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.

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

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

@[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 x => ¬↑(RingEquiv.symm (RingEquiv.symm (RingEquiv.ofBijective (algebraMap R A) h))) (↑(↑(LinearEquiv.restrictScalars A b.repr) a) x) = 0) (↑(LinearEquiv.restrictScalars A b.repr) a).support.val

@[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) (↑(↑(LinearEquiv.restrictScalars A b.repr) a) a_1)

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 = Basis.mapCoeffs b (RingEquiv.ofBijective (algebraMap R A) h) (_ : ∀ (c : R) (x : M), ↑(algebraMap R A) c • x = c • x)

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} [CommSemiring R] [Semiring S] [AddCommMonoid A] [Algebra 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 => b p.fst • c p.snd

noncomputable def Basis.smul {R : Type u} {S : Type v} {A : Type w} [CommSemiring R] [Semiring S] [AddCommMonoid A] [Algebra R S] [Module S A] [Module R A] [IsScalarTower R S A] {ι : Type v₁} {ι' : Type w₁} (b : Basis ι R S) (c : Basis ι' S A) : Basis (ι × ι') R A

Basis.SMul (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.

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

@[simp]

theorem Basis.smul_repr {R : Type u} {S : Type v} {A : Type w} [CommSemiring R] [Semiring S] [AddCommMonoid A] [Algebra R S] [Module S A] [Module R A] [IsScalarTower R S A] {ι : Type v₁} {ι' : Type w₁} (b : Basis ι R S) (c : Basis ι' S A) (x : A) (ij : ι × ι') : ↑(↑(Basis.smul b c).repr x) ij = ↑(↑b.repr (↑(↑c.repr x) ij.snd)) ij.fst

theorem Basis.smul_repr_mk {R : Type u} {S : Type v} {A : Type w} [CommSemiring R] [Semiring S] [AddCommMonoid A] [Algebra R S] [Module S A] [Module R A] [IsScalarTower R S A] {ι : Type v₁} {ι' : Type w₁} (b : Basis ι R S) (c : Basis ι' S A) (x : A) (i : ι) (j : ι') : ↑(↑(Basis.smul b c).repr x) (i, j) = ↑(↑b.repr (↑(↑c.repr x) j)) i

@[simp]

theorem Basis.smul_apply {R : Type u} {S : Type v} {A : Type w} [CommSemiring R] [Semiring S] [AddCommMonoid A] [Algebra R S] [Module S A] [Module R A] [IsScalarTower R S A] {ι : Type v₁} {ι' : Type w₁} (b : Basis ι R S) (c : Basis ι' S A) (ij : ι × ι') : ↑(Basis.smul b c) ij = ↑b ij.fst • ↑c ij.snd

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] (f : C →ₐ[A] D) [CommSemiring B] [Algebra A B] [Algebra B C] [IsScalarTower A B C] : B →ₐ[A] D

Restrict the domain of an AlgHom.

Equations

• AlgHom.restrictDomain B f = AlgHom.comp f (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] (f : C →ₐ[A] D) [CommSemiring B] [Algebra A B] [Algebra B C] [IsScalarTower A B C] : C →ₐ[B] D

Extend the scalars of an AlgHom.

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

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.