generalise finite adele ring arguments to AKLB setup

Author: kbuzzard

FLT/NumberField/AdeleRing.lean

@@ -1,10 +1,9 @@
 import Mathlib.RingTheory.DedekindDomain.FiniteAdeleRing
 import Mathlib.NumberTheory.NumberField.Basic
+import Mathlib
 
 universe u
 
-variable (K : Type*) [Field K] [NumberField K]
-
 section PRed
 -- Don't try anything in this section because we are missing a definition.
 
@@ -13,6 +12,8 @@ def NumberField.AdeleRing (K : Type u) [Field K] [NumberField K] : Type u := sor
 
 open NumberField
 
+variable (K : Type*) [Field K] [NumberField K]
+
 -- All these are already done in mathlib PRs, so can be assumed for now.
 -- When they're in mathlib master we can update mathlib and then get these theorems/definitions
 -- for free.
@@ -41,6 +42,8 @@ section TODO
 -- these theorems can't be worked on yet because we don't have an actual definition
 -- of the adele ring. We can work out a strategy when we do.
 
+variable (K : Type*) [Field K] [NumberField K]
+
 -- Maybe this isn't even stated in the best way?
 theorem NumberField.AdeleRing.discrete : ∀ k : K, ∃ U : Set (AdeleRing K),
     IsOpen U ∧ (algebraMap K (AdeleRing K)) ⁻¹' U = {k} := sorry
@@ -56,48 +59,89 @@ end TODO
 
 -- *************** Sorries which are not impossible start here *****************
 
--- recall we already had `variable (K : Type*) [Field K] [NumberField K]`
+variable (A K L B : Type*) [CommRing A] [CommRing B] [Algebra A B] [Field K] [Field L]
+    [Algebra A K] [IsFractionRing A K] [Algebra B L] [IsDedekindDomain A]
+    [Algebra K L] [Algebra A L] [IsScalarTower A B L] [IsScalarTower A K L]
+    [IsIntegralClosure B A L] [FiniteDimensional K L]
+
+variable [IsDomain B] -- is this necessary?
+variable [Algebra.IsSeparable K L]
 
-variable (L : Type*) [Field L] [Algebra K L] [FiniteDimensional K L]
+-- example : IsDedekindDomain B := IsIntegralClosure.isDedekindDomain A K L B
 
-instance : NumberField L := sorry -- inferInstance fails
+variable [IsDedekindDomain B]
 
--- We want the map `L ⊗[K] 𝔸_K → 𝔸_L`, but we don't have a definition of adeles.
--- However we do have a definition of finite adeles, so let's work there.
+-- example : IsFractionRing B L := IsIntegralClosure.isFractionRing_of_finite_extension A K L B
 
+variable [IsFractionRing B L]
+
+-- example : Algebra.IsIntegral A B := IsIntegralClosure.isIntegral_algebra A L
+
+variable [Algebra.IsIntegral A B]
+
+-- Conjecture: in this generality there's a natural isomorphism `L ⊗[K] 𝔸_K^∞ → 𝔸_L^∞`
+-- Note however that we don't have a definition of adeles in this generality.
+
+-- One key input is the purely local statement that if L/K
+-- is a finite extension of number fields, and v is a place of K,
+-- then `L ⊗[K] Kᵥ ≅ ⊕_w|v L_w`, where the sum is over the places w
+-- of L above v.
 namespace DedekindDomain
 
-open scoped TensorProduct NumberField -- ⊗ notation for tensor product and 𝓞 K notation for ring of ints
+open scoped TensorProduct -- ⊗ notation for tensor product
 
-noncomputable instance : Algebra K (ProdAdicCompletions (𝓞 L) L) := RingHom.toAlgebra <|
-  (algebraMap L (ProdAdicCompletions (𝓞 L) L)).comp (algebraMap K L)
+noncomputable instance : Algebra K (ProdAdicCompletions B L) := RingHom.toAlgebra <|
+  (algebraMap L (ProdAdicCompletions B L)).comp (algebraMap K L)
 
--- Need to pull back elements of HeightOneSpectrum (𝓞 L) to HeightOneSpectrum (𝓞 K)
+-- Need to pull back elements of HeightOneSpectrum B to HeightOneSpectrum A
 open IsDedekindDomain
 
-variable {L} in
-def HeightOneSpectrum.comap (w : HeightOneSpectrum (𝓞 L)) : HeightOneSpectrum (𝓞 K) where
-  asIdeal := w.asIdeal.comap (algebraMap (𝓞 K) (𝓞 L))
-  isPrime := Ideal.comap_isPrime (algebraMap (𝓞 K) (𝓞 L)) w.asIdeal
-  ne_bot := sorry -- pullback of nonzero prime is nonzero? Certainly true for number fields.
-  -- Idea of proof: show that if B/A is integral and B is an integral domain, then
-  -- any nonzero element of B divides a nonzero element of A.
-  -- In fact this definition should be made in that generality.
-  -- Use `Algebra.exists_dvd_nonzero_if_isIntegral` in the FLT file MathlibExperiments/FrobeniusRiou.lean
-
-variable {L} in
-noncomputable def HeightOneSpectrum.of_comap (w : HeightOneSpectrum (𝓞 L)) :
-    (HeightOneSpectrum.adicCompletion K (comap K w)) →+* (HeightOneSpectrum.adicCompletion L w) :=
-  letI : UniformSpace K := (comap K w).adicValued.toUniformSpace;
+namespace HeightOneSpectrum
+-- first need a map from finite places of 𝓞L to finite places of 𝓞K
+variable {B L} in
+def comap (w : HeightOneSpectrum B) : HeightOneSpectrum A where
+  asIdeal := w.asIdeal.comap (algebraMap A B)
+  isPrime := Ideal.comap_isPrime (algebraMap A B) w.asIdeal
+  ne_bot := mt Ideal.eq_bot_of_comap_eq_bot w.ne_bot
+
+open scoped algebraMap
+
+-- homework
+def valuation_comap (w : HeightOneSpectrum B) (x : K) :
+    (comap A w).valuation x = w.valuation (algebraMap K L x) ^ (Ideal.ramificationIdx (algebraMap A B) (comap A w).asIdeal w.asIdeal) := sorry
+
+-- Say w is a finite place of L lying above v a finite places
+-- of K. Then there's a ring hom K_v -> L_w.
+variable {B L} in
+noncomputable def of_comap (w : HeightOneSpectrum B) :
+    (HeightOneSpectrum.adicCompletion K (comap A w)) →+* (HeightOneSpectrum.adicCompletion L w) :=
+  letI : UniformSpace K := (comap A w).adicValued.toUniformSpace;
   letI : UniformSpace L := w.adicValued.toUniformSpace;
   UniformSpace.Completion.mapRingHom (algebraMap K L) <| by
+  -- question is the following:
+  -- if L/K is a finite extension of sensible fields (e.g. number fields)
+  -- and if w is a finite place of L lying above v a finite place of K,
+  -- and if we give L the w-adic topology and K the v-adic topology,
+  -- then the map K → L is continuous
+  refine continuous_of_continuousAt_zero (algebraMap K L) ?hf
+  delta ContinuousAt
+  simp only [map_zero]
+  rw [(@Valued.hasBasis_nhds_zero K _ _ _ (comap A w).adicValued).tendsto_iff
+    (@Valued.hasBasis_nhds_zero L _ _ _ w.adicValued)]
+  simp only [HeightOneSpectrum.adicValued_apply, Set.mem_setOf_eq, true_and, true_implies]
+  refine fun i ↦ ⟨i ^ (1 / Ideal.ramificationIdx (algebraMap A B) (comap A w).asIdeal w.asIdeal), fun _ h ↦ ?_⟩
+  -- last line is nonsense but use `valuation_comap` to finish
+  push_cast at h
   sorry
 
-open HeightOneSpectrum in
+end HeightOneSpectrum
+
+open HeightOneSpectrum
+
 noncomputable def ProdAdicCompletions.baseChange :
-    L ⊗[K] ProdAdicCompletions (𝓞 K) K →ₗ[K] ProdAdicCompletions (𝓞 L) L := TensorProduct.lift <| {
+    L ⊗[K] ProdAdicCompletions A K →ₗ[K] ProdAdicCompletions B L := TensorProduct.lift <| {
   toFun := fun l ↦ {
-    toFun := fun kv w ↦ l • (of_comap K w (kv (comap K w)))
+    toFun := fun kv w ↦ l • (of_comap A K w (kv (comap A w)))
     map_add' := sorry
     map_smul' := sorry
   }
@@ -106,8 +150,10 @@ noncomputable def ProdAdicCompletions.baseChange :
 }
 
 -- hard but hopefully enough (this proof will be a lot of work)
-theorem ProdAdicCompletions.baseChange_iso (x : ProdAdicCompletions (𝓞 K) K) :
+theorem ProdAdicCompletions.baseChange_iso (x : ProdAdicCompletions A K) :
   ProdAdicCompletions.IsFiniteAdele x ↔
-  ProdAdicCompletions.IsFiniteAdele (ProdAdicCompletions.baseChange K L (1 ⊗ₜ x)) := sorry
+  ProdAdicCompletions.IsFiniteAdele (ProdAdicCompletions.baseChange A K L B (1 ⊗ₜ x)) := sorry
+
+end DedekindDomain
 
 -- Can we now write down the isomorphism L ⊗ 𝔸_K^∞ = 𝔸_L^∞ ?