_right,
    intros hs r hr,
    rw set.not_subset at hs,
    rcases hs with ⟨s, hs1, hs2⟩,
    apply (ideal.is_prime.mem_or_mem (by apply_instance) _).resolve_left hs2,
    apply h,
    split,
    { exact ideal.mul_mem_left _ hr },
    { exact ideal.mul_mem_right _ hs1 } },
  { rintro (h|h),
    all_goals
    { rw mem_zero_locus at h ⊢,
      refine set.subset.trans _ h,
      intros r hr, cases hr, assumption } }
end

lemma 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 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

/-- 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. -/
instance 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,
    let f : Zs → set R := λ i, classical.some (h i.2),
    have hf : ∀ i : Zs, ↑i = zero_locus (f i) := λ i, (classical.some_spec (h i.2)).symm,
    simp only [hf],
    exact ⟨_, zero_locus_Union _⟩
  end
  (by { rintro _ _ ⟨s, rfl⟩ ⟨t, rfl⟩, exact ⟨_, (union_zero_locus s t).symm⟩ })

lemma is_open_iff (U : set (prime_spectrum R)) :
  is_open U ↔ ∃ s, Uᶜ = zero_locus s :=
by simp only [@eq_comm _ Uᶜ]; refl

lemma is_closed_iff_zero_locus (Z : set (prime_spectrum R)) :
  is_closed Z ↔ ∃ s, Z = zero_locus s :=
by rw [is_closed, is_open_iff, set.compl_compl]

lemma is_closed_zero_locus (s : set R) :
  is_closed (zero_locus s) :=
by { rw [is_closed_iff_zero_locus], exact ⟨s, rfl⟩ }

section comap
variables {S : Type v} [comm_ring S] {S' : Type*} [comm_ring S']

/-- The function between prime spectra of commutative rings induced by a ring homomorphism.
This function is continuous. -/
def comap (f : R →+* S) : prime_spectrum S → prime_spectrum R :=
λ y, ⟨ideal.comap f y.as_ideal, by exact ideal.is_prime.comap _⟩

variables (f : R →+* S)

@[simp] lemma comap_as_ideal (y : prime_spectrum S) :
  (comap f y).as_ideal = ideal.comap f y.as_ideal :=
rfl

@[simp] lemma comap_id : comap (ring_hom.id R) = id :=
funext $ λ x, ext.mpr $ by { rw [comap_as_ideal], apply ideal.ext, intros r, simp }

@[simp] lemma comap_comp (f : R →+* S) (g : S →+* S') :
  comap (g.comp f) = comap f ∘ comap g :=
funext $ λ x, ext.mpr $ by { simp, refl }

@[simp] lemma preimage_comap_zero_locus (s : set R) :
  (comap f) ⁻¹' (zero_locus s) = zero_locus (f '' s) :=
begin
  ext x,
  simp only [mem_zero_locus, set.mem_preimage, comap_as_ideal, set.image_subset_iff],
  refl
end

lemma comap_continuous (f : R →+* S) : continuous (comap f) :=
begin
  rw continuous_iff_is_closed,
  simp only [is_closed_iff_zero_locus],
  rintro _ ⟨s, rfl⟩,
  exact ⟨_, preimage_comap_zero_locus f s⟩
end

end comap

lemma 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,
    calc fs ⊆ vanishing_ideal t : ht
        ... ⊆ x.as_ideal        : hx },
  { rw (is_closed_zero_locus _).closure_subset_iff,
    exact subset_zero_locus_vanishing_ideal t }
end

/-- The prime spectrum of a commutative ring is a compact topological space. -/
instance : compact_space (prime_spectrum R) :=
begin
  apply compact_space_of_finite_subfamily_closed,
  intros ι Z hZc hZ,
  let I : ι → ideal R := λ i, vanishing_ideal (Z i),
  have hI : ∀ i, Z i = zero_locus (I i),
  { intro i,
    rw [zero_locus_vanishing_ideal_eq_closure, is_closed.closure_eq],
    exact hZc i },
  have one_mem : (1:R) ∈ ⨆ (i : ι), I i,
  { rw [← ideal.eq_top_iff_one, ← zero_locus_empty_iff_eq_top, zero_locus_supr],
    simpa only [hI] using hZ },
  obtain ⟨s, hs⟩ : ∃ s : finset ι, (1:R) ∈ ⨆ i ∈ s, I i :=
    submodule.exists_finset_of_mem_supr I one_mem,
  show ∃ t : finset ι, (⋂ i ∈ t, Z i) = ∅,
  use s,
  rw [← ideal.eq_top_iff_one, ←zero_locus_empty_iff_eq_top] at hs,
  simpa only [zero_locus_supr, hI] using hs
end

section basic_open

/-- `basic_open r` is the open subset containing all prime ideals not containing `r`. -/
def basic_open (r : R) : topological_space.opens (prime_spectrum R) :=
{ val := { x | r ∉ x.as_ideal },
  property := ⟨{r}, set.ext $ λ x, set.singleton_subset_iff.trans $ not_not.symm⟩ }

end basic_open

end prime_spectrum