Finiteness conditions in commutative algebra

In this file we define a notion of finiteness that is common in commutative algebra.

Main declarations

• Submodule.FG, Ideal.FG These express that some object is finitely generated as submodule over some base ring.

• Module.Finite, RingHom.Finite, AlgHom.Finite all of these express that some object is finitely generated as module over some base ring.

Main results

• exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul is Nakayama's lemma, in the following form: if N is a finitely generated submodule of an ambient R-module M and I is an ideal of R such that N ⊆ IN, then there exists r ∈ 1 + I such that rN = 0.

def Submodule.FG {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] (N : Submodule R M) : Prop

A submodule of M is finitely generated if it is the span of a finite subset of M.

Equations

• Submodule.FG N = ∃ S, Submodule.span R ↑S = N

theorem Submodule.fg_def {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {N : Submodule R M} : Submodule.FG N ↔ ∃ S, Set.Finite S ∧ Submodule.span R S = N

theorem Submodule.fg_iff_addSubmonoid_fg {M : Type u_2} [AddCommMonoid M] (P : Submodule ℕ M) : Submodule.FG P ↔ AddSubmonoid.FG P.toAddSubmonoid

theorem Submodule.fg_iff_add_subgroup_fg {G : Type u_3} [AddCommGroup G] (P : Submodule ℤ G) : Submodule.FG P ↔ AddSubgroup.FG (Submodule.toAddSubgroup P)

theorem Submodule.fg_iff_exists_fin_generating_family {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {N : Submodule R M} : Submodule.FG N ↔ ∃ n s, Submodule.span R (Set.range s) = N

theorem Submodule.exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul {R : Type u_3} [CommRing R] {M : Type u_4} [AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : Submodule.FG N) (hin : N ≤ I • N) : ∃ r, r - 1 ∈ I ∧ ∀ (n : M), n ∈ N → r • n = 0

Nakayama's Lemma. Atiyah-Macdonald 2.5, Eisenbud 4.7, Matsumura 2.2, Stacks 00DV

theorem Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul {R : Type u_3} [CommRing R] {M : Type u_4} [AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : Submodule.FG N) (hin : N ≤ I • N) : ∃ r, r ∈ I ∧ ∀ (n : M), n ∈ N → r • n = n

theorem Submodule.fg_bot {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] : Submodule.FG ⊥

theorem Subalgebra.fg_bot_toSubmodule {R : Type u_3} {A : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] : Submodule.FG (↑Subalgebra.toSubmodule ⊥)

theorem Submodule.fg_unit {R : Type u_3} {A : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] (I : (Submodule R A)ˣ) : Submodule.FG ↑I

theorem Submodule.fg_of_isUnit {R : Type u_3} {A : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] {I : Submodule R A} (hI : IsUnit I) : Submodule.FG I

theorem Submodule.fg_span {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {s : Set M} (hs : Set.Finite s) : Submodule.FG (Submodule.span R s)

theorem Submodule.fg_span_singleton {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] (x : M) : Submodule.FG (Submodule.span R {x})

theorem Submodule.FG.sup {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {N₁ : Submodule R M} {N₂ : Submodule R M} (hN₁ : Submodule.FG N₁) (hN₂ : Submodule.FG N₂) : Submodule.FG (N₁ ⊔ N₂)

theorem Submodule.fg_finset_sup {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {ι : Type u_3} (s : Finset ι) (N : ι → Submodule R M) (h : ∀ (i : ι), i ∈ s → Submodule.FG (N i)) : Submodule.FG (Finset.sup s N)

theorem Submodule.fg_biSup {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {ι : Type u_3} (s : Finset ι) (N : ι → Submodule R M) (h : ∀ (i : ι), i ∈ s → Submodule.FG (N i)) : Submodule.FG (⨆ (i : ι) (_ : i ∈ s), N i)

theorem Submodule.fg_iSup {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {ι : Type u_3} [Finite ι] (N : ι → Submodule R M) (h : ∀ (i : ι), Submodule.FG (N i)) : Submodule.FG (iSup N)

theorem Submodule.FG.map {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {P : Type u_3} [AddCommMonoid P] [Module R P] (f : M →ₗ[R] P) {N : Submodule R M} (hs : Submodule.FG N) : Submodule.FG (Submodule.map f N)

theorem Submodule.fg_of_fg_map_injective {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {P : Type u_3} [AddCommMonoid P] [Module R P] (f : M →ₗ[R] P) (hf : Function.Injective ↑f) {N : Submodule R M} (hfn : Submodule.FG (Submodule.map f N)) : Submodule.FG N

theorem Submodule.fg_of_fg_map {R : Type u_4} {M : Type u_5} {P : Type u_6} [Ring R] [AddCommGroup M] [Module R M] [AddCommGroup P] [Module R P] (f : M →ₗ[R] P) (hf : LinearMap.ker f = ⊥) {N : Submodule R M} (hfn : Submodule.FG (Submodule.map f N)) : Submodule.FG N

theorem Submodule.fg_top {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] (N : Submodule R M) : Submodule.FG ⊤ ↔ Submodule.FG N

theorem Submodule.fg_of_linearEquiv {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {P : Type u_3} [AddCommMonoid P] [Module R P] (e : M ≃ₗ[R] P) (h : Submodule.FG ⊤) : Submodule.FG ⊤

theorem Submodule.FG.prod {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {P : Type u_3} [AddCommMonoid P] [Module R P] {sb : Submodule R M} {sc : Submodule R P} (hsb : Submodule.FG sb) (hsc : Submodule.FG sc) : Submodule.FG (Submodule.prod sb sc)

theorem Submodule.fg_pi {R : Type u_1} [Semiring R] {ι : Type u_4} {M : ι → Type u_5} [Finite ι] [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] {p : (i : ι) → Submodule R (M i)} (hsb : ∀ (i : ι), Submodule.FG (p i)) : Submodule.FG (Submodule.pi Set.univ p)

theorem Submodule.fg_of_fg_map_of_fg_inf_ker {R : Type u_4} {M : Type u_5} {P : Type u_6} [Ring R] [AddCommGroup M] [Module R M] [AddCommGroup P] [Module R P] (f : M →ₗ[R] P) {s : Submodule R M} (hs1 : Submodule.FG (Submodule.map f s)) (hs2 : Submodule.FG (s ⊓ LinearMap.ker f)) : Submodule.FG s

If 0 → M' → M → M'' → 0 is exact and M' and M'' are finitely generated then so is M.

theorem Submodule.fg_induction (R : Type u_4) (M : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] (P : Submodule R M → Prop) (h₁ : (x : M) → P (Submodule.span R {x})) (h₂ : (M₁ M₂ : Submodule R M) → P M₁ → P M₂ → P (M₁ ⊔ M₂)) (N : Submodule R M) (hN : Submodule.FG N) : P N

theorem Submodule.fg_ker_comp {R : Type u_4} {M : Type u_5} {N : Type u_6} {P : Type u_7} [Ring R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [AddCommGroup P] [Module R P] (f : M →ₗ[R] N) (g : N →ₗ[R] P) (hf1 : Submodule.FG (LinearMap.ker f)) (hf2 : Submodule.FG (LinearMap.ker g)) (hsur : Function.Surjective ↑f) : Submodule.FG (LinearMap.ker (LinearMap.comp g f))

The kernel of the composition of two linear maps is finitely generated if both kernels are and the first morphism is surjective.

theorem Submodule.fg_restrictScalars {R : Type u_4} {S : Type u_5} {M : Type u_6} [CommSemiring R] [Semiring S] [Algebra R S] [AddCommGroup M] [Module S M] [Module R M] [IsScalarTower R S M] (N : Submodule S M) (hfin : Submodule.FG N) (h : Function.Surjective ↑(algebraMap R S)) : Submodule.FG (Submodule.restrictScalars R N)

theorem Submodule.FG.stablizes_of_iSup_eq {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {M' : Submodule R M} (hM' : Submodule.FG M') (N : ℕ →o Submodule R M) (H : iSup ↑N = M') : ∃ n, M' = ↑N n

theorem Submodule.fg_iff_compact {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] (s : Submodule R M) : Submodule.FG s ↔ CompleteLattice.IsCompactElement s

Finitely generated submodules are precisely compact elements in the submodule lattice.

theorem Submodule.FG.map₂ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : M →ₗ[R] N →ₗ[R] P) {p : Submodule R M} {q : Submodule R N} (hp : Submodule.FG p) (hq : Submodule.FG q) : Submodule.FG (Submodule.map₂ f p q)

theorem Submodule.FG.mul {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] {M : Submodule R A} {N : Submodule R A} (hm : Submodule.FG M) (hn : Submodule.FG N) : Submodule.FG (M * N)

theorem Submodule.FG.pow {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] {M : Submodule R A} (h : Submodule.FG M) (n : ℕ) : Submodule.FG (M ^ n)

def Ideal.FG {R : Type u_1} [Semiring R] (I : Ideal R) : Prop

An ideal of R is finitely generated if it is the span of a finite subset of R.

This is defeq to Submodule.FG, but unfolds more nicely.

Equations

• Ideal.FG I = ∃ S, Ideal.span ↑S = I

theorem Ideal.FG.map {R : Type u_3} {S : Type u_4} [Semiring R] [Semiring S] {I : Ideal R} (h : Ideal.FG I) (f : R →+* S) : Ideal.FG (Ideal.map f I)

The image of a finitely generated ideal is finitely generated.

This is the Ideal version of Submodule.FG.map.

theorem Ideal.fg_ker_comp {R : Type u_3} {S : Type u_4} {A : Type u_5} [CommRing R] [CommRing S] [CommRing A] (f : R →+* S) (g : S →+* A) (hf : Ideal.FG (RingHom.ker f)) (hg : Ideal.FG (RingHom.ker g)) (hsur : Function.Surjective ↑f) : Ideal.FG (RingHom.ker (RingHom.comp g f))

theorem Ideal.exists_radical_pow_le_of_fg {R : Type u_3} [CommSemiring R] (I : Ideal R) (h : Ideal.FG (Ideal.radical I)) : ∃ n, Ideal.radical I ^ n ≤ I

class Module.Finite (R : Type u_1) (M : Type u_4) [Semiring R] [AddCommMonoid M] [Module R M] : Prop

• out : Submodule.FG ⊤

  A module over a semiring is Finite if it is finitely generated as a module.

A module over a semiring is Finite if it is finitely generated as a module.

theorem Module.finite_def {R : Type u_6} {M : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] : Module.Finite R M ↔ Submodule.FG ⊤

theorem Module.Finite.iff_addMonoid_fg {M : Type u_6} [AddCommMonoid M] : Module.Finite ℕ M ↔ AddMonoid.FG M

theorem Module.Finite.iff_addGroup_fg {G : Type u_6} [AddCommGroup G] : Module.Finite ℤ G ↔ AddGroup.FG G

theorem Module.Finite.exists_fin {R : Type u_1} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] [Module.Finite R M] : ∃ n s, Submodule.span R (Set.range s) = ⊤

See also Module.Finite.exists_fin'.

theorem Module.Finite.exists_fin' (R : Type u_6) (M : Type u_7) [CommSemiring R] [AddCommMonoid M] [Module R M] [Module.Finite R M] : ∃ n f, Function.Surjective ↑f

theorem Module.Finite.of_surjective {R : Type u_1} {M : Type u_4} {N : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [hM : Module.Finite R M] (f : M →ₗ[R] N) (hf : Function.Surjective ↑f) : Module.Finite R N

instance Module.Finite.range {R : Type u_1} {M : Type u_4} {N : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [Module.Finite R M] (f : M →ₗ[R] N) : Module.Finite R { x // x ∈ LinearMap.range f }

The range of a linear map from a finite module is finite.

Equations

• @Module.Finite.range = @Module.Finite.range.proof_1

instance Module.Finite.map {R : Type u_1} {M : Type u_4} {N : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (p : Submodule R M) [Module.Finite R { x // x ∈ p }] (f : M →ₗ[R] N) : Module.Finite R { x // x ∈ Submodule.map f p }

Pushforwards of finite submodules are finite.

Equations

• @Module.Finite.map = @Module.Finite.map.proof_1

instance Module.Finite.self (R : Type u_1) [Semiring R] : Module.Finite R R

Equations

• Module.Finite.self = Module.Finite.self.proof_1

theorem Module.Finite.of_restrictScalars_finite (R : Type u_6) (A : Type u_7) (M : Type u_8) [CommSemiring R] [Semiring A] [AddCommMonoid M] [Module R M] [Module A M] [Algebra R A] [IsScalarTower R A M] [hM : Module.Finite R M] : Module.Finite A M

instance Module.Finite.prod {R : Type u_1} {M : Type u_4} {N : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [hM : Module.Finite R M] [hN : Module.Finite R N] : Module.Finite R (M × N)

Equations

• @Module.Finite.prod = @Module.Finite.prod.proof_1

instance Module.Finite.pi {R : Type u_1} [Semiring R] {ι : Type u_6} {M : ι → Type u_7} [Finite ι] [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] [h : ∀ (i : ι), Module.Finite R (M i)] : Module.Finite R ((i : ι) → M i)

Equations

• @Module.Finite.pi = @Module.Finite.pi.proof_1

theorem Module.Finite.equiv {R : Type u_1} {M : Type u_4} {N : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [Module.Finite R M] (e : M ≃ₗ[R] N) : Module.Finite R N

instance Module.Finite.ulift {R : Type u_1} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] [Module.Finite R M] : Module.Finite R (ULift.{u_6, u_4} M)

Equations

• @Module.Finite.ulift = @Module.Finite.ulift.proof_1

theorem Module.Finite.trans {R : Type u_6} (A : Type u_7) (M : Type u_8) [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [Module.Finite R A] [Module.Finite A M] : Module.Finite R M

def instModuleTensorProductToAddCommMonoidToNonUnitalNonAssocSemiringToNonAssocSemiringToModuleAddCommMonoid (R : Type u_1) (A : Type u_2) (M : Type u_4) [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] : Module A (TensorProduct R A M)

Porting note: reminding Lean about this instance for Module.Finite.base_change

Equations

• instModuleTensorProductToAddCommMonoidToNonUnitalNonAssocSemiringToNonAssocSemiringToModuleAddCommMonoid R A M = TensorProduct.leftModule

instance Module.Finite.base_change (R : Type u_1) (A : Type u_2) (M : Type u_4) [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [h : Module.Finite R M] : Module.Finite A (TensorProduct R A M)

Equations

• Module.Finite.base_change = Module.Finite.base_change.proof_1

instance Module.Finite.tensorProduct (R : Type u_1) (M : Type u_4) (N : Type u_5) [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [hM : Module.Finite R M] [hN : Module.Finite R N] : Module.Finite R (TensorProduct R M N)

Equations

• Module.Finite.tensorProduct = Module.Finite.tensorProduct.proof_1

def RingHom.Finite {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] (f : A →+* B) : Prop

A ring morphism A →+* B is Finite if B is finitely generated as A-module.

Equations

• RingHom.Finite f = Module.Finite A B

theorem RingHom.Finite.id (A : Type u_1) [CommRing A] : RingHom.Finite (RingHom.id A)

theorem RingHom.Finite.of_surjective {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] (f : A →+* B) (hf : Function.Surjective ↑f) : RingHom.Finite f

theorem RingHom.Finite.comp {A : Type u_1} {B : Type u_2} {C : Type u_3} [CommRing A] [CommRing B] [CommRing C] {g : B →+* C} {f : A →+* B} (hg : RingHom.Finite g) (hf : RingHom.Finite f) : RingHom.Finite (RingHom.comp g f)

theorem RingHom.Finite.of_comp_finite {A : Type u_1} {B : Type u_2} {C : Type u_3} [CommRing A] [CommRing B] [CommRing C] {f : A →+* B} {g : B →+* C} (h : RingHom.Finite (RingHom.comp g f)) : RingHom.Finite g

def AlgHom.Finite {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommRing R] [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : Prop

An algebra morphism A →ₐ[R] B is finite if it is finite as ring morphism. In other words, if B is finitely generated as A-module.

Equations

• AlgHom.Finite f = RingHom.Finite ↑f

theorem AlgHom.Finite.id (R : Type u_1) (A : Type u_2) [CommRing R] [CommRing A] [Algebra R A] : AlgHom.Finite (AlgHom.id R A)

theorem AlgHom.Finite.comp {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommRing R] [CommRing A] [CommRing B] [CommRing C] [Algebra R A] [Algebra R B] [Algebra R C] {g : B →ₐ[R] C} {f : A →ₐ[R] B} (hg : AlgHom.Finite g) (hf : AlgHom.Finite f) : AlgHom.Finite (AlgHom.comp g f)

theorem AlgHom.Finite.of_surjective {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommRing R] [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (hf : Function.Surjective ↑f) : AlgHom.Finite f

theorem AlgHom.Finite.of_comp_finite {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommRing R] [CommRing A] [CommRing B] [CommRing C] [Algebra R A] [Algebra R B] [Algebra R C] {f : A →ₐ[R] B} {g : B →ₐ[R] C} (h : AlgHom.Finite (AlgHom.comp g f)) : AlgHom.Finite g