feat(LinearAlgebra/QuadraticForm/TensorProduct): base change of quadratic forms (#6636)

Author: eric-wieser

Mathlib.lean

@@ -2262,6 +2262,7 @@ import Mathlib.LinearAlgebra.QuadraticForm.Complex
 import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
 import Mathlib.LinearAlgebra.QuadraticForm.Prod
 import Mathlib.LinearAlgebra.QuadraticForm.Real
+import Mathlib.LinearAlgebra.QuadraticForm.TensorProduct
 import Mathlib.LinearAlgebra.Quotient
 import Mathlib.LinearAlgebra.QuotientPi
 import Mathlib.LinearAlgebra.Ray

Mathlib/LinearAlgebra/QuadraticForm/Basic.lean

@@ -883,6 +883,11 @@ theorem associated_linMulLin (f g : M →ₗ[R₁] R₁) :
   ring
 #align quadratic_form.associated_lin_mul_lin QuadraticForm.associated_linMulLin
 
+@[simp]
+lemma associated_sq : associated (R₁ := R₁) sq = LinearMap.toBilin (LinearMap.mul R₁ R₁) :=
+  (associated_linMulLin (LinearMap.id) (LinearMap.id)).trans <|
+    by simp only [smul_add, invOf_two_smul_add_invOf_two_smul]; rfl
+
 end Associated
 
 section Anisotropic

Mathlib/LinearAlgebra/QuadraticForm/TensorProduct.lean

@@ -0,0 +1,90 @@
+/-
+Copyright (c) 2023 Eric Wieser. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eric Wieser
+-/
+import Mathlib.LinearAlgebra.BilinearForm.TensorProduct
+import Mathlib.LinearAlgebra.QuadraticForm.Basic
+
+/-!
+# The quadratic form on a tensor product
+
+## Main definitions
+
+* `QuadraticForm.tensorDistrib (Q₁ ⊗ₜ Q₂)`: the quadratic form on `M₁ ⊗ M₂` constructed by applying
+  `Q₁` on `M₁` and `Q₂` on `M₂`. This construction is not available in characteristic two.
+
+-/
+
+universe uR uA uM₁ uM₂
+
+variable {R : Type uR} {A : Type uA} {M₁ : Type uM₁} {M₂ : Type uM₂}
+
+open TensorProduct
+
+namespace QuadraticForm
+
+section CommRing
+variable [CommRing R] [CommRing A]
+variable [AddCommGroup M₁] [AddCommGroup M₂]
+variable [Algebra R A] [Module R M₁] [Module A M₁]
+variable [SMulCommClass R A M₁] [SMulCommClass A R M₁] [IsScalarTower R A M₁]
+variable [Module R M₂] [Invertible (2 : R)]
+
+
+variable (R A) in
+/-- The tensor product of two quadratic forms injects into quadratic forms on tensor products.
+
+Note this is heterobasic; the quadratic form on the left can take values in a larger ring than
+the one on the right. -/
+def tensorDistrib : QuadraticForm A M₁ ⊗[R] QuadraticForm R M₂ →ₗ[A] QuadraticForm A (M₁ ⊗[R] M₂) :=
+  letI : Invertible (2 : A) := (Invertible.map (algebraMap R A) 2).copy 2 (map_ofNat _ _).symm
+  -- while `letI`s would produce a better term than `let`, they would make this already-slow
+  -- definition even slower.
+  let toQ := BilinForm.toQuadraticFormLinearMap A A (M₁ ⊗[R] M₂)
+  let tmulB := BilinForm.tensorDistrib R A (M₁ := M₁) (M₂ := M₂)
+  let toB := AlgebraTensorModule.map
+      (QuadraticForm.associated : QuadraticForm A M₁ →ₗ[A] BilinForm A M₁)
+      (QuadraticForm.associated : QuadraticForm R M₂ →ₗ[R] BilinForm R M₂)
+  toQ ∘ₗ tmulB ∘ₗ toB
+
+-- TODO: make the RHS `MulOpposite.op (Q₂ m₂) • Q₁ m₁` so that this has a nicer defeq for
+-- `R = A` of `Q₁ m₁ * Q₂ m₂`.
+@[simp]
+theorem tensorDistrib_tmul (Q₁ : QuadraticForm A M₁) (Q₂ : QuadraticForm R M₂) (m₁ : M₁) (m₂ : M₂) :
+    tensorDistrib R A (Q₁ ⊗ₜ Q₂) (m₁ ⊗ₜ m₂) = Q₂ m₂ • Q₁ m₁ :=
+  letI : Invertible (2 : A) := (Invertible.map (algebraMap R A) 2).copy 2 (map_ofNat _ _).symm
+  (BilinForm.tensorDistrib_tmul _ _ _ _ _ _).trans <| congr_arg₂ _
+    (associated_eq_self_apply _ _ _) (associated_eq_self_apply _ _ _)
+
+/-- The tensor product of two quadratic forms, a shorthand for dot notation. -/
+protected abbrev tmul (Q₁ : QuadraticForm A M₁) (Q₂ : QuadraticForm R M₂) :
+    QuadraticForm A (M₁ ⊗[R] M₂) :=
+  tensorDistrib R A (Q₁ ⊗ₜ[R] Q₂)
+
+theorem associated_tmul [Invertible (2 : A)] (Q₁ : QuadraticForm A M₁) (Q₂ : QuadraticForm R M₂) :
+    associated (R₁ := A) (Q₁.tmul Q₂)
+      = (associated (R₁ := A) Q₁).tmul (associated (R₁ := R) Q₂) := by
+  rw [QuadraticForm.tmul, tensorDistrib, BilinForm.tmul]
+  dsimp
+  convert associated_left_inverse A ((associated_isSymm A Q₁).tmul (associated_isSymm R Q₂))
+
+variable (A) in
+/-- The base change of a quadratic form. -/
+protected def baseChange (Q : QuadraticForm R M₂) : QuadraticForm A (A ⊗[R] M₂) :=
+  QuadraticForm.tmul (R := R) (A := A) (M₁ := A) (M₂ := M₂) (QuadraticForm.sq (R := A)) Q
+
+@[simp]
+theorem baseChange_tmul (Q : QuadraticForm R M₂) (a : A) (m₂ : M₂) :
+    Q.baseChange A (a ⊗ₜ m₂) = Q m₂ • (a * a) :=
+  tensorDistrib_tmul _ _ _ _
+
+
+theorem associated_baseChange [Invertible (2 : A)] (Q : QuadraticForm R M₂)  :
+    associated (R₁ := A) (Q.baseChange A) = (associated (R₁ := R) Q).baseChange A := by
+  dsimp only [QuadraticForm.baseChange, BilinForm.baseChange]
+  rw [associated_tmul (QuadraticForm.sq (R := A)) Q, associated_sq]
+
+end CommRing
+
+end QuadraticForm