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ring_theory.finiteness

Finiteness conditions in commutative algebra #

In this file we define several notions of finiteness that are common in commutative algebra.

Main declarations #

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@[class]
def module.finite (R : Type u_1) (M : Type u_4) [comm_ring R] [add_comm_group M] [module R M] :
Prop

A module over a commutative ring is finite if it is finitely generated as a module.

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@[class]
def algebra.finite_type (R : Type u_1) (A : Type u_2) [comm_ring R] [comm_ring A] [algebra R A] :
Prop

An algebra over a commutative ring is of finite_type if it is finitely generated over the base ring as algebra.

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def algebra.finite_presentation (R : Type u_1) (A : Type u_2) [comm_ring R] [comm_ring A] [algebra R A] :
Prop

An algebra over a commutative ring is finite_presentation if it is the quotient of a polynomial ring in n variables by a finitely generated ideal.

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theorem module.finite_def {R : Type u_1} {M : Type u_4} [comm_ring R] [add_comm_group M] [module R M] :
module.finite R M ↔ ⊤.fg
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@[instance]
def module.is_noetherian.finite (R : Type u_1) (M : Type u_4) [comm_ring R] [add_comm_group M] [module R M] [is_noetherian R M] :
module.finite R M
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theorem module.finite.of_surjective {R : Type u_1} {M : Type u_4} {N : Type u_5} [comm_ring R] [add_comm_group M] [module R M] [add_comm_group N] [module R N] [hM : module.finite R M] (f : M →ₗ[R] N) (hf : function.surjective ⇑f) :
module.finite R N
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theorem module.finite.of_injective {R : Type u_1} {M : Type u_4} {N : Type u_5} [comm_ring R] [add_comm_group M] [module R M] [add_comm_group N] [module R N] [is_noetherian R N] (f : M →ₗ[R] N) (hf : function.injective ⇑f) :
module.finite R M
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@[instance]
def module.finite.self (R : Type u_1) [comm_ring R] :
module.finite R R
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@[instance]
def module.finite.prod {R : Type u_1} {M : Type u_4} {N : Type u_5} [comm_ring R] [add_comm_group M] [module R M] [add_comm_group N] [module R N] [hM : module.finite R M] [hN : module.finite R N] :
module.finite R (M × N)
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theorem module.finite.equiv {R : Type u_1} {M : Type u_4} {N : Type u_5} [comm_ring R] [add_comm_group M] [module R M] [add_comm_group N] [module R N] [hM : module.finite R M] (e : M ≃ₗ[R] N) :
module.finite R N
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theorem module.finite.trans {R : Type u_1} (A : Type u_2) (B : Type u_3) [comm_ring R] [comm_ring A] [algebra R A] [comm_ring B] [algebra R B] [algebra A B] [is_scalar_tower R A B] [hRA : module.finite R A] [hAB : module.finite A B] :
module.finite R B
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@[instance]
def module.finite.finite_type {R : Type u_1} (A : Type u_2) [comm_ring R] [comm_ring A] [algebra R A] [hRA : module.finite R A] :
algebra.finite_type R A
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theorem algebra.finite_type.self (R : Type u_1) [comm_ring R] :
algebra.finite_type R R
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theorem algebra.finite_type.mv_polynomial (R : Type u_1) [comm_ring R] (ι : Type u_2) [fintype ι] :
algebra.finite_type R (mv_polynomial ι R)
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theorem algebra.finite_type.of_surjective {R : Type u_1} {A : Type u_2} {B : Type u_3} [comm_ring R] [comm_ring A] [algebra R A] [comm_ring B] [algebra R B] (hRA : algebra.finite_type R A) (f : A →ₐ[R] B) (hf : function.surjective ⇑f) :
algebra.finite_type R B
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theorem algebra.finite_type.equiv {R : Type u_1} {A : Type u_2} {B : Type u_3} [comm_ring R] [comm_ring A] [algebra R A] [comm_ring B] [algebra R B] (hRA : algebra.finite_type R A) (e : A ≃ₐ[R] B) :
algebra.finite_type R B
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theorem algebra.finite_type.trans {R : Type u_1} {A : Type u_2} {B : Type u_3} [comm_ring R] [comm_ring A] [algebra R A] [comm_ring B] [algebra R B] [algebra A B] [is_scalar_tower R A B] (hRA : algebra.finite_type R A) (hAB : algebra.finite_type A B) :
algebra.finite_type R B
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theorem algebra.finite_type.iff_quotient_mv_polynomial {R : Type u_1} {A : Type u_2} [comm_ring R] [comm_ring A] [algebra R A] :
algebra.finite_type R A ↔ ∃ (s : finset A) (f : mv_polynomial {x // x ∈ s} R →ₐ[R] A), function.surjective ⇑f

An algebra is finitely generated if and only if it is a quotient of a polynomial ring whose variables are indexed by a finset.

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theorem algebra.finite_type.iff_quotient_mv_polynomial' {R : Type u_1} {A : Type u_2} [comm_ring R] [comm_ring A] [algebra R A] :
algebra.finite_type R A ↔ ∃ (ι : Type u_2) [_inst_10 : fintype ι] (f : mv_polynomial ι R →ₐ[R] A), function.surjective ⇑f

An algebra is finitely generated if and only if it is a quotient of a polynomial ring whose variables are indexed by a fintype.

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theorem algebra.finite_type.iff_quotient_mv_polynomial'' {R : Type u_1} {A : Type u_2} [comm_ring R] [comm_ring A] [algebra R A] :
algebra.finite_type R A ↔ ∃ (n : ℕ) (f : mv_polynomial (fin n) R →ₐ[R] A), function.surjective ⇑f

An algebra is finitely generated if and only if it is a quotient of a polynomial ring in n variables.

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theorem algebra.finite_type.of_finite_presentation {R : Type u_1} {A : Type u_2} [comm_ring R] [comm_ring A] [algebra R A] :
algebra.finite_presentation R A → algebra.finite_type R A

A finitely presented algebra is of finite type.

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theorem algebra.finite_presentation.of_finite_type {R : Type u_1} {A : Type u_2} [comm_ring R] [comm_ring A] [algebra R A] [is_noetherian_ring R] :
algebra.finite_type R A ↔ algebra.finite_presentation R A

An algebra over a Noetherian ring is finitely generated if and only if it is finitely presented.

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theorem algebra.finite_presentation.equiv {R : Type u_1} {A : Type u_2} {B : Type u_3} [comm_ring R] [comm_ring A] [algebra R A] [comm_ring B] [algebra R B] (hfp : algebra.finite_presentation R A) (e : A ≃ₐ[R] B) :
algebra.finite_presentation R B

If e : A ≃ₐ[R] B and A is finitely presented, then so is B.

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theorem algebra.finite_presentation.mv_polynomial (R : Type u_1) [comm_ring R] (ι : Type u_2) [fintype ι] :
algebra.finite_presentation R (mv_polynomial ι R)

The ring of polynomials in finitely many variables is finitely presented.

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theorem algebra.finite_presentation.self (R : Type u_1) [comm_ring R] :
algebra.finite_presentation R R

R is finitely presented as R-algebra.

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theorem algebra.finite_presentation.quotient {R : Type u_1} {A : Type u_2} [comm_ring R] [comm_ring A] [algebra R A] {I : ideal A} (h : submodule.fg I) (hfp : algebra.finite_presentation R A) :
algebra.finite_presentation R I.quotient

The quotient of a finitely presented algebra by a finitely generated ideal is finitely presented.

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theorem algebra.finite_presentation.of_surjective {R : Type u_1} {A : Type u_2} {B : Type u_3} [comm_ring R] [comm_ring A] [algebra R A] [comm_ring B] [algebra R B] {f : A →ₐ[R] B} (hf : function.surjective ⇑f) (hker : submodule.fg f.to_ring_hom.ker) (hfp : algebra.finite_presentation R A) :
algebra.finite_presentation R B

If f : A →ₐ[R] B is surjective with finitely generated kernel and A is finitely presented, then so is B.

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theorem algebra.finite_presentation.iff_quotient_mv_polynomial' {R : Type u_1} {A : Type u_2} [comm_ring R] [comm_ring A] [algebra R A] :
algebra.finite_presentation R A ↔ ∃ (ι : Type u_2) [_inst_10 : fintype ι] (f : mv_polynomial ι R →ₐ[R] A), function.surjective ⇑f ∧ submodule.fg f.to_ring_hom.ker

An algebra is finitely presented if and only if it is a quotient of a polynomial ring whose variables are indexed by a fintype by a finitely generated ideal.

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def ring_hom.finite {A : Type u_1} {B : Type u_2} [comm_ring A] [comm_ring B] (f : A →+* B) :
Prop

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

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def ring_hom.finite_type {A : Type u_1} {B : Type u_2} [comm_ring A] [comm_ring B] (f : A →+* B) :
Prop

A ring morphism A →+* B is of finite_type if B is finitely generated as A-algebra.

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def ring_hom.finite_presentation {A : Type u_1} {B : Type u_2} [comm_ring A] [comm_ring B] (f : A →+* B) :
Prop

A ring morphism A →+* B is of finite_presentation if B is finitely presented as A-algebra.

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theorem ring_hom.finite.id (A : Type u_1) [comm_ring A] :
(ring_hom.id A).finite
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theorem ring_hom.finite.of_surjective {A : Type u_1} {B : Type u_2} [comm_ring A] [comm_ring B] (f : A →+* B) (hf : function.surjective ⇑f) :
f.finite
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theorem ring_hom.finite.comp {A : Type u_1} {B : Type u_2} {C : Type u_3} [comm_ring A] [comm_ring B] [comm_ring C] {g : B →+* C} {f : A →+* B} (hg : g.finite) (hf : f.finite) :
(g.comp f).finite
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theorem ring_hom.finite.finite_type {A : Type u_1} {B : Type u_2} [comm_ring A] [comm_ring B] {f : A →+* B} (hf : f.finite) :
f.finite_type
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theorem ring_hom.finite_type.id (A : Type u_1) [comm_ring A] :
(ring_hom.id A).finite_type
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theorem ring_hom.finite_type.comp_surjective {A : Type u_1} {B : Type u_2} {C : Type u_3} [comm_ring A] [comm_ring B] [comm_ring C] {f : A →+* B} {g : B →+* C} (hf : f.finite_type) (hg : function.surjective ⇑g) :
(g.comp f).finite_type
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theorem ring_hom.finite_type.of_surjective {A : Type u_1} {B : Type u_2} [comm_ring A] [comm_ring B] (f : A →+* B) (hf : function.surjective ⇑f) :
f.finite_type
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theorem ring_hom.finite_type.comp {A : Type u_1} {B : Type u_2} {C : Type u_3} [comm_ring A] [comm_ring B] [comm_ring C] {g : B →+* C} {f : A →+* B} (hg : g.finite_type) (hf : f.finite_type) :
(g.comp f).finite_type
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theorem ring_hom.finite_type.of_finite_presentation {A : Type u_1} {B : Type u_2} [comm_ring A] [comm_ring B] {f : A →+* B} (hf : f.finite_presentation) :
f.finite_type
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theorem ring_hom.finite_presentation.id (A : Type u_1) [comm_ring A] :
(ring_hom.id A).finite_presentation
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theorem ring_hom.finite_presentation.comp_surjective {A : Type u_1} {B : Type u_2} {C : Type u_3} [comm_ring A] [comm_ring B] [comm_ring C] {f : A →+* B} {g : B →+* C} (hf : f.finite_presentation) (hg : function.surjective ⇑g) (hker : submodule.fg g.ker) :
(g.comp f).finite_presentation
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theorem ring_hom.finite_presentation.of_surjective {A : Type u_1} {B : Type u_2} [comm_ring A] [comm_ring B] (f : A →+* B) (hf : function.surjective ⇑f) (hker : submodule.fg f.ker) :
f.finite_presentation
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theorem ring_hom.finite_presentation.of_finite_type {A : Type u_1} {B : Type u_2} [comm_ring A] [comm_ring B] [is_noetherian_ring A] {f : A →+* B} :
f.finite_type ↔ f.finite_presentation
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def alg_hom.finite {R : Type u_1} {A : Type u_2} {B : Type u_3} [comm_ring R] [comm_ring A] [comm_ring 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.

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def alg_hom.finite_type {R : Type u_1} {A : Type u_2} {B : Type u_3} [comm_ring R] [comm_ring A] [comm_ring B] [algebra R A] [algebra R B] (f : A →ₐ[R] B) :
Prop

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

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def alg_hom.finite_presentation {R : Type u_1} {A : Type u_2} {B : Type u_3} [comm_ring R] [comm_ring A] [comm_ring B] [algebra R A] [algebra R B] (f : A →ₐ[R] B) :
Prop

An algebra morphism A →ₐ[R] B is of finite_presentation if it is of finite presentation as ring morphism. In other words, if B is finitely presented as A-algebra.

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theorem alg_hom.finite.id (R : Type u_1) (A : Type u_2) [comm_ring R] [comm_ring A] [algebra R A] :
(alg_hom.id R A).finite
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theorem alg_hom.finite.comp {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [comm_ring R] [comm_ring A] [comm_ring B] [comm_ring C] [algebra R A] [algebra R B] [algebra R C] {g : B →ₐ[R] C} {f : A →ₐ[R] B} (hg : g.finite) (hf : f.finite) :
(g.comp f).finite
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theorem alg_hom.finite.of_surjective {R : Type u_1} {A : Type u_2} {B : Type u_3} [comm_ring R] [comm_ring A] [comm_ring B] [algebra R A] [algebra R B] (f : A →ₐ[R] B) (hf : function.surjective ⇑f) :
f.finite
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theorem alg_hom.finite.finite_type {R : Type u_1} {A : Type u_2} {B : Type u_3} [comm_ring R] [comm_ring A] [comm_ring B] [algebra R A] [algebra R B] {f : A →ₐ[R] B} (hf : f.finite) :
f.finite_type
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theorem alg_hom.finite_type.id (R : Type u_1) (A : Type u_2) [comm_ring R] [comm_ring A] [algebra R A] :
(alg_hom.id R A).finite_type
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theorem alg_hom.finite_type.comp {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [comm_ring R] [comm_ring A] [comm_ring B] [comm_ring C] [algebra R A] [algebra R B] [algebra R C] {g : B →ₐ[R] C} {f : A →ₐ[R] B} (hg : g.finite_type) (hf : f.finite_type) :
(g.comp f).finite_type
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theorem alg_hom.finite_type.comp_surjective {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [comm_ring R] [comm_ring A] [comm_ring B] [comm_ring C] [algebra R A] [algebra R B] [algebra R C] {f : A →ₐ[R] B} {g : B →ₐ[R] C} (hf : f.finite_type) (hg : function.surjective ⇑g) :
(g.comp f).finite_type
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theorem alg_hom.finite_type.of_surjective {R : Type u_1} {A : Type u_2} {B : Type u_3} [comm_ring R] [comm_ring A] [comm_ring B] [algebra R A] [algebra R B] (f : A →ₐ[R] B) (hf : function.surjective ⇑f) :
f.finite_type
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theorem alg_hom.finite_type.of_finite_presentation {R : Type u_1} {A : Type u_2} {B : Type u_3} [comm_ring R] [comm_ring A] [comm_ring B] [algebra R A] [algebra R B] {f : A →ₐ[R] B} (hf : f.finite_presentation) :
f.finite_type
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theorem alg_hom.finite_presentation.id (R : Type u_1) (A : Type u_2) [comm_ring R] [comm_ring A] [algebra R A] :
(alg_hom.id R A).finite_presentation
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theorem alg_hom.finite_presentation.comp_surjective {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [comm_ring R] [comm_ring A] [comm_ring B] [comm_ring C] [algebra R A] [algebra R B] [algebra R C] {f : A →ₐ[R] B} {g : B →ₐ[R] C} (hf : f.finite_presentation) (hg : function.surjective ⇑g) (hker : submodule.fg g.to_ring_hom.ker) :
(g.comp f).finite_presentation
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theorem alg_hom.finite_presentation.of_surjective {R : Type u_1} {A : Type u_2} {B : Type u_3} [comm_ring R] [comm_ring A] [comm_ring B] [algebra R A] [algebra R B] (f : A →ₐ[R] B) (hf : function.surjective ⇑f) (hker : submodule.fg f.to_ring_hom.ker) :
f.finite_presentation
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theorem alg_hom.finite_presentation.of_finite_type {R : Type u_1} {A : Type u_2} {B : Type u_3} [comm_ring R] [comm_ring A] [comm_ring B] [algebra R A] [algebra R B] [is_noetherian_ring A] {f : A →ₐ[R] B} :
f.finite_type ↔ f.finite_presentation