mathlib documentation

ring_theory.algebra_tower

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 is_scalar_tower R S A, the later asserting the compatibility condition (r • s) • a = r • (s • a).

In field_theory/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 a 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.

source
def is_scalar_tower.invertible.algebra_tower (R : Type u) (S : Type v) (A : Type w) [comm_semiring R] [comm_semiring S] [semiring A] [algebra R S] [algebra S A] [algebra R A] [is_scalar_tower R S A] (r : R) [invertible (⇑(algebra_map R S) r)] :
invertible (⇑(algebra_map 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
source
def is_scalar_tower.invertible_algebra_coe_nat (R : Type u) (A : Type w) [comm_semiring 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
source
theorem algebra.adjoin_algebra_map' {R : Type u} {S : Type v} {A : Type w} [comm_ring R] [comm_ring S] [comm_ring A] [algebra R S] [algebra S A] (s : set S) :
algebra.adjoin R (⇑(algebra_map S (restrict_scalars R S A)) '' s) = (algebra.adjoin R s).map (alg_hom.restrict_scalars R (algebra.of_id S (restrict_scalars R S A)))
source
theorem algebra.adjoin_algebra_map (R : Type u) (S : Type v) (A : Type w) [comm_ring R] [comm_ring S] [comm_ring A] [algebra R S] [algebra S A] [algebra R A] [is_scalar_tower R S A] (s : set S) :
algebra.adjoin R (⇑(algebra_map S A) '' s) = (algebra.adjoin R s).map (is_scalar_tower.to_alg_hom R S A)
source
theorem algebra.adjoin_res (C : Type u_1) (D : Type u_2) (E : Type u_3) [comm_semiring C] [comm_semiring D] [comm_semiring E] [algebra C D] [algebra C E] [algebra D E] [is_scalar_tower C D E] (S : set E) :
subalgebra.res C (algebra.adjoin D S) = (⊤.map (is_scalar_tower.to_alg_hom C D E)).under (algebra.adjoin ↥(⊤.map (is_scalar_tower.to_alg_hom C D E)) S)
source
theorem algebra.adjoin_res_eq_adjoin_res (C : Type u_1) (D : Type u_2) (E : Type u_3) (F : Type u_4) [comm_semiring C] [comm_semiring D] [comm_semiring E] [comm_semiring F] [algebra C D] [algebra C E] [algebra C F] [algebra D F] [algebra E F] [is_scalar_tower C D F] [is_scalar_tower C E F] {S : set D} {T : set E} (hS : algebra.adjoin C S = ⊤) (hT : algebra.adjoin C T = ⊤) :
subalgebra.res C (algebra.adjoin E (⇑(algebra_map D F) '' S)) = subalgebra.res C (algebra.adjoin D (⇑(algebra_map E F) '' T))
source
theorem algebra.fg_trans' {R : Type u_1} {S : Type u_2} {A : Type u_3} [comm_ring R] [comm_ring S] [comm_ring A] [algebra R S] [algebra S A] [algebra R A] [is_scalar_tower R S A] (hRS : ⊤.fg) (hSA : ⊤.fg) :
⊤.fg
source
theorem linear_independent_smul {R : Type u} {S : Type v} {A : Type w} [comm_ring R] [ring S] [add_comm_group A] [algebra R S] [module S A] [module R A] [is_scalar_tower R S A] {ι : Type v₁} {b : ι → S} {ι' : Type w₁} {c : ι' → A} (hb : linear_independent R b) (hc : linear_independent S c) :
linear_independent R (λ (p : ι × ι'), b p.fst • c p.snd)
source
def basis.smul {R : Type u} {S : Type v} {A : Type w} [comm_ring R] [ring S] [add_comm_group A] [algebra R S] [module S A] [module R A] [is_scalar_tower 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.

Equations
source
@[simp]
theorem basis.smul_repr {R : Type u} {S : Type v} {A : Type w} [comm_ring R] [ring S] [add_comm_group A] [algebra R S] [module S A] [module R A] [is_scalar_tower R S 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.snd)) ij.fst
source
theorem basis.smul_repr_mk {R : Type u} {S : Type v} {A : Type w} [comm_ring R] [ring S] [add_comm_group A] [algebra R S] [module S A] [module R A] [is_scalar_tower 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') :
⇑(⇑((b.smul c).repr) x) (i, j) = ⇑(⇑(b.repr) (⇑(⇑(c.repr) x) j)) i
source
@[simp]
theorem basis.smul_apply {R : Type u} {S : Type v} {A : Type w} [comm_ring R] [ring S] [add_comm_group A] [algebra R S] [module S A] [module R A] [is_scalar_tower R S 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.fst • ⇑c ij.snd
source
theorem exists_subalgebra_of_fg (A : Type w) (B : Type u₁) (C : Type u_1) [comm_ring A] [comm_ring B] [comm_ring C] [algebra A B] [algebra B C] [algebra A C] [is_scalar_tower A B C] (hAC : ⊤.fg) (hBC : ⊤.fg) :
∃ (B₀ : subalgebra A B), B₀.fg ∧ ⊤.fg
source
theorem fg_of_fg_of_fg (A : Type w) (B : Type u₁) (C : Type u_1) [comm_ring A] [comm_ring B] [comm_ring C] [algebra A B] [algebra B C] [algebra A C] [is_scalar_tower A B C] [is_noetherian_ring A] (hAC : ⊤.fg) (hBC : ⊤.fg) (hBCi : function.injective ⇑(algebra_map B C)) :
⊤.fg

Artin--Tate lemma: if A ⊆ B ⊆ C is a chain of subrings of commutative rings, and A is noetherian, and C is algebra-finite over A, and C is module-finite over B, then B is algebra-finite over A.

References: Atiyah--Macdonald Proposition 7.8; Stacks 00IS; Altman--Kleiman 16.17.

source
def alg_hom.restrict_domain {A : Type w} (B : Type u₁) {C : Type u_1} {D : Type u_2} [comm_semiring A] [comm_semiring C] [comm_semiring D] [algebra A C] [algebra A D] (f : C →ₐ[A] D) [comm_semiring B] [algebra A B] [algebra B C] [is_scalar_tower A B C] :
B →ₐ[A] D

Restrict the domain of an alg_hom.

Equations
source
def alg_hom.extend_scalars {A : Type w} (B : Type u₁) {C : Type u_1} {D : Type u_2} [comm_semiring A] [comm_semiring C] [comm_semiring D] [algebra A C] [algebra A D] (f : C →ₐ[A] D) [comm_semiring B] [algebra A B] [algebra B C] [is_scalar_tower A B C] :
C →ₐ[B] D

Extend the scalars of an alg_hom.

Equations
source
def alg_hom_equiv_sigma {A : Type w} {B : Type u₁} {C : Type u_1} {D : Type u_2} [comm_semiring A] [comm_semiring C] [comm_semiring D] [algebra A C] [algebra A D] [comm_semiring B] [algebra A B] [algebra B C] [is_scalar_tower A B C] :
(C →ₐ[A] D) ≃ Σ (f : B →ₐ[A] D), C →ₐ[B] D

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

Equations