Additively-graded algebra structures on ⨁ i, A i

This file provides R-algebra structures on external direct sums of R-modules.

Recall that if A i are a family of AddCommMonoids indexed by an AddMonoid, then an instance of DirectSum.GMonoid A is a multiplication A i → A j → A (i + j) giving ⨁ i, A i the structure of a semiring. In this file, we introduce the DirectSum.GAlgebra R A class for the case where all A i are R-modules. This is the extra structure needed to promote ⨁ i, A i to an R-algebra.

Main definitions

• DirectSum.GAlgebra R A, the typeclass.
• DirectSum.toAlgebra extends DirectSum.toSemiring to produce an AlgHom.

class DirectSum.GAlgebra {ι : Type uι} (R : Type uR) (A : ι → Type uA) [DecidableEq ι] [CommSemiring R] [(i : ι) → AddCommMonoid (A i)] [(i : ι) → Module R (A i)] [AddMonoid ι] [DirectSum.GSemiring A] : Type (max uA uR)

• toFun : R →+ A 0
• map_one : ↑DirectSum.GAlgebra.toFun 1 = GradedMonoid.GOne.one
• map_mul : ∀ (r s_1 : R), GradedMonoid.mk 0 (↑DirectSum.GAlgebra.toFun (r * s_1)) = { fst := 0 + 0, snd := GradedMonoid.GMul.mul (↑DirectSum.GAlgebra.toFun r) (↑DirectSum.GAlgebra.toFun s_1) }
• commutes : ∀ (r : R) (x : GradedMonoid A), GradedMonoid.mk 0 (↑DirectSum.GAlgebra.toFun r) * x = x * { fst := 0, snd := ↑DirectSum.GAlgebra.toFun r }
• smul_def : ∀ (r : R) (x : GradedMonoid A), GradedMonoid.mk x.fst (r • x.snd) = { fst := 0, snd := ↑DirectSum.GAlgebra.toFun r } * x

A graded version of Algebra. An instance of DirectSum.GAlgebra R A endows (⨁ i, A i) with an R-algebra structure.

instance DirectSum.instAlgebraDirectSumSemiring {ι : Type uι} (R : Type uR) (A : ι → Type uA) [DecidableEq ι] [CommSemiring R] [(i : ι) → AddCommMonoid (A i)] [(i : ι) → Module R (A i)] [AddMonoid ι] [DirectSum.GSemiring A] [DirectSum.GAlgebra R A] : Algebra R (⨁ (i : ι), A i)

Equations

• One or more equations did not get rendered due to their size.

theorem DirectSum.algebraMap_apply {ι : Type uι} (R : Type uR) (A : ι → Type uA) [DecidableEq ι] [CommSemiring R] [(i : ι) → AddCommMonoid (A i)] [(i : ι) → Module R (A i)] [AddMonoid ι] [DirectSum.GSemiring A] [DirectSum.GAlgebra R A] (r : R) : ↑(algebraMap R (⨁ (i : ι), A i)) r = ↑(DirectSum.of A 0) (↑DirectSum.GAlgebra.toFun r)

theorem DirectSum.algebraMap_toAddMonoid_hom {ι : Type uι} (R : Type uR) (A : ι → Type uA) [DecidableEq ι] [CommSemiring R] [(i : ι) → AddCommMonoid (A i)] [(i : ι) → Module R (A i)] [AddMonoid ι] [DirectSum.GSemiring A] [DirectSum.GAlgebra R A] : ↑(algebraMap R (⨁ (i : ι), A i)) = AddMonoidHom.comp (DirectSum.of A 0) DirectSum.GAlgebra.toFun

@[simp]

theorem DirectSum.toAlgebra_apply {ι : Type uι} (R : Type uR) (A : ι → Type uA) {B : Type uB} [DecidableEq ι] [CommSemiring R] [(i : ι) → AddCommMonoid (A i)] [(i : ι) → Module R (A i)] [AddMonoid ι] [DirectSum.GSemiring A] [Semiring B] [DirectSum.GAlgebra R A] [Algebra R B] (f : (i : ι) → A i →ₗ[R] B) (hone : ↑(f 0) GradedMonoid.GOne.one = 1) (hmul : ∀ {i j : ι} (ai : A i) (aj : A j), ↑(f (i + j)) (GradedMonoid.GMul.mul ai aj) = ↑(f i) ai * ↑(f j) aj) (hcommutes : ∀ (r : R), ↑(f 0) (↑DirectSum.GAlgebra.toFun r) = ↑(algebraMap R B) r) (a : ⨁ (i : ι), A i) : ↑(DirectSum.toAlgebra R A f hone hmul hcommutes) a = ↑(DirectSum.toSemiring (fun i => LinearMap.toAddMonoidHom (f i)) hone hmul) a

def DirectSum.toAlgebra {ι : Type uι} (R : Type uR) (A : ι → Type uA) {B : Type uB} [DecidableEq ι] [CommSemiring R] [(i : ι) → AddCommMonoid (A i)] [(i : ι) → Module R (A i)] [AddMonoid ι] [DirectSum.GSemiring A] [Semiring B] [DirectSum.GAlgebra R A] [Algebra R B] (f : (i : ι) → A i →ₗ[R] B) (hone : ↑(f 0) GradedMonoid.GOne.one = 1) (hmul : ∀ {i j : ι} (ai : A i) (aj : A j), ↑(f (i + j)) (GradedMonoid.GMul.mul ai aj) = ↑(f i) ai * ↑(f j) aj) (hcommutes : ∀ (r : R), ↑(f 0) (↑DirectSum.GAlgebra.toFun r) = ↑(algebraMap R B) r) : (⨁ (i : ι), A i) →ₐ[R] B

A family of LinearMaps preserving DirectSum.GOne.one and DirectSum.GMul.mul describes an AlgHom on ⨁ i, A i. This is a stronger version of DirectSum.toSemiring.

Of particular interest is the case when A i are bundled subojects, f is the family of coercions such as Submodule.subtype (A i), and the [GMonoid A] structure originates from DirectSum.GMonoid.ofAddSubmodules, in which case the proofs about GOne and GMul can be discharged by rfl.

Equations

• One or more equations did not get rendered due to their size.

theorem DirectSum.algHom_ext' {ι : Type uι} (R : Type uR) (A : ι → Type uA) {B : Type uB} [DecidableEq ι] [CommSemiring R] [(i : ι) → AddCommMonoid (A i)] [(i : ι) → Module R (A i)] [AddMonoid ι] [DirectSum.GSemiring A] [Semiring B] [DirectSum.GAlgebra R A] [Algebra R B] ⦃f : (⨁ (i : ι), A i) →ₐ[R] B⦄ ⦃g : (⨁ (i : ι), A i) →ₐ[R] B⦄ (h : ∀ (i : ι), LinearMap.comp (AlgHom.toLinearMap f) (DirectSum.lof R ι A i) = LinearMap.comp (AlgHom.toLinearMap g) (DirectSum.lof R ι A i)) : f = g

Two AlgHoms out of a direct sum are equal if they agree on the generators.

See note [partially-applied ext lemmas].

theorem DirectSum.algHom_ext {ι : Type uι} (R : Type uR) (A : ι → Type uA) {B : Type uB} [DecidableEq ι] [CommSemiring R] [(i : ι) → AddCommMonoid (A i)] [(i : ι) → Module R (A i)] [AddMonoid ι] [DirectSum.GSemiring A] [Semiring B] [DirectSum.GAlgebra R A] [Algebra R B] ⦃f : (⨁ (i : ι), A i) →ₐ[R] B⦄ ⦃g : (⨁ (i : ι), A i) →ₐ[R] B⦄ (h : ∀ (i : ι) (x : A i), ↑f (↑(DirectSum.of A i) x) = ↑g (↑(DirectSum.of A i) x)) : f = g

Concrete instances

@[simp]

theorem Algebra.directSumGAlgebra_toFun_apply {ι : Type uι} {R : Type u_1} {A : Type u_2} [DecidableEq ι] [AddMonoid ι] [CommSemiring R] [Semiring A] [Algebra R A] : ∀ (a : R), ↑DirectSum.GAlgebra.toFun a = OneHom.toFun (↑↑(algebraMap R A)) a

instance Algebra.directSumGAlgebra {ι : Type uι} {R : Type u_1} {A : Type u_2} [DecidableEq ι] [AddMonoid ι] [CommSemiring R] [Semiring A] [Algebra R A] : DirectSum.GAlgebra R fun x => A

A direct sum of copies of an Algebra inherits the algebra structure.

Equations

• One or more equations did not get rendered due to their size.