formal/lean/mathlib/linear_algebra/invariant_basis_number.lean

/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Scott Morrison
-/
import ring_theory.ideal.quotient
import ring_theory.principal_ideal_domain

/-!
# Invariant basis number property

We say that a ring `R` satisfies the invariant basis number property if there is a well-defined
notion of the rank of a finitely generated free (left) `R`-module. Since a finitely generated free
module with a basis consisting of `n` elements is linearly equivalent to `fin n → R`, it is
sufficient that `(fin n → R) ≃ₗ[R] (fin m → R)` implies `n = m`.

It is also useful to consider two stronger conditions, namely the rank condition,
that a surjective linear map `(fin n → R) →ₗ[R] (fin m → R)` implies `m ≤ n` and
the strong rank condition, that an injective linear map `(fin n → R) →ₗ[R] (fin m → R)`
implies `n ≤ m`.

The strong rank condition implies the rank condition, and the rank condition implies
the invariant basis number property.

## Main definitions

`strong_rank_condition R` is a type class stating that `R` satisfies the strong rank condition.
`rank_condition R` is a type class stating that `R` satisfies the rank condition.
`invariant_basis_number R` is a type class stating that `R` has the invariant basis number property.

## Main results

We show that every nontrivial left-noetherian ring satisfies the strong rank condition,
(and so in particular every division ring or field),
and then use this to show every nontrivial commutative ring has the invariant basis number property.

More generally, every commutative ring satisfies the strong rank condition. This is proved in
`linear_algebra/free_module/strong_rank_condition`. We keep
`invariant_basis_number_of_nontrivial_of_comm_ring` here since it imports fewer files.

## Future work

So far, there is no API at all for the `invariant_basis_number` class. There are several natural
ways to formulate that a module `M` is finitely generated and free, for example
`M ≃ₗ[R] (fin n → R)`, `M ≃ₗ[R] (ι → R)`, where `ι` is a fintype, or providing a basis indexed by
a finite type. There should be lemmas applying the invariant basis number property to each
situation.

The finite version of the invariant basis number property implies the infinite analogue, i.e., that
`(ι →₀ R) ≃ₗ[R] (ι' →₀ R)` implies that `cardinal.mk ι = cardinal.mk ι'`. This fact (and its
variants) should be formalized.

## References

* https://en.wikipedia.org/wiki/Invariant_basis_number
* https://mathoverflow.net/a/2574/

## Tags

free module, rank, invariant basis number, IBN

-/

noncomputable theory

open_locale classical big_operators
open function

universes u v w

section
variables (R : Type u) [semiring R]

/-- We say that `R` satisfies the strong rank condition if `(fin n → R) →ₗ[R] (fin m → R)` injective
    implies `n ≤ m`. -/
@[mk_iff]
class strong_rank_condition : Prop :=
(le_of_fin_injective : ∀ {n m : ℕ} (f : (fin n → R) →ₗ[R] (fin m → R)), injective f → n ≤ m)

lemma le_of_fin_injective [strong_rank_condition R] {n m : ℕ} (f : (fin n → R) →ₗ[R] (fin m → R)) :
  injective f → n ≤ m :=
strong_rank_condition.le_of_fin_injective f

/-- A ring satisfies the strong rank condition if and only if, for all `n : ℕ`, any linear map
`(fin (n + 1) → R) →ₗ[R] (fin n → R)` is not injective. -/
lemma strong_rank_condition_iff_succ : strong_rank_condition R ↔
  ∀ (n : ℕ) (f : (fin (n + 1) → R) →ₗ[R] (fin n → R)), ¬function.injective f :=
begin
  refine ⟨λ h n, λ f hf, _, λ h, ⟨λ n m f hf, _⟩⟩,
  { letI : strong_rank_condition R := h,
    exact nat.not_succ_le_self n (le_of_fin_injective R f hf) },
  { by_contra H,
    exact h m (f.comp (function.extend_by_zero.linear_map R (fin.cast_le (not_le.1 H))))
      (hf.comp (function.extend_injective (rel_embedding.injective _) 0)) }
end

lemma card_le_of_injective [strong_rank_condition R] {α β : Type*} [fintype α] [fintype β]
  (f : (α → R) →ₗ[R] (β → R)) (i : injective f) : fintype.card α ≤ fintype.card β :=
begin
  let P := linear_equiv.fun_congr_left R R (fintype.equiv_fin α),
  let Q := linear_equiv.fun_congr_left R R (fintype.equiv_fin β),
  exact le_of_fin_injective R ((Q.symm.to_linear_map.comp f).comp P.to_linear_map)
    (((linear_equiv.symm Q).injective.comp i).comp (linear_equiv.injective P)),
end

lemma card_le_of_injective' [strong_rank_condition R] {α β : Type*} [fintype α] [fintype β]
  (f : (α →₀ R) →ₗ[R] (β →₀ R)) (i : injective f) : fintype.card α ≤ fintype.card β :=
begin
  let P := (finsupp.linear_equiv_fun_on_fintype R R β),
  let Q := (finsupp.linear_equiv_fun_on_fintype R R α).symm,
  exact card_le_of_injective R ((P.to_linear_map.comp f).comp Q.to_linear_map)
    ((P.injective.comp i).comp Q.injective)
end

/-- We say that `R` satisfies the rank condition if `(fin n → R) →ₗ[R] (fin m → R)` surjective
    implies `m ≤ n`. -/
class rank_condition : Prop :=
(le_of_fin_surjective : ∀ {n m : ℕ} (f : (fin n → R) →ₗ[R] (fin m → R)), surjective f → m ≤ n)

lemma le_of_fin_surjective [rank_condition R] {n m : ℕ} (f : (fin n → R) →ₗ[R] (fin m → R)) :
  surjective f → m ≤ n :=
rank_condition.le_of_fin_surjective f

lemma card_le_of_surjective [rank_condition R] {α β : Type*} [fintype α] [fintype β]
  (f : (α → R) →ₗ[R] (β → R)) (i : surjective f) : fintype.card β ≤ fintype.card α :=
begin
  let P := linear_equiv.fun_congr_left R R (fintype.equiv_fin α),
  let Q := linear_equiv.fun_congr_left R R (fintype.equiv_fin β),
  exact le_of_fin_surjective R ((Q.symm.to_linear_map.comp f).comp P.to_linear_map)
    (((linear_equiv.symm Q).surjective.comp i).comp (linear_equiv.surjective P)),
end

lemma card_le_of_surjective' [rank_condition R] {α β : Type*} [fintype α] [fintype β]
  (f : (α →₀ R) →ₗ[R] (β →₀ R)) (i : surjective f) : fintype.card β ≤ fintype.card α :=
begin
  let P := (finsupp.linear_equiv_fun_on_fintype R R β),
  let Q := (finsupp.linear_equiv_fun_on_fintype R R α).symm,
  exact card_le_of_surjective R ((P.to_linear_map.comp f).comp Q.to_linear_map)
    ((P.surjective.comp i).comp Q.surjective)
end

/--
By the universal property for free modules, any surjective map `(fin n → R) →ₗ[R] (fin m → R)`
has an injective splitting `(fin m → R) →ₗ[R] (fin n → R)`
from which the strong rank condition gives the necessary inequality for the rank condition.
-/
@[priority 100]
instance rank_condition_of_strong_rank_condition [strong_rank_condition R] : rank_condition R :=
{ le_of_fin_surjective := λ n m f s,
    le_of_fin_injective R _ (f.splitting_of_fun_on_fintype_surjective_injective s), }

/-- We say that `R` has the invariant basis number property if `(fin n → R) ≃ₗ[R] (fin m → R)`
    implies `n = m`. This gives rise to a well-defined notion of rank of a finitely generated free
    module. -/
class invariant_basis_number : Prop :=
(eq_of_fin_equiv : ∀ {n m : ℕ}, ((fin n → R) ≃ₗ[R] (fin m → R)) → n = m)

@[priority 100]
instance invariant_basis_number_of_rank_condition [rank_condition R] : invariant_basis_number R :=
{ eq_of_fin_equiv := λ n m e, le_antisymm
    (le_of_fin_surjective R e.symm.to_linear_map e.symm.surjective)
    (le_of_fin_surjective R e.to_linear_map e.surjective) }

end

section
variables (R : Type u) [semiring R] [invariant_basis_number R]

lemma eq_of_fin_equiv {n m : ℕ} : ((fin n → R) ≃ₗ[R] (fin m → R)) → n = m :=
invariant_basis_number.eq_of_fin_equiv

lemma card_eq_of_lequiv {α β : Type*} [fintype α] [fintype β]
  (f : (α → R) ≃ₗ[R] (β → R)) : fintype.card α = fintype.card β :=
eq_of_fin_equiv R (((linear_equiv.fun_congr_left R R (fintype.equiv_fin α)).trans f) ≪≫ₗ
  ((linear_equiv.fun_congr_left R R (fintype.equiv_fin β)).symm))

lemma nontrivial_of_invariant_basis_number : nontrivial R :=
begin
  by_contra h,
  refine zero_ne_one (eq_of_fin_equiv R _),
  haveI := not_nontrivial_iff_subsingleton.1 h,
  haveI : subsingleton (fin 1 → R) := ⟨λ a b, funext $ λ x, subsingleton.elim _ _⟩,
  refine { .. }; { intros, exact 0 } <|> tidy
end

end

section
variables (R : Type u) [ring R] [nontrivial R] [is_noetherian_ring R]

/--
Any nontrivial noetherian ring satisfies the strong rank condition.

An injective map `((fin n ⊕ fin (1 + m)) → R) →ₗ[R] (fin n → R)` for some left-noetherian `R`
would force `fin (1 + m) → R ≃ₗ punit` (via `is_noetherian.equiv_punit_of_prod_injective`),
which is not the case!
-/
-- Note this includes fields,
-- and we use this below to show any commutative ring has invariant basis number.
@[priority 100]
instance noetherian_ring_strong_rank_condition : strong_rank_condition R :=
begin
  fsplit,
  intros m n f i,
  by_contradiction h,
  rw [not_le, ←nat.add_one_le_iff, le_iff_exists_add] at h,
  obtain ⟨m, rfl⟩ := h,
  let e : fin (n + 1 + m) ≃ fin n ⊕ fin (1 + m) :=
    (fin_congr (add_assoc _ _ _)).trans fin_sum_fin_equiv.symm,
  let f' := f.comp ((linear_equiv.sum_arrow_lequiv_prod_arrow _ _ R R).symm.trans
    (linear_equiv.fun_congr_left R R e)).to_linear_map,
  have i' : injective f' := i.comp (linear_equiv.injective _),
  apply @zero_ne_one (fin (1 + m) → R) _ _,
  apply (is_noetherian.equiv_punit_of_prod_injective f' i').injective,
  ext,
end

end

/-!
  We want to show that nontrivial commutative rings have invariant basis number. The idea is to
  take a maximal ideal `I` of `R` and use an isomorphism `R^n ≃ R^m` of `R` modules to produce an
  isomorphism `(R/I)^n ≃ (R/I)^m` of `R/I`-modules, which will imply `n = m` since `R/I` is a field
  and we know that fields have invariant basis number.

  We construct the isomorphism in two steps:
  1. We construct the ring `R^n/I^n`, show that it is an `R/I`-module and show that there is an
     isomorphism of `R/I`-modules `R^n/I^n ≃ (R/I)^n`. This isomorphism is called
    `ideal.pi_quot_equiv` and is located in the file `ring_theory/ideals.lean`.
  2. We construct an isomorphism of `R/I`-modules `R^n/I^n ≃ R^m/I^m` using the isomorphism
     `R^n ≃ R^m`.
-/

section
variables {R : Type u} [comm_ring R] (I : ideal R) {ι : Type v} [fintype ι] {ι' : Type w}

/-- An `R`-linear map `R^n → R^m` induces a function `R^n/I^n → R^m/I^m`. -/
private def induced_map (I : ideal R) (e : (ι → R) →ₗ[R] (ι' → R)) :
  (ι → R) ⧸ (I.pi ι) → (ι' → R) ⧸ I.pi ι' :=
λ x, quotient.lift_on' x (λ y, ideal.quotient.mk _ (e y))
begin
  refine λ a b hab, ideal.quotient.eq.2 (λ h, _),
  rw submodule.quotient_rel_r_def at hab,
  rw ←linear_map.map_sub,
  exact ideal.map_pi _ _ hab e h,
end

/-- An isomorphism of `R`-modules `R^n ≃ R^m` induces an isomorphism of `R/I`-modules
    `R^n/I^n ≃ R^m/I^m`. -/
private def induced_equiv [fintype ι'] (I : ideal R) (e : (ι → R) ≃ₗ[R] (ι' → R)) :
  ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] (ι' → R) ⧸ I.pi ι' :=
begin
  refine { to_fun := induced_map I e, inv_fun := induced_map I e.symm, .. },
  all_goals { rintro ⟨a⟩ ⟨b⟩ <|> rintro ⟨a⟩,
    change ideal.quotient.mk _ _ = ideal.quotient.mk _ _,
    congr, simp }
end

end

section
local attribute [instance] ideal.quotient.field

/-- Nontrivial commutative rings have the invariant basis number property.

In fact, any nontrivial commutative ring satisfies the strong rank condition, see
`comm_ring_strong_rank_condition`. We prove this instance separately to avoid dependency on
`linear_algebra.charpoly.basic`. -/
@[priority 100]
instance invariant_basis_number_of_nontrivial_of_comm_ring {R : Type u} [comm_ring R]
  [nontrivial R] : invariant_basis_number R :=
⟨λ n m e, let ⟨I, hI⟩ := ideal.exists_maximal R in
  by exactI eq_of_fin_equiv (R ⧸ I)
    ((ideal.pi_quot_equiv _ _).symm ≪≫ₗ ((induced_equiv _ e) ≪≫ₗ (ideal.pi_quot_equiv _ _)))⟩

end