Action of the polynomial ring on module induced by an algebra element.

Given an element a in an R-algebra A and an A-module M we define an R[X]-module Module.AEval R M a, which is a type synonym of M with the action of a polynomial f given by f • m = Polynomial.aeval a f • m. In particular X • m = a • m.

In the special case that A = M →ₗ[R] M and φ : M →ₗ[R] M, the module Module.AEval R M a is abbreviated Module.AEval' φ. In this module we have X • m = ↑φ m.

def Module.AEval (R : Type u_1) (M : Type u_2) {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] : A → Type u_2

Suppose a is an element of an R-algebra A and M is an A-module. Loosely speaking, Module.AEval R M a is the R[X]-module with elements m : M, where the action of a polynomial $f$ is given by $f • m = f(a) • m$.

More precisely, Module.AEval R M a has elements Module.AEval.of R M a m for m : M, and the action of f is f • (of R M a m) = of R M a ((aeval a f) • m).

Equations

• Module.AEval R M x✝ = M

instance Module.AEval.instAddCommGroup {R : Type u_1} {A : Type u_2} {M : Type u_3} [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommGroup M] [Module A M] [Module R M] [IsScalarTower R A M] : AddCommGroup (AEval R M a)

Equations

• Module.AEval.instAddCommGroup a = inferInstanceAs (AddCommGroup M)

instance Module.AEval.instAddCommMonoid {R : Type u_1} {A : Type u_2} {M : Type u_3} [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] : AddCommMonoid (AEval R M a)

Equations

• Module.AEval.instAddCommMonoid a = inferInstanceAs (AddCommMonoid M)

instance Module.AEval.instModuleOrig {R : Type u_1} {A : Type u_2} {M : Type u_3} [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] : Module R (AEval R M a)

Equations

• Module.AEval.instModuleOrig a = inferInstanceAs (Module R M)

instance Module.AEval.instFiniteOrig {R : Type u_1} {A : Type u_3} {M : Type u_2} [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] [Module.Finite R M] : Module.Finite R (AEval R M a)

instance Module.AEval.instModulePolynomial {R : Type u_1} {A : Type u_2} {M : Type u_3} [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] : Module (Polynomial R) (AEval R M a)

Equations

• Module.AEval.instModulePolynomial a = Module.compHom M (Polynomial.aeval a).toRingHom

def Module.AEval.of (R : Type u_1) {A : Type u_2} (M : Type u_3) [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] : M ≃ₗ[R] AEval R M a

The canonical linear equivalence between M and Module.AEval R M a as an R-module.

Equations

• Module.AEval.of R M a = LinearEquiv.refl R M

theorem Module.AEval.of_aeval_smul {R : Type u_1} {A : Type u_3} {M : Type u_2} [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] (f : Polynomial R) (m : M) : (of R M a) ((Polynomial.aeval a) f • m) = f • (of R M a) m

@[simp]

theorem Module.AEval.of_symm_smul {R : Type u_1} {A : Type u_3} {M : Type u_2} [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] (f : Polynomial R) (m : AEval R M a) : (of R M a).symm (f • m) = (Polynomial.aeval a) f • (of R M a).symm m

@[simp]

theorem Module.AEval.C_smul {R : Type u_1} {A : Type u_3} {M : Type u_2} [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] (t : R) (m : AEval R M a) : Polynomial.C t • m = t • m

theorem Module.AEval.X_smul_of {R : Type u_2} {A : Type u_3} {M : Type u_1} [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] (m : M) : Polynomial.X • (of R M a) m = (of R M a) (a • m)

theorem Module.AEval.of_symm_X_smul {R : Type u_1} {A : Type u_3} {M : Type u_2} [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] (m : AEval R M a) : (of R M a).symm (Polynomial.X • m) = a • (of R M a).symm m

instance Module.AEval.instIsScalarTowerOrigPolynomial {R : Type u_1} {A : Type u_3} {M : Type u_2} [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] : IsScalarTower R (Polynomial R) (AEval R M a)

instance Module.AEval.instFinitePolynomial {R : Type u_1} {A : Type u_3} {M : Type u_2} [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] [Module.Finite R M] : Module.Finite (Polynomial R) (AEval R M a)

def LinearMap.ofAEval {R : Type u_1} {A : Type u_2} {M : Type u_3} [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] {N : Type u_4} [AddCommMonoid N] [Module R N] [Module (Polynomial R) N] [IsScalarTower R (Polynomial R) N] (f : M →ₗ[R] N) (hf : ∀ (m : M), f (a • m) = Polynomial.X • f m) : Module.AEval R M a →ₗ[Polynomial R] N

Construct an R[X]-linear map out of AEval R M a from a R-linear map out of M.

Equations

• LinearMap.ofAEval a f hf = { toAddHom := (f ∘ₗ ↑(Module.AEval.of R M a).symm).toAddHom, map_smul' := ⋯ }

def LinearEquiv.ofAEval {R : Type u_1} {A : Type u_2} {M : Type u_3} [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] {N : Type u_4} [AddCommMonoid N] [Module R N] [Module (Polynomial R) N] [IsScalarTower R (Polynomial R) N] (f : M ≃ₗ[R] N) (hf : ∀ (m : M), f (a • m) = Polynomial.X • f m) : Module.AEval R M a ≃ₗ[Polynomial R] N

Construct an R[X]-linear equivalence out of AEval R M a from a R-linear map out of M.

Equations

• LinearEquiv.ofAEval a f hf = { toLinearMap := LinearMap.ofAEval a (↑f) hf, invFun := ⇑(Module.AEval.of R M a) ∘ ⇑f.symm, left_inv := ⋯, right_inv := ⋯ }

theorem Module.AEval.annihilator_eq_ker_aeval {R : Type u_3} {A : Type u_1} {M : Type u_2} [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] [FaithfulSMul A M] : annihilator (Polynomial R) (AEval R M a) = RingHom.ker (Polynomial.aeval a)

@[simp]

theorem Module.AEval.annihilator_top_eq_ker_aeval {R : Type u_3} {A : Type u_1} {M : Type u_2} [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] [FaithfulSMul A M] : ⊤.annihilator = RingHom.ker (Polynomial.aeval a)

def Module.AEval.mapSubmodule (R : Type u_1) {A : Type u_2} (M : Type u_3) [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] : ↥((Algebra.lsmul R R M) a).invtSubmodule ≃o Submodule (Polynomial R) (AEval R M a)

The natural order isomorphism between the two ways to represent invariant submodules.

Equations

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

@[simp]

theorem Module.AEval.mem_mapSubmodule_apply (R : Type u_2) {A : Type u_3} (M : Type u_1) [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] {p : ↥((Algebra.lsmul R R M) a).invtSubmodule} {m : AEval R M a} : m ∈ (mapSubmodule R M a) p ↔ (of R M a).symm m ∈ ↑p

@[simp]

theorem Module.AEval.mem_mapSubmodule_symm_apply (R : Type u_1) {A : Type u_3} (M : Type u_2) [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] {q : Submodule (Polynomial R) (AEval R M a)} {m : M} : m ∈ ↑((mapSubmodule R M a).symm q) ↔ (of R M a) m ∈ q

def Module.AEval.equiv_mapSubmodule {R : Type u_1} {A : Type u_2} {M : Type u_3} [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] (p : Submodule R M) (hp : p ∈ ((Algebra.lsmul R R M) a).invtSubmodule) : ↥p ≃ₗ[R] ↥((mapSubmodule R M a) ⟨p, hp⟩)

The natural R-linear equivalence between the two ways to represent an invariant submodule.

Equations

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

noncomputable def Module.AEval.restrict_equiv_mapSubmodule {R : Type u_1} {A : Type u_2} {M : Type u_3} [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] (p : Submodule R M) (hp : p ∈ ((Algebra.lsmul R R M) a).invtSubmodule) : AEval R (↥p) (LinearMap.restrict ((Algebra.lsmul R R M) a) hp) ≃ₗ[Polynomial R] ↥((mapSubmodule R M a) ⟨p, hp⟩)

The natural R[X]-linear equivalence between the two ways to represent an invariant submodule.

Equations

• Module.AEval.restrict_equiv_mapSubmodule a p hp = LinearEquiv.ofAEval (LinearMap.restrict ((Algebra.lsmul R R M) a) hp) (Module.AEval.equiv_mapSubmodule a p hp) ⋯

@[reducible, inline]

abbrev Module.AEval' {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (φ : M →ₗ[R] M) : Type u_2

Given and R-module M and a linear map φ : M →ₗ[R] M, Module.AEval' φ is loosely speaking the R[X]-module with elements m : M, where the action of a polynomial $f$ is given by $f • m = f(a) • m$.

More precisely, Module.AEval' φ has elements Module.AEval'.of φ m for m : M, and the action of f is f • (of φ m) = of φ ((aeval φ f) • m).

Module.AEval' is defined as a special case of Module.AEval in which the R-algebra is M →ₗ[R] M. Lemmas involving Module.AEval may be applied to Module.AEval'.

Equations

• Module.AEval' φ = Module.AEval R M φ

@[reducible, inline]

abbrev Module.AEval'.of {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (φ : M →ₗ[R] M) : M ≃ₗ[R] AEval' φ

The canonical linear equivalence between M and Module.AEval' φ as an R-module, where φ : M →ₗ[R] M.

Equations

• Module.AEval'.of φ = Module.AEval.of R M φ

theorem Module.AEval'_def {R : Type u_2} {M : Type u_1} [CommSemiring R] [AddCommMonoid M] [Module R M] (φ : M →ₗ[R] M) : AEval' φ = AEval R M φ

theorem Module.AEval'.X_smul_of {R : Type u_2} {M : Type u_1} [CommSemiring R] [AddCommMonoid M] [Module R M] (φ : M →ₗ[R] M) (m : M) : Polynomial.X • (of φ) m = (of φ) (φ m)

theorem Module.AEval'.of_symm_X_smul {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (φ : M →ₗ[R] M) (m : AEval' φ) : (of φ).symm (Polynomial.X • m) = φ ((of φ).symm m)

instance Module.instFinitePolynomialAEval' {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (φ : M →ₗ[R] M) [Module.Finite R M] : Module.Finite (Polynomial R) (AEval' φ)