Merge pull request #127 from ImperialCollegeLondon/Ore-Localization-for-algebras

Add algebra instances for localizations

Author: kbuzzard

FLT/ForMathlib/Algebra.lean

@@ -0,0 +1,192 @@
+import Mathlib.RingTheory.OreLocalization.Ring
+
+-- extend localization theory to algebras
+
+/-!
+
+# Algebra instances of Ore Localizations
+
+If `R` is a commutative ring with submonoid `S`, and if `A` is a commutative `R`-algebra,
+then my guess is that `A[S⁻¹]` is an `R[S⁻¹]`-algebra. Let's see if I'm right and if so,
+in what generality.
+
+-/
+
+namespace OreLocalization
+
+section CommMagma
+
+variable {R A : Type*} [CommMonoid R] [CommMagma A] [MulAction R A] [IsScalarTower R A A]
+  {S : Submonoid R}
+
+@[to_additive]
+private def mul' (a₁ : A) (s₁ : S) (a₂ : A) (s₂ : S) : A[S⁻¹] :=
+  a₁ * a₂ /ₒ (s₂ * s₁)
+
+@[to_additive]
+private def mul''
+  (a : A) (s : S) : A[S⁻¹] → A[S⁻¹] :=
+  liftExpand (mul' a s) fun a₁ r₁ ⟨s₁, hs₁⟩ hr₁s₁ => by
+  unfold OreLocalization.mul'
+  simp only at hr₁s₁ ⊢
+  rw [oreDiv_eq_iff]
+  use ⟨s₁, hs₁⟩, r₁ * s₁
+  simp only [Submonoid.mk_smul, Submonoid.coe_mul]
+  constructor
+  · rw [← smul_mul_assoc]
+    rw [← smul_mul_assoc]
+    rw [mul_comm]
+    rw [smul_mul_assoc, smul_mul_assoc]
+    rw [mul_comm]
+    -- it's like a bloody Rubik's cube
+    rw [smul_mul_assoc]
+    rw [← mul_smul]
+  · obtain ⟨s₂, hs₂⟩ := s
+    simpa only [Submonoid.mk_smul, Submonoid.coe_mul] using mul_left_comm s₁ (r₁ * s₁) s₂
+
+@[to_additive]
+protected def mul : A[S⁻¹] → A[S⁻¹] → A[S⁻¹] :=
+  liftExpand mul'' fun a₁ r₁ s hs => by
+  obtain ⟨s₁, hs₁⟩ := s
+  simp only at hs
+  unfold OreLocalization.mul''
+  simp
+  unfold OreLocalization.mul'
+  dsimp
+  ext sa
+  induction sa
+  rename_i a s_temp
+  obtain ⟨s, hs⟩ := s_temp
+  rw [liftExpand_of]
+  rw [liftExpand_of]
+  rw [oreDiv_eq_iff]
+  simp only [Submonoid.mk_smul, Submonoid.coe_mul]
+  use ⟨s₁, hs₁⟩, r₁ * s₁
+  simp only [Submonoid.mk_smul, Submonoid.coe_mul]
+  constructor
+  · rw [smul_mul_assoc]
+    rw [← mul_smul]
+    rw [mul_comm]
+  · repeat rw [mul_assoc]
+    repeat rw [mul_left_comm s₁]
+    rw [mul_left_comm s]
+
+instance : Mul (A[S⁻¹]) where
+  mul := OreLocalization.mul
+
+protected def mul_def (a : A) (s : { x // x ∈ S }) (b : A) (t : { x // x ∈ S }) :
+  a /ₒ s * (b /ₒ t) = a * b /ₒ (t * s) := rfl
+
+unseal OreLocalization.smul in
+example (as bt : R[S⁻¹]) : as * bt = as • bt := rfl
+
+end CommMagma
+
+section One
+
+variable {R A : Type*} [CommMonoid R] [MulAction R A] [One A] --[MulAction R A] [IsScalarTower R A A]
+  {S : Submonoid R}
+
+instance : One (A[S⁻¹]) where
+  one := 1 /ₒ 1
+
+protected lemma one_def' : (1 : A[S⁻¹]) = 1 /ₒ 1 := rfl
+
+end One
+
+section CommMonoid
+
+variable {R A : Type*} [CommMonoid R] [CommMonoid A] [MulAction R A] [IsScalarTower R A A]
+  {S : Submonoid R}
+
+instance commMonoid : CommMonoid (A[S⁻¹]) where
+  mul_assoc a b c := by
+    induction' a with a
+    induction' b with b
+    induction' c with c
+    change (a * b * c) /ₒ _ = (a * (b * c)) /ₒ _
+    simp [mul_assoc]
+  one_mul a := by
+    induction' a with a
+    change (1 * a) /ₒ _ = a /ₒ _
+    simp
+  mul_one a := by
+    induction' a with a s
+    simp [OreLocalization.mul_def, OreLocalization.one_def']
+  mul_comm a b := by
+    induction' a with a
+    induction' b with b
+    simp only [OreLocalization.mul_def, mul_comm]
+
+end CommMonoid
+
+section CommSemiring
+
+variable {R A : Type*} [CommMonoid R] [CommRing A] [DistribMulAction R A] [IsScalarTower R A A]
+  {S : Submonoid R}
+
+lemma left_distrib' (a b c : A[S⁻¹]) :
+    a * (b + c) = a * b + a * c := by
+  induction' a with a s
+  induction' b with b t
+  induction' c with c u
+  rcases oreDivAddChar' b c t u with ⟨r₁, s₁, h₁, q⟩; rw [q]; clear q
+  simp only [OreLocalization.mul_def]
+  rcases oreDivAddChar' (a * b) (a * c) (t * s) (u * s) with ⟨r₂, s₂, h₂, q⟩; rw [q]; clear q
+  rw [oreDiv_eq_iff]
+  sorry
+
+instance : CommSemiring A[S⁻¹] where
+  __ := commMonoid
+  left_distrib := left_distrib'
+  right_distrib a b c := by
+    rw [mul_comm, mul_comm a, mul_comm b, left_distrib']
+  zero_mul a := by
+    induction' a with a s
+    rw [OreLocalization.zero_def, OreLocalization.mul_def]
+    simp
+  mul_zero a := by
+    induction' a with a s
+    rw [OreLocalization.zero_def, OreLocalization.mul_def]
+    simp
+
+end CommSemiring
+
+
+ -- fails on mathlib master;
+-- works if irreducibiliy of OreLocalization.smul is removed
+
+-- Next job: make API so I can prove `Algebra (R[S⁻¹]) A[S⁻¹]`
+-- Might also want `numeratorAlgHom : A →ₐ[R] A[S⁻¹]`
+-- Might also want some universal property a la
+/-
+variable {T : Type*} [Semiring T]
+variable (f : R →+* T) (fS : S →* Units T)
+variable (hf : ∀ s : S, f s = fS s)
+
+/-- The universal lift from a ring homomorphism `f : R →+* T`, which maps elements in `S` to
+units of `T`, to a ring homomorphism `R[S⁻¹] →+* T`. This extends the construction on
+monoids. -/
+def universalHom : R[S⁻¹] →+* T :=
+-/
+
+
+section first_goal
+
+variable {R A : Type*} [CommRing R] [CommRing A] [Algebra R A] {S : Submonoid R}
+
+abbrev SR := R[S⁻¹]
+abbrev SA := A[S⁻¹]
+
+
+--unseal OreLocalization.smul in
+--instance : Algebra (R[S⁻¹]) (A[S⁻¹]) where
+/-
+error:
+failed to synthesize
+  Semiring (OreLocalization S A)
+-/
+
+end first_goal
+
+end OreLocalization