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feat: add semialgebra maps and refactor finiteadelering basechange
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| import Mathlib | ||
| import FLT.Mathlib.Algebra.Algebra.Hom | ||
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| variable {R S : Type*} [CommSemiring R] [CommSemiring S] {φ : R →+* S} | ||
| {A B : Type*} [CommSemiring A] [Semiring B] [Algebra R A] [Algebra S B] | ||
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| open scoped TensorProduct in | ||
| noncomputable | ||
| def SemialgHom.baseChange_of_algebraMap [Algebra R S] (ψ : A →ₛₐ[algebraMap R S] B) : | ||
| S ⊗[R] A →ₐ[S] B := | ||
| letI : Algebra R B := Algebra.compHom _ (algebraMap R S) | ||
| have : IsScalarTower R S B := .of_algebraMap_eq fun _ ↦ rfl | ||
| let ρ : A →ₐ[R] B := { | ||
| toRingHom := ψ.toRingHom | ||
| commutes' := ψ.commutes | ||
| } | ||
| Algebra.TensorProduct.lift (Algebra.ofId S _) ρ fun s a ↦ Algebra.commutes s (ρ a) |
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| import Mathlib.Algebra.Algebra.Hom | ||
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| section semialghom | ||
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| /-- Let `φ : R →+* S` be a ring homomorphism, let `A` be an `R`-algebra and let `B` be | ||
| an `S`-algebra. Then `SemialgHom φ A B` or `A →ₛₐ[φ] B` is the ring homomorphisms `ψ : A →+* B` | ||
| making lying above `φ` (i.e. such that `ψ (r • a) = φ r • ψ a`). | ||
| -/ | ||
| structure SemialgHom {R S : Type*} [CommSemiring R] [CommSemiring S] (φ : R →+* S) | ||
| (A B : Type*) [Semiring A] [Semiring B] [Algebra R A] [Algebra S B] | ||
| extends A →ₛₗ[φ] B, RingHom A B | ||
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| @[inherit_doc SemialgHom] | ||
| infixr:25 " →ₛₐ " => SemialgHom _ | ||
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| @[inherit_doc] | ||
| notation:25 A " →ₛₐ[" φ:25 "] " B:0 => SemialgHom φ A B | ||
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| variable {R S : Type*} [CommSemiring R] [CommSemiring S] (φ : R →+* S) | ||
| (A B : Type*) [Semiring A] [Semiring B] [Algebra R A] [Algebra S B] | ||
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| instance instFunLike : FunLike (A →ₛₐ[φ] B) A B where | ||
| coe f := f.toFun | ||
| coe_injective' f g h := by | ||
| cases f | ||
| cases g | ||
| congr | ||
| exact DFunLike.coe_injective' h | ||
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| variable {φ} {A} {B} in | ||
| lemma SemialgHom.map_smul (ψ : A →ₛₐ[φ] B) (m : R) (x : A) : ψ (m • x) = φ m • ψ x := | ||
| LinearMap.map_smul' ψ.toLinearMap m x | ||
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| end semialghom | ||
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| section semialghomclass | ||
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| class SemialgHomClass (F : Type*) {R S : outParam Type*} | ||
| [CommSemiring R] [CommSemiring S] (φ : outParam (R →+* S)) (A B : outParam Type*) | ||
| [Semiring A] [Semiring B] [Algebra R A] [Algebra S B] | ||
| [FunLike F A B] extends SemilinearMapClass F φ A B, RingHomClass F A B | ||
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| variable (F : Type*) {R S : Type*} | ||
| [CommSemiring R] [CommSemiring S] (φ : R →+* S) (A B : outParam Type*) | ||
| [Semiring A] [Semiring B] [Algebra R A] [Algebra S B] | ||
| [FunLike F A B] [SemialgHomClass F φ A B] | ||
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| instance SemialgHomClass.instSemialgHom : SemialgHomClass (A →ₛₐ[φ] B) φ A B where | ||
| map_add ψ := ψ.map_add | ||
| map_smulₛₗ ψ := ψ.map_smulₛₗ | ||
| map_mul ψ := ψ.map_mul | ||
| map_one ψ := ψ.map_one | ||
| map_zero ψ := ψ.map_zero | ||
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| end semialghomclass | ||
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| section semialghom | ||
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| variable {R S : Type*} [CommSemiring R] [CommSemiring S] {φ : R →+* S} | ||
| {A B : Type*} [Semiring A] [Semiring B] [Algebra R A] [Algebra S B] | ||
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| lemma SemialgHom.commutes (ψ : A →ₛₐ[φ] B) (r : R) : | ||
| ψ (algebraMap R A r) = algebraMap S B (φ r) := by | ||
| have := ψ.map_smul r 1 | ||
| rw [Algebra.smul_def, mul_one, map_one] at this | ||
| rw [this, Algebra.smul_def, mul_one] | ||
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| end semialghom |
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| import Mathlib.Algebra.Algebra.Pi | ||
| import FLT.Mathlib.Algebra.Algebra.Hom | ||
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| def Pi.semialgHom {I : Type*} {R S : Type*} (f : I → Type*) [CommSemiring R] [CommSemiring S] | ||
| (φ : R →+* S) [s : (i : I) → Semiring (f i)] [(i : I) → Algebra S (f i)] {A : Type*} | ||
| [Semiring A] [Algebra R A] (g : (i : I) → A →ₛₐ[φ] f i) : | ||
| A →ₛₐ[φ] (i : I) → f i where | ||
| __ := Pi.ringHom fun i ↦ (g i).toRingHom | ||
| map_smul' r a := by ext; simp |
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