finish the first lemma

Whysoserioushah · Sep 26, 2024 · b85251f · b85251f
1 parent 953ca01
commit b85251f
Showing 1 changed file with 97 additions and 37 deletions.
diff --git a/BrauerGroup/DoubleCentralizer.lean b/BrauerGroup/DoubleCentralizer.lean
@@ -14,10 +14,101 @@ section lemma1
 variable {F A A' : Type u}
 variable [Field F] [Ring A] [Algebra F A] [Ring A'] [Algebra F A']
 variable (B : Subalgebra F A) (B' : Subalgebra F A')
+variable {ι ι' : Type*} (𝒜 : Basis ι F A) (𝒜' : Basis ι' F A')
 
-#check Subalgebra.mem_centralizer_iff
-lemma centralizer_tensor_centralizer
-    {ι ι' : Type*} (𝒜 : Basis ι F A) (𝒜' : Basis ι' F A') :
+include 𝒜' in
+lemma centralizer_inclusionLeft :
+    Subalgebra.centralizer F (A := A ⊗[F] A')
+      (Algebra.TensorProduct.includeLeft (R := F) (S := F) (A := A) (B := A') |>.comp
+        B.val).range =
+      (Algebra.TensorProduct.map (Subalgebra.centralizer F B).val (AlgHom.id F A')).range := by
+  refine le_antisymm ?_ ?_
+  · rintro x hx
+    obtain ⟨s, a, rfl⟩ := IsCentralSimple.TensorProduct.eq_repr_basis_right F A A' 𝒜' x
+    refine Subalgebra.sum_mem _ fun i hi => ?_
+    simp only [AlgHom.mem_range]
+    refine ⟨⟨a i, fun b hb => ?_⟩ ⊗ₜ 𝒜' i, by simp⟩
+    have eq := hx (b ⊗ₜ 1) (by simpa using ⟨b, hb, rfl⟩)
+    simp only [Finset.mul_sum, Algebra.TensorProduct.tmul_mul_tmul, one_mul, Finset.sum_mul,
+      mul_one] at eq
+    rw [← sub_eq_zero, ← Finset.sum_sub_distrib] at eq
+    simp_rw [← sub_tmul] at eq
+    have := IsCentralSimple.TensorProduct.sum_tmul_basis_right_eq_zero (h := eq)
+    specialize this i hi
+    rw [sub_eq_zero] at this
+    exact this
+
+  · rintro _ ⟨x, rfl⟩ _ ⟨y, rfl⟩
+    induction x using TensorProduct.induction_on with
+    | zero => simp
+    | tmul a b =>
+      simp only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe, AlgHom.coe_comp, Subalgebra.coe_val,
+        Function.comp_apply, Algebra.TensorProduct.includeLeft_apply,
+        Algebra.TensorProduct.map_tmul, AlgHom.coe_id, id_eq,
+        Algebra.TensorProduct.tmul_mul_tmul, one_mul, mul_one]
+      congr 1
+      exact a.2 _ y.2
+    | add x y hx hy =>
+      simp only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe, AlgHom.coe_comp, Subalgebra.coe_val,
+        Function.comp_apply, Algebra.TensorProduct.includeLeft_apply, map_add] at hx hy ⊢
+      simp [mul_add, hx, hy, add_mul]
+
+include 𝒜 in
+lemma centralizer_inclusionRight :
+    Subalgebra.centralizer F (A := A ⊗[F] A')
+      (Algebra.TensorProduct.includeRight (R := F) (A := A) (B := A') |>.comp
+        B'.val).range =
+      (Algebra.TensorProduct.map (AlgHom.id F A) (Subalgebra.centralizer F B').val).range := by
+  refine le_antisymm ?_ ?_
+  · rintro x hx
+    obtain ⟨s, a, rfl⟩ := IsCentralSimple.TensorProduct.eq_repr_basis_left F A A' 𝒜 x
+    refine Subalgebra.sum_mem _ fun i hi => ?_
+    simp only [AlgHom.mem_range]
+    refine ⟨𝒜 i ⊗ₜ ⟨a i, fun b hb => ?_⟩, by simp⟩
+    have eq := hx (1 ⊗ₜ b) (by simpa using ⟨b, hb, rfl⟩)
+    simp only [Finset.sum_mul, Algebra.TensorProduct.tmul_mul_tmul, one_mul, Finset.mul_sum,
+      mul_one] at eq
+    rw [← sub_eq_zero, ← Finset.sum_sub_distrib] at eq
+    simp_rw [← tmul_sub] at eq
+    have := IsCentralSimple.TensorProduct.sum_tmul_basis_left_eq_zero (h := eq)
+    specialize this i hi
+    rw [sub_eq_zero] at this
+    exact this
+
+  · rintro _ ⟨x, rfl⟩ _ ⟨y, rfl⟩
+    induction x using TensorProduct.induction_on with
+    | zero => simp
+    | tmul a b =>
+      simp only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe, AlgHom.coe_comp, Subalgebra.coe_val,
+        Function.comp_apply, Algebra.TensorProduct.includeRight_apply,
+        Algebra.TensorProduct.map_tmul, AlgHom.coe_id, id_eq,
+        Algebra.TensorProduct.tmul_mul_tmul, one_mul, mul_one]
+      congr 1
+      exact b.2 _ y.2
+    | add x y hx hy =>
+      simp only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe, AlgHom.coe_comp, Subalgebra.coe_val,
+        Function.comp_apply, Algebra.TensorProduct.includeRight_apply, map_add] at hx hy ⊢
+      simp [mul_add, hx, hy, add_mul]
+
+lemma centralizer_tensor_le_inf_centralizer :
+    Subalgebra.centralizer F (A := A ⊗[F] A')
+      ((Algebra.TensorProduct.map B.val B'.val).range :
+        Subalgebra F (A ⊗[F] A')) ≤
+    Subalgebra.centralizer F (A := A ⊗[F] A')
+      (Algebra.TensorProduct.includeLeft (R := F) (S := F) (A := A) (B := A') |>.comp
+        B.val).range ⊓
+    Subalgebra.centralizer F (A := A ⊗[F] A')
+      (Algebra.TensorProduct.includeRight (R := F) (A := A) (B := A') |>.comp
+        B'.val).range := by
+  intro x hx
+  rw [← IsCentralSimple.Subalgebra.centralizer_sup]
+  intro y hy
+  apply hx
+  rw [Algebra.TensorProduct.map_range]
+  exact hy
+
+include 𝒜 𝒜' in
+lemma centralizer_tensor_centralizer :
     Subalgebra.centralizer F (A := A ⊗[F] A')
       ((Algebra.TensorProduct.map B.val B'.val).range :
           Subalgebra F (A ⊗[F] A')) =
@@ -36,50 +127,19 @@ lemma centralizer_tensor_centralizer
         Subalgebra.centralizer F (A := A ⊗[F] A')
           (Algebra.TensorProduct.includeRight (R := F) (A := A) (B := A') |>.comp
             B'.val).range := by
-      sorry
+      apply centralizer_tensor_le_inf_centralizer
 
     have eq1 : Subalgebra.centralizer F (A := A ⊗[F] A')
           (Algebra.TensorProduct.includeLeft (R := F) (S := F) (A := A) (B := A') |>.comp
             B.val).range =
           (Algebra.TensorProduct.map (Subalgebra.centralizer F B).val (AlgHom.id F A')).range := by
-      refine le_antisymm ?_ ?_
-      · sorry
-
-      · rintro _ ⟨x, rfl⟩ _ ⟨y, rfl⟩
-        induction x using TensorProduct.induction_on with
-        | zero => simp
-        | tmul a b =>
-          simp only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe, AlgHom.coe_comp, Subalgebra.coe_val,
-            Function.comp_apply, Algebra.TensorProduct.includeLeft_apply,
-            Algebra.TensorProduct.map_tmul, AlgHom.coe_id, id_eq,
-            Algebra.TensorProduct.tmul_mul_tmul, one_mul, mul_one]
-          congr 1
-          exact a.2 _ y.2
-        | add x y hx hy =>
-          simp only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe, AlgHom.coe_comp, Subalgebra.coe_val,
-            Function.comp_apply, Algebra.TensorProduct.includeLeft_apply, map_add] at hx hy ⊢
-          simp [mul_add, hx, hy, add_mul]
+      apply centralizer_inclusionLeft (𝒜' := 𝒜')
 
     have eq2 : Subalgebra.centralizer F (A := A ⊗[F] A')
           (Algebra.TensorProduct.includeRight (R := F) (A := A) (B := A') |>.comp
             B'.val).range =
           (Algebra.TensorProduct.map (AlgHom.id F A) (Subalgebra.centralizer F B').val).range := by
-      refine le_antisymm ?_ ?_
-      · sorry
-      · rintro _ ⟨x, rfl⟩ _ ⟨y, rfl⟩
-        induction x using TensorProduct.induction_on with
-        | zero => simp
-        | tmul a b =>
-          simp only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe, AlgHom.coe_comp, Subalgebra.coe_val,
-            Function.comp_apply, Algebra.TensorProduct.includeRight_apply,
-            Algebra.TensorProduct.map_tmul, AlgHom.coe_id, id_eq,
-            Algebra.TensorProduct.tmul_mul_tmul, one_mul, mul_one]
-          congr 1
-          exact b.2 _ y.2
-        | add x y hx hy =>
-          simp only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe, AlgHom.coe_comp, Subalgebra.coe_val,
-            Function.comp_apply, Algebra.TensorProduct.includeRight_apply, map_add] at hx hy ⊢
-          simp [mul_add, hx, hy, add_mul]
+      apply centralizer_inclusionRight (𝒜 := 𝒜)
 
 
     refine ineq1.trans ?_