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Mathlib.RingTheory.Flat.EquationalCriterion

The equational criterion for flatness #

Let M be a module over a commutative ring R. Let us say that a relation iιfixi=0 in M is trivial (Module.IsTrivialRelation) if there exist a finite index type κ, elements (yj)jκ of M, and elements (aij)iι,jκ of R such that for all i, xi=jaijyj and for all j, ifiaij=0.

The equational criterion for flatness Stacks 00HK (Module.Flat.iff_forall_isTrivialRelation) states that M is flat if and only if every relation in M is trivial.

The equational criterion for flatness can be stated in the following form (Module.Flat.iff_forall_exists_factorization). Let M be an R-module. Then the following two conditions are equivalent:

Of course, the module Rι in this statement can be replaced by an arbitrary free module (Module.Flat.exists_factorization_of_apply_eq_zero_of_free).

We also have the following strengthening of the equational criterion for flatness (Module.Flat.exists_factorization_of_comp_eq_zero_of_free): Let M be a flat module. Let K and N be finite R-modules with N free, and let f:KN and x:NM be homomorphisms such that xf=0. Then there exist a finite index type κ and module homomorphisms a:NRκ and y:RκM such that x=ya and af=0. We recover the usual equational criterion for flatness if K=R and N=Rι. This is used in the proof of Lazard's theorem.

We conclude that every homomorphism from a finitely presented module to a flat module factors through a finite free module (Module.Flat.exists_factorization_of_isFinitelyPresented).

References #

@[reducible, inline]
abbrev Module.IsTrivialRelation {R : Type u} {M : Type u} [CommRing R] [AddCommGroup M] [Module R M] {ι : Type u} [Fintype ι] (f : ιR) (x : ιM) :

The proposition that the relation ifixi=0 in M is trivial. That is, there exist a finite index type κ, elements (yj)jκ of M, and elements (aij)iι,jκ of R such that for all i, xi=jaijyj and for all j, ifiaij=0. By Module.sum_smul_eq_zero_of_isTrivialRelation, this condition implies ifixi=0.

Equations
theorem Module.sum_smul_eq_zero_of_isTrivialRelation {R : Type u} {M : Type u} [CommRing R] [AddCommGroup M] [Module R M] {ι : Type u} [Fintype ι] {f : ιR} {x : ιM} (h : Module.IsTrivialRelation f x) :
i : ι, f i x i = 0

If the relation given by (fi)iι and (xi)iι is trivial, then ifixi is actually equal to 0.

theorem Module.Flat.tfae_equational_criterion (R : Type u) (M : Type u) [CommRing R] [AddCommGroup M] [Module R M] :
[Module.Flat R M, ∀ (I : Ideal R), Function.Injective (LinearMap.rTensor M (Submodule.subtype I)), ∀ {ι : Type u} [inst : Fintype ι] {f : ιR} {x : ιM}, i : ι, f i ⊗ₜ[R] x i = 0TensorProduct.VanishesTrivially R f x, ∀ {ι : Type u} [inst : Fintype ι] {f : ιR} {x : ιM}, i : ι, f i x i = 0Module.IsTrivialRelation f x, ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0∃ (κ : Type u) (x_1 : Fintype κ) (a : (ι →₀ R) →ₗ[R] κ →₀ R) (y : (κ →₀ R) →ₗ[R] M), x = y ∘ₗ a a f = 0, ∀ {N : Type u} [inst : AddCommGroup N] [inst_1 : Module R N] [inst_2 : Module.Free R N] [inst_3 : Module.Finite R N] {f : N} {x : N →ₗ[R] M}, x f = 0∃ (κ : Type u) (x_1 : Fintype κ) (a : N →ₗ[R] κ →₀ R) (y : (κ →₀ R) →ₗ[R] M), x = y ∘ₗ a a f = 0].TFAE

Equational criterion for flatness Stacks 00HK, combined form.

Let M be a module over a commutative ring R. The following are equivalent:

  • M is flat.
  • For all ideals IR, the map IMM is injective.
  • Every ifixi that vanishes in RM vanishes trivially.
  • Every relation ifixi=0 in M is trivial.
  • For all finite index types ι, all elements fRι, and all homomorphisms x:RιM such that x(f)=0, there exist a finite index type κ and module homomorphisms a:RιRκ and y:RκM such that x=ya and a(f)=0.
  • For all finite free modules N, all elements fN, and all homomorphisms x:NM such that x(f)=0, there exist a finite index type κ and module homomorphisms a:NRκ and y:RκM such that x=ya and a(f)=0.
theorem Module.Flat.iff_forall_isTrivialRelation {R : Type u} {M : Type u} [CommRing R] [AddCommGroup M] [Module R M] :
Module.Flat R M ∀ {ι : Type u} [inst : Fintype ι] {f : ιR} {x : ιM}, i : ι, f i x i = 0Module.IsTrivialRelation f x

Equational criterion for flatness Stacks 00HK.

A module M is flat if and only if every relation ifixi=0 in M is trivial.

theorem Module.Flat.isTrivialRelation_of_sum_smul_eq_zero {R : Type u} {M : Type u} [CommRing R] [AddCommGroup M] [Module R M] [Module.Flat R M] {ι : Type u} [Fintype ι] {f : ιR} {x : ιM} (h : i : ι, f i x i = 0) :

Equational criterion for flatness Stacks 00HK, forward direction.

If M is flat, then every relation ifixi=0 in M is trivial.

theorem Module.Flat.of_forall_isTrivialRelation {R : Type u} {M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (hfx : ∀ {ι : Type u} [inst : Fintype ι] {f : ιR} {x : ιM}, i : ι, f i x i = 0Module.IsTrivialRelation f x) :

Equational criterion for flatness Stacks 00HK, backward direction.

If every relation ifixi=0 in M is trivial, then M is flat.

theorem Module.Flat.iff_forall_exists_factorization {R : Type u} {M : Type u} [CommRing R] [AddCommGroup M] [Module R M] :
Module.Flat R M ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0∃ (κ : Type u) (x_1 : Fintype κ) (a : (ι →₀ R) →ₗ[R] κ →₀ R) (y : (κ →₀ R) →ₗ[R] M), x = y ∘ₗ a a f = 0

Equational criterion for flatness Stacks 00HK, alternate form.

A module M is flat if and only if for all finite free modules Rι, all fRι, and all homomorphisms x:RιM such that x(f)=0, there exist a finite free module Rκ and homomorphisms a:RιRκ and y:RκM such that x=ya and a(f)=0.

theorem Module.Flat.exists_factorization_of_apply_eq_zero {R : Type u} {M : Type u} [CommRing R] [AddCommGroup M] [Module R M] [Module.Flat R M] {ι : Type u} [Finite ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M} (h : x f = 0) :
∃ (κ : Type u) (x_1 : Fintype κ) (a : (ι →₀ R) →ₗ[R] κ →₀ R) (y : (κ →₀ R) →ₗ[R] M), x = y ∘ₗ a a f = 0

Equational criterion for flatness Stacks 00HK, forward direction, alternate form.

Let M be a flat module. Let Rι be a finite free module, let fRι be an element, and let x:RιM be a homomorphism such that x(f)=0. Then there exist a finite free module Rκ and homomorphisms a:RιRκ and y:RκM such that x=ya and a(f)=0.

theorem Module.Flat.of_forall_exists_factorization {R : Type u} {M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (h : ∀ {ι : Type u} [inst : Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0∃ (κ : Type u) (x_1 : Fintype κ) (a : (ι →₀ R) →ₗ[R] κ →₀ R) (y : (κ →₀ R) →ₗ[R] M), x = y ∘ₗ a a f = 0) :

Equational criterion for flatness Stacks 00HK, backward direction, alternate form.

Let M be a module over a commutative ring R. Suppose that for all finite free modules Rι, all fRι, and all homomorphisms x:RιM such that x(f)=0, there exist a finite free module Rκ and homomorphisms a:RιRκ and y:RκM such that x=ya and a(f)=0. Then M is flat.

theorem Module.Flat.exists_factorization_of_apply_eq_zero_of_free {R : Type u} {M : Type u} [CommRing R] [AddCommGroup M] [Module R M] [Module.Flat R M] {N : Type u} [AddCommGroup N] [Module R N] [Module.Free R N] [Module.Finite R N] {f : N} {x : N →ₗ[R] M} (h : x f = 0) :
∃ (κ : Type u) (x_1 : Fintype κ) (a : N →ₗ[R] κ →₀ R) (y : (κ →₀ R) →ₗ[R] M), x = y ∘ₗ a a f = 0

Equational criterion for flatness Stacks 00HK, forward direction, second alternate form.

Let M be a flat module over a commutative ring R. Let N be a finite free module over R, let fN, and let x:NM be a homomorphism such that x(f)=0. Then there exist a finite index type κ and module homomorphisms a:NRκ and y:RκM such that x=ya and a(f)=0.

theorem Module.Flat.exists_factorization_of_comp_eq_zero_of_free {R : Type u} {M : Type u} [CommRing R] [AddCommGroup M] [Module R M] [Module.Flat R M] {K : Type u} {N : Type u} [AddCommGroup K] [Module R K] [Module.Finite R K] [AddCommGroup N] [Module R N] [Module.Free R N] [Module.Finite R N] {f : K →ₗ[R] N} {x : N →ₗ[R] M} (h : x ∘ₗ f = 0) :
∃ (κ : Type u) (x_1 : Fintype κ) (a : N →ₗ[R] κ →₀ R) (y : (κ →₀ R) →ₗ[R] M), x = y ∘ₗ a a ∘ₗ f = 0

Let M be a flat module. Let K and N be finite R-modules with N free, and let f:KN and x:NM be homomorphisms such that xf=0. Then there exist a finite index type κ and module homomorphisms a:NRκ and y:RκM such that x=ya and af=0.

theorem Module.Flat.exists_factorization_of_isFinitelyPresented {R : Type u} {M : Type u} [CommRing R] [AddCommGroup M] [Module R M] [Module.Flat R M] {P : Type u} [AddCommGroup P] [Module R P] [Module.FinitePresentation R P] (h₁ : P →ₗ[R] M) :
∃ (κ : Type u) (x : Fintype κ) (h₂ : P →ₗ[R] κ →₀ R) (h₃ : (κ →₀ R) →ₗ[R] M), h₁ = h₃ ∘ₗ h₂

Every homomorphism from a finitely presented module to a flat module factors through a finite free module.