# Documentation

Mathlib.RingTheory.AlgebraTower

# 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.

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.

Instances For
@[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 => ¬↑() (↑(↑(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} [] [] [] [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 = ↑() (↑(↑(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} [] [] [] [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.

Instances For
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} [] [] [] [Algebra 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 => b p.fst c p.snd
noncomputable def Basis.smul {R : Type u} {S : Type v} {A : Type w} [] [] [] [Algebra 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.

Instances For
@[simp]
theorem Basis.smul_repr {R : Type u} {S : Type v} {A : Type w} [] [] [] [Algebra 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') :
↑(().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} [] [] [] [Algebra 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') :
↑(().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} [] [] [] [Algebra 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') :
↑() ij = b ij.fst c ij.snd
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.

Instances For
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.

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.

Instances For