# 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) [] [] [] [Algebra R S] [Algebra S A] [Algebra R A] [] (r : R) [Invertible (() r)] :
Invertible (() 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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Instances For
def IsScalarTower.invertibleAlgebraCoeNat (R : Type u) (A : Type w) [] [] [Algebra R A] (n : ) [inv : ] :

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

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@[simp]
theorem Basis.algebraMapCoeffs_repr_apply_toFun {R : Type u} (A : Type w) {ι : Type u_1} {M : Type u_2} [] [] [] [Algebra R A] [Module A M] [Module R M] [] (b : Basis ι R M) (h : Function.Bijective ()) :
∀ (a : M) (a_1 : ι), (().repr a) a_1 = () ((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} [] [] [] [Algebra R A] [Module A M] [Module R M] [] (b : Basis ι R M) (h : Function.Bijective ()) :
∀ (a : M), (().repr a).support.val = Multiset.filter (fun (x : ι) => ¬().symm.symm ((b.repr a) x) = 0) (b.repr a).support.val
noncomputable def Basis.algebraMapCoeffs {R : Type u} (A : Type w) {ι : Type u_1} {M : Type u_2} [] [] [] [Algebra R A] [Module A M] [Module R M] [] (b : Basis ι R M) (h : Function.Bijective ()) :
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.

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• = b.mapCoeffs ()
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theorem Basis.algebraMapCoeffs_apply {R : Type u} (A : Type w) {ι : Type u_1} {M : Type u_2} [] [] [] [Algebra R A] [Module A M] [Module R M] [] (b : Basis ι R M) (h : Function.Bijective ()) (i : ι) :
() i = b i
@[simp]
theorem Basis.coe_algebraMapCoeffs {R : Type u} (A : Type w) {ι : Type u_1} {M : Type u_2} [] [] [] [Algebra R A] [Module A M] [Module R M] [] (b : Basis ι R M) (h : Function.Bijective ()) :
() = b
theorem linearIndependent_smul {R : Type u} {S : Type v} {A : Type w} [] [] [] [Module R S] [Module S A] [Module R A] [] {ι : Type v₁} {b : ιS} {ι' : Type w₁} {c : ι'A} (hb : ) (hc : ) :
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} [] [] [] [Module R S] [Module S A] [Module R A] [] {ι : Type u_1} [] (b : Basis ι S A) :
theorem Basis.isScalarTower_finsupp (R : Type u) {S : Type v} {A : Type w} [] [] [] [Module R S] [Module S A] [Module R A] [] {ι : Type u_1} (b : Basis ι S A) :
noncomputable def Basis.smul {R : Type u} {S : Type v} {A : Type w} [] [] [] [Module R S] [Module S A] [Module R 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.
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@[simp]
theorem Basis.smul_repr {R : Type u} {S : Type v} {A : Type w} [] [] [] [Module R S] [Module S A] [Module R A] [] {ι : Type v₁} {ι' : Type w₁} (b : Basis ι R S) (c : Basis ι' S A) (x : A) (ij : ι × ι') :
((b.smul c).repr x) ij = (b.repr ((c.repr x) ij.2)) ij.1
theorem Basis.smul_repr_mk {R : Type u} {S : Type v} {A : Type w} [] [] [] [Module R S] [Module S A] [Module R A] [] {ι : Type v₁} {ι' : Type w₁} (b : Basis ι R S) (c : Basis ι' S A) (x : A) (i : ι) (j : ι') :
((b.smul 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} [] [] [] [Module R S] [Module S A] [Module R A] [] {ι : Type v₁} {ι' : Type w₁} (b : Basis ι R S) (c : Basis ι' S A) (ij : ι × ι') :
(b.smul c) ij = b ij.1 c ij.2
theorem Basis.algebraMap_injective {R : Type u} {S : Type v} [] [Ring S] [Algebra R S] {ι : Type u_1} [] [] (b : Basis ι R S) :
def AlgHom.restrictDomain {A : Type w} (B : Type u₁) {C : Type u_1} {D : Type u_2} [] [] [] [Algebra A C] [Algebra A D] (f : C →ₐ[A] D) [] [Algebra A B] [Algebra B C] [] :

Restrict the domain of an AlgHom.

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

Extend the scalars of an AlgHom.

Equations
• = let __spread.0 := ().toAlgebra; { toFun := (f.toRingHom).toFun, map_one' := , map_mul' := , map_zero' := , map_add' := , commutes' := }
Instances For
def algHomEquivSigma {A : Type w} {B : Type u₁} {C : Type u_1} {D : Type u_2} [] [] [] [Algebra A C] [Algebra A D] [] [Algebra A B] [Algebra 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.
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