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import continuous_valuations
import Huber_pair
/-!
# The adic spectrum as a topological space
In this file we define a structure (`rational_open_data`) that will parameterise
a basis for the topology on the adic spectrum of a Huber pair.
-/
open_locale classical
local attribute [instance] set.pointwise_mul_comm_semiring
local attribute [instance] set.smul_set_action
local postfix `⁺` : 66 := λ A : Huber_pair, A.plus
variables {Γ₀ : Type*} [linear_ordered_comm_group_with_zero Γ₀]
-- We reserve the name `Spa` (with upper case `S`) for the bundled adic spectrum (`adic_space.lean`)
/-- The space underlying the adic spectrum of a Huber pair (A,A⁺)
consists of all the equivalence classes of valuations that are continuous
and whose value on the ring A⁺ is1. [Wedhorn, Def 7.23]. -/
definition spa (A : Huber_pair) : Type :=
{v : Spv A // v.is_continuous ∧ ∀ r : A⁺, v (algebra_map A r) ≤ 1}
/--The equivalence class of a valuation is contained in spa
if and only if the valuation is continuous and its values on the ring A⁺ are ≤ 1,
since these properties are constant on equivalence classes.-/
lemma mk_mem_spa {A : Huber_pair} {v : valuation A Γ₀} :
Spv.mk v ∈ {v : Spv A | v.is_continuous ∧ ∀ r : A⁺, v (algebra_map A r) ≤ 1} ↔
v.is_continuous ∧ ∀ r : A⁺, v (algebra_map A r) ≤ 1 :=
begin
apply and_congr,
{ exact (Spv.out_mk v).is_continuous_iff, },
{ apply forall_congr,
intro r,
simpa using (Spv.out_mk v) (algebra_map A r) 1, }
end
namespace spa
open set algebra
variables {A : Huber_pair}
/-- The coercion from the adic spectrum of a Huber pair to the ambient valuation spectrum.-/
instance : has_coe (spa A) (Spv A) := ⟨subtype.val⟩
@[ext]
lemma ext (v₁ v₂ : spa A) (h : (Spv.out ↑v₁).is_equiv (Spv.out (↑v₂ : Spv A))) :
v₁ = v₂ :=
subtype.val_injective $ Spv.ext _ _ h
lemma ext_iff {v₁ v₂ : spa A} :
v₁ = v₂ ↔ ((Spv.out ↑v₁).is_equiv (Spv.out (↑v₂ : Spv A))) :=
by rw [subtype.coe_ext, Spv.ext_iff]
/-- The value monoid of a random representative valuation of a point in the adic spectrum. -/
abbreviation out_Γ₀ (v : spa A) := Spv.out_Γ₀ (v : Spv A)
/-- A valuation in the adic spectrum is continuous. -/
lemma is_continuous (v : spa A) : Spv.is_continuous (v : Spv A) := v.property.left
/-- The valuation of an integral element is at most 1. -/
lemma map_plus (v : spa A) (a : (A⁺)) : v (algebra_map A a) ≤ 1 := v.property.right a
/-- The valuation of a unit of the ring of integral elements is 1. -/
@[simp] lemma map_unit (v : spa A) (u : units (A⁺)) :
v ((algebra_map A : (A⁺) → A) u) = 1 :=
begin
have h₁ := map_plus v u,
have h₂ := map_plus v (u⁻¹ : _),
have := actual_ordered_comm_monoid.mul_eq_one_iff_of_le_one' h₁ h₂,
apply (this.mp _).left,
erw ← valuation.map_mul,
rw ← is_ring_hom.map_mul (algebra_map A : (A⁺) → A),
simp only [units.mul_inv, algebra.map_one, valuation.map_one]
end
-- We are now going to setup the topology on `spa A`.
-- A basis of the topology is indexed by the following data:
/--A rational open subset of `spa A` is indexed by:
* an element s of A, and
* a finite set T ⊆ A that generates an open ideal in A.
In the literature, these sets are commonly denoted by D(T,s).-/
structure rational_open_data (A : Huber_pair) :=
(s : A)
(T : set A)
[Tfin : fintype T]
(Hopen : is_open ((ideal.span T) : set A))
namespace rational_open_data
variables (r : rational_open_data A)
attribute [instance] Tfin
@[ext]
lemma ext {r₁ r₂ : rational_open_data A} (hs : r₁.s = r₂.s) (hT : r₁.T = r₂.T) :
r₁ = r₂ :=
begin
cases r₁, cases r₂,
congr; assumption
end
/--The subset of the adic spectrum associated with the data for a rational open subset.
In the literature, these sets are commonly denoted by D(T,s).-/
def open_set (r : rational_open_data A) : set (spa A) :=
{v : spa A | (∀ t ∈ r.T, (v t ≤ v r.s)) ∧ (v r.s ≠ 0)}
variable (A)
/--The rational open subset covering the entire adic spectrum.-/
def univ : rational_open_data A :=
{ s := 1,
T := {1},
Hopen := by { rw ideal.span_singleton_one, exact is_open_univ } }
variable {A}
@[simp] lemma univ_s : (univ A).s = 1 := rfl
@[simp] lemma univ_T : (univ A).T = {1} := rfl
@[simp] lemma univ_open_set :
(univ A).open_set = set.univ :=
begin
rw eq_univ_iff_forall,
intros v,
split,
{ intros t ht,
erw mem_singleton_iff at ht,
rw [ht, univ_s], },
{ erw [univ_s, Spv.map_one],
exact one_ne_zero }
end
/--The rational open subset D(T,s) is the same as D(T ∪ {s}, s).-/
noncomputable def insert_s (r : rational_open_data A) : rational_open_data A :=
{ s := r.s,
T := insert r.s r.T,
Hopen := submodule.is_open_of_open_submodule
⟨ideal.span (r.T), r.Hopen, ideal.span_mono $ set.subset_insert _ _⟩ }
@[simp] lemma insert_s_s (r : rational_open_data A) :
(insert_s r).s = r.s := rfl
@[simp] lemma insert_s_T (r : rational_open_data A) :
(insert_s r).T = insert r.s r.T := rfl
@[simp] lemma insert_s_open_set (r : rational_open_data A) :
(insert_s r).open_set = r.open_set :=
begin
ext v,
split; rintros ⟨h₁, h₂⟩; split; try { exact h₂ }; intros t ht,
{ apply h₁ t,
exact mem_insert_of_mem _ ht },
{ cases ht,
{ rw [ht, insert_s_s], },
{ exact h₁ t ht } },
end
lemma mem_insert_s (r : rational_open_data A) :
r.s ∈ (insert_s r).T := by {left, refl}
/-- Auxilliary definition for the intersection of two rational open sets.-/
noncomputable def inter_aux (r1 r2 : rational_open_data A) : rational_open_data A :=
{ s := r1.s * r2.s,
T := r1.T * r2.T,
Tfin := set.pointwise_mul_fintype _ _,
Hopen :=
begin
rcases Huber_ring.exists_pod_subset _ (mem_nhds_sets r1.Hopen $ ideal.zero_mem $ ideal.span r1.T)
with ⟨A₀, _, _, _, ⟨_, emb, I, fg, top⟩, hI⟩,
dsimp only at hI,
resetI,
rw is_ideal_adic_iff at top,
cases top.2 (algebra_map A ⁻¹' ↑(ideal.span r2.T)) _ with n hn,
{ apply submodule.is_open_of_open_submodule,
use ideal.map (of_id A₀ A) (I^(n+1)),
refine ⟨is_open_ideal_map_open_embedding emb _ (top.1 (n+1)), _⟩,
delta ideal.span,
erw [pow_succ, ideal.map_mul, ← submodule.span_mul_span],
apply submodule.mul_le_mul,
{ exact (ideal.span_le.mpr hI) },
{ rw ← image_subset_iff at hn,
exact (ideal.span_le.mpr hn) } },
{ apply emb.continuous.tendsto,
rw show algebra.to_fun A (0:A₀) = 0,
{ haveI : is_ring_hom (algebra.to_fun A : A₀ → A) := algebra.is_ring_hom,
apply is_ring_hom.map_zero },
exact (mem_nhds_sets r2.Hopen $ ideal.zero_mem $ ideal.span r2.T) }
end }
/--The intersection of two rational open sets is a rational open set.-/
noncomputable def inter (r1 r2 : rational_open_data A) : rational_open_data A :=
inter_aux (rational_open_data.insert_s r1) (rational_open_data.insert_s r2)
@[simp] lemma inter_s (r1 r2 : rational_open_data A) :
(r1.inter r2).s = r1.s * r2.s := rfl
@[simp] lemma inter_T (r1 r2 : rational_open_data A) :
(r1.inter r2).T = (insert r1.s r1.T) * (insert r2.s r2.T) := rfl
lemma inter_open_set (r1 r2 : rational_open_data A) :
(inter r1 r2).open_set = r1.open_set ∩ r2.open_set :=
begin
rw [← insert_s_open_set r1, ← insert_s_open_set r2],
apply le_antisymm,
{ rintros v ⟨hv, hs⟩,
have vmuls : v (r1.s * r2.s) = v r1.s * v r2.s := valuation.map_mul _ _ _,
have hs₁ : v r1.s ≠ 0 := λ H, by simpa [-coe_fn_coe_base, vmuls, H] using hs,
have hs₂ : v r2.s ≠ 0 := λ H, by simpa [-coe_fn_coe_base, vmuls, H] using hs,
split; split; try { assumption };
intros t ht,
{ suffices H : v t * v r2.s ≤ v r1.s * v r2.s,
{ simpa [hs₂, mul_assoc, -coe_fn_coe_base] using
linear_ordered_structure.mul_le_mul_right H (group_with_zero.mk₀ _ hs₂)⁻¹, },
{ simpa using hv (t * r2.s) ⟨t, ht, r2.s, mem_insert_s r2, rfl⟩, } },
{ suffices H : v r1.s * v t ≤ v r1.s * v r2.s,
{ simpa [hs₁, mul_assoc, -coe_fn_coe_base] using
linear_ordered_structure.mul_le_mul_left H (group_with_zero.mk₀ _ hs₁)⁻¹, },
{ simpa using hv (r1.s * t) ⟨r1.s, mem_insert_s r1, t, ht, rfl⟩, } } },
{ rintros v ⟨⟨hv₁, hs₁⟩, ⟨hv₂, hs₂⟩⟩,
split,
{ rintros t ⟨t₁, ht₁, t₂, ht₂, rfl⟩,
convert le_trans
(linear_ordered_structure.mul_le_mul_right (hv₁ t₁ ht₁) _)
(linear_ordered_structure.mul_le_mul_left (hv₂ t₂ ht₂) _);
apply valuation.map_mul },
{ assume eq_zero, simp at eq_zero, tauto }, }
end
lemma inter_symm (r1 r2 : rational_open_data A) :
r1.inter r2 = r2.inter r1 :=
ext (mul_comm _ _) (mul_comm _ _)
end rational_open_data
variable (A)
/--The basis for the topology on the adic spectrum, consisting of rational open sets.-/
def rational_basis := {U : set (spa A) | ∃ r : rational_open_data A, U = r.open_set}
/--The topology on the adic spectrum, generated by rational open sets.-/
instance : topological_space (spa A) :=
topological_space.generate_from (rational_basis A)
variable {A}
/--The rational open sets form a basis for the topology on the adic spectrum.-/
lemma rational_basis.is_basis : topological_space.is_topological_basis (rational_basis A) :=
begin
refine ⟨_, _, rfl⟩,
{ rintros _ ⟨r₁, rfl⟩ _ ⟨r₂, rfl⟩ x hx,
refine ⟨_, ⟨_, (rational_open_data.inter_open_set r₁ r₂).symm⟩, hx, subset.refl _⟩, },
{ apply subset.antisymm (subset_univ _) (subset_sUnion_of_mem _),
exact ⟨_, rational_open_data.univ_open_set.symm⟩ }
end
end spa