Towers of algebras

We set up the basic theory of algebra towers. An algebra tower A/S/R is expressed by having instances of Algebra A S, Algebra R S, Algebra R A and IsScalarTower R S A, the later asserting the compatibility condition (r • s) • a = r • (s • a).

In FieldTheory/Tower.lean we use this to prove the tower law for finite extensions, that if R and S are both fields, then [A:R] = [A:S] [S:A].

In this file we prepare the main lemma: if {bi | i ∈ I} is an R-basis of S and {cj | j ∈ J} is an S-basis of A, then {bi cj | i ∈ I, j ∈ J} is an R-basis of A. This statement does not require the base rings to be a field, so we also generalize the lemma to rings in this file.

def IsScalarTower.Invertible.algebraTower (R : Type u) (S : Type v) (A : Type w) [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A] (r : R) [Invertible ((algebraMap R S) r)] : Invertible ((algebraMap R A) r)

Suppose that R → S → A is a tower of algebras. If an element r : R is invertible in S, then it is invertible in A.

Equations

• IsScalarTower.Invertible.algebraTower R S A r = (Invertible.map (algebraMap S A) ((algebraMap R S) r)).copy ((algebraMap R A) r) ⋯

def IsScalarTower.invertibleAlgebraCoeNat (R : Type u) (A : Type w) [CommSemiring R] [Semiring A] [Algebra R A] (n : ℕ) [inv : Invertible ↑n] : Invertible ↑n

A natural number that is invertible when coerced to R is also invertible when coerced to any R-algebra.

Equations

• IsScalarTower.invertibleAlgebraCoeNat R A n = IsScalarTower.Invertible.algebraTower ℕ R A n

noncomputable def Basis.algebraMapCoeffs {R : Type u} (A : Type w) {ι : Type u_1} {M : Type u_2} [CommSemiring R] [Semiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] [IsScalarTower R A M] (b : Basis ι R M) (h : Function.Bijective ⇑(algebraMap R A)) : Basis ι A M

If R and A have a bijective algebraMap R A and act identically on M, then a basis for M as R-module is also a basis for M as R'-module.

Equations

• Basis.algebraMapCoeffs A b h = b.mapCoeffs (RingEquiv.ofBijective (algebraMap R A) h) ⋯

@[simp]

theorem Basis.algebraMapCoeffs_repr_apply_toFun {R : Type u} (A : Type w) {ι : Type u_1} {M : Type u_2} [CommSemiring R] [Semiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] [IsScalarTower R A M] (b : Basis ι R M) (h : Function.Bijective ⇑(algebraMap R A)) (a✝ : M) (a✝¹ : ι) : ((Basis.algebraMapCoeffs A b h).repr a✝) a✝¹ = (algebraMap R A) ((b.repr a✝) a✝¹)

@[simp]

theorem Basis.algebraMapCoeffs_repr_apply_support_val {R : Type u} (A : Type w) {ι : Type u_1} {M : Type u_2} [CommSemiring R] [Semiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] [IsScalarTower R A M] (b : Basis ι R M) (h : Function.Bijective ⇑(algebraMap R A)) (a✝ : M) : ((Basis.algebraMapCoeffs A b h).repr a✝).support.val = Multiset.filter (fun (a : ι) => ¬(algebraMap R A) ((b.repr a✝) a) = 0) (b.repr a✝).support.val

theorem Basis.algebraMapCoeffs_apply {R : Type u} (A : Type w) {ι : Type u_1} {M : Type u_2} [CommSemiring R] [Semiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] [IsScalarTower R A M] (b : Basis ι R M) (h : Function.Bijective ⇑(algebraMap R A)) (i : ι) : (Basis.algebraMapCoeffs A b h) i = b i

@[simp]

theorem Basis.coe_algebraMapCoeffs {R : Type u} (A : Type w) {ι : Type u_1} {M : Type u_2} [CommSemiring R] [Semiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] [IsScalarTower R A M] (b : Basis ι R M) (h : Function.Bijective ⇑(algebraMap R A)) : ⇑(Basis.algebraMapCoeffs A b h) = ⇑b

theorem linearIndependent_smul {R : Type u} {S : Type v} {A : Type w} [Ring R] [Ring S] [AddCommGroup A] [Module R S] [Module S A] [Module R A] [IsScalarTower R S A] {ι : Type v₁} {b : ι → S} {ι' : Type w₁} {c : ι' → A} (hb : LinearIndependent R b) (hc : LinearIndependent S c) : LinearIndependent R fun (p : ι × ι') => b p.1 • c p.2

theorem Basis.isScalarTower_of_nonempty (R : Type u) {S : Type v} {A : Type w} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R S] [Module S A] [Module R A] [IsScalarTower R S A] {ι : Type u_1} [Nonempty ι] (b : Basis ι S A) : IsScalarTower R S S

theorem Basis.isScalarTower_finsupp (R : Type u) {S : Type v} {A : Type w} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R S] [Module S A] [Module R A] [IsScalarTower R S A] {ι : Type u_1} (b : Basis ι S A) : IsScalarTower R S (ι →₀ S)

noncomputable def Basis.smulTower {R : Type u} {S : Type v} {A : Type w} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R S] [Module S A] [Module R A] [IsScalarTower R S A] {ι : Type u_1} {ι' : Type u_2} (b : Basis ι R S) (c : Basis ι' S A) : Basis (ι × ι') R A

Basis.smulTower (b : Basis ι R S) (c : Basis ι S A) is the R-basis on A where the (i, j)th basis vector is b i • c j.

Equations

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

@[simp]

theorem Basis.smulTower_repr {R : Type u} {S : Type v} {A : Type w} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R S] [Module S A] [Module R A] [IsScalarTower R S A] {ι : Type u_1} {ι' : Type u_2} (b : Basis ι R S) (c : Basis ι' S A) (x : A) (ij : ι × ι') : ((b.smulTower c).repr x) ij = (b.repr ((c.repr x) ij.2)) ij.1

theorem Basis.smulTower_repr_mk {R : Type u} {S : Type v} {A : Type w} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R S] [Module S A] [Module R A] [IsScalarTower R S A] {ι : Type u_1} {ι' : Type u_2} (b : Basis ι R S) (c : Basis ι' S A) (x : A) (i : ι) (j : ι') : ((b.smulTower c).repr x) (i, j) = (b.repr ((c.repr x) j)) i

@[simp]

theorem Basis.smulTower_apply {R : Type u} {S : Type v} {A : Type w} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R S] [Module S A] [Module R A] [IsScalarTower R S A] {ι : Type u_1} {ι' : Type u_2} (b : Basis ι R S) (c : Basis ι' S A) (ij : ι × ι') : (b.smulTower c) ij = b ij.1 • c ij.2

noncomputable def Basis.smulTower' {R : Type u} {S : Type v} {A : Type w} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R S] [Module S A] [Module R A] [IsScalarTower R S A] {ι : Type u_1} {ι' : Type u_2} (b : Basis ι R S) (c : Basis ι' S A) : Basis (ι' × ι) R A

Basis.smulTower (b : Basis ι R S) (c : Basis ι S A) is the R-basis on A where the (i, j)th basis vector is b j • c i.

Equations

• b.smulTower' c = (b.smulTower c).reindex (Equiv.prodComm ι ι')

theorem Basis.smulTower'_repr {R : Type u} {S : Type v} {A : Type w} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R S] [Module S A] [Module R A] [IsScalarTower R S A] {ι : Type u_1} {ι' : Type u_2} (b : Basis ι R S) (c : Basis ι' S A) (x : A) (ij : ι' × ι) : ((b.smulTower' c).repr x) ij = (b.repr ((c.repr x) ij.1)) ij.2

theorem Basis.smulTower'_repr_mk {R : Type u} {S : Type v} {A : Type w} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R S] [Module S A] [Module R A] [IsScalarTower R S A] {ι : Type u_1} {ι' : Type u_2} (b : Basis ι R S) (c : Basis ι' S A) (x : A) (i : ι') (j : ι) : ((b.smulTower' c).repr x) (i, j) = (b.repr ((c.repr x) i)) j

theorem Basis.smulTower'_apply {R : Type u} {S : Type v} {A : Type w} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R S] [Module S A] [Module R A] [IsScalarTower R S A] {ι : Type u_1} {ι' : Type u_2} (b : Basis ι R S) (c : Basis ι' S A) (ij : ι' × ι) : (b.smulTower' c) ij = b ij.2 • c ij.1

theorem Basis.algebraMap_injective {R : Type u} {S : Type v} [CommRing R] [Ring S] [Algebra R S] {ι : Type u_1} [NoZeroDivisors R] [Nontrivial S] (b : Basis ι R S) : Function.Injective ⇑(algebraMap R S)

def AlgHom.restrictDomain {A : Type w} (B : Type u₁) {C : Type u_1} {D : Type u_2} [CommSemiring A] [CommSemiring C] [CommSemiring D] [Algebra A C] [Algebra A D] [CommSemiring B] [Algebra A B] [Algebra B C] [IsScalarTower A B C] (f : C →ₐ[A] D) : B →ₐ[A] D

Restrict the domain of an AlgHom.

Equations

• AlgHom.restrictDomain B f = f.comp (IsScalarTower.toAlgHom A B C)

def AlgHom.extendScalars {A : Type w} (B : Type u₁) {C : Type u_1} {D : Type u_2} [CommSemiring A] [CommSemiring C] [CommSemiring D] [Algebra A C] [Algebra A D] [CommSemiring B] [Algebra A B] [Algebra B C] [IsScalarTower A B C] (f : C →ₐ[A] D) : C →ₐ[B] D

Extend the scalars of an AlgHom.

Equations

• AlgHom.extendScalars B f = { toFun := (↑↑f.toRingHom).toFun, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯, commutes' := ⋯ }

def algHomEquivSigma {A : Type w} {B : Type u₁} {C : Type u_1} {D : Type u_2} [CommSemiring A] [CommSemiring C] [CommSemiring D] [Algebra A C] [Algebra A D] [CommSemiring B] [Algebra A B] [Algebra B C] [IsScalarTower A B C] : (C →ₐ[A] D) ≃ (f : B →ₐ[A] D) × (C →ₐ[B] D)

AlgHoms from the top of a tower are equivalent to a pair of AlgHoms.

Equations

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