ring_theory.finiteness - mathlib docs

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@[class]

structure module.finite (R : Type u_6) (M : Type u_7) [comm_semiring R] [add_comm_monoid M] [module R M] :

Prop

out : ⊤.fg

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

Instances

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@[class]

structure algebra.finite_type (R : Type u_1) (A : Type u_2) [comm_ring R] [comm_ring A] [algebra R A] :

Prop

out : ⊤.fg

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

Instances

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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.

Equations

algebra.finite_presentation R A = ∃ (n : ℕ) (f : mv_polynomial (fin n) R →ₐ[R] A), function.surjective ⇑f ∧ submodule.fg f.to_ring_hom.ker

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theorem module.finite_def {R : Type u_1} {M : Type u_2} [comm_semiring R] [add_comm_monoid 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.iff_add_monoid_fg {M : Type u_1} [add_comm_monoid M] :

module.finite ℕ M ↔ add_monoid.fg M

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theorem module.finite.iff_add_group_fg {G : Type u_1} [add_comm_group G] :

module.finite ℤ G ↔ add_group.fg G

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theorem module.finite.exists_fin {R : Type u_1} {M : Type u_4} [comm_ring R] [add_comm_group M] [module R M] [module.finite R M] :

∃ (n : ℕ) (s : fin n → M), submodule.span R (set.range s) = ⊤

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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] [module.finite R A] [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 {R : Type u_1} {A : Type u_2} [comm_ring R] [comm_ring A] [algebra R A] :

algebra.finite_presentation R A ↔ ∃ (n : ℕ) (I : ideal (mv_polynomial (fin n) R)) (e : I.quotient ≃ₐ[R] A), submodule.fg I

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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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theorem algebra.finite_presentation.mv_polynomial_of_finite_presentation {R : Type u_1} {A : Type u_2} [comm_ring R] [comm_ring A] [algebra R A] (hfp : algebra.finite_presentation R A) (ι : Type u_3) [fintype ι] :

algebra.finite_presentation R (mv_polynomial ι A)

If A is a finitely presented R-algebra, then mv_polynomial (fin n) A is finitely presented as R-algebra.

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theorem algebra.finite_presentation.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] (hfpA : algebra.finite_presentation R A) (hfpB : algebra.finite_presentation A B) :

algebra.finite_presentation R B

If A is an R-algebra and S is an A-algebra, both finitely presented, then S is finitely presented as R-algebra.

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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.

Equations

f.finite = let _inst : algebra A B := f.to_algebra in module.finite A B

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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.

Equations

f.finite_type = algebra.finite_type A B

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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.

Equations

f.finite_presentation = algebra.finite_presentation A B

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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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theorem ring_hom.finite_presentation.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_presentation) (hf : f.finite_presentation) :

(g.comp 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.

Equations

f.finite = f.to_ring_hom.finite

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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.

Equations

f.finite_type = f.to_ring_hom.finite_type

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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.

Equations

f.finite_presentation = f.to_ring_hom.finite_presentation

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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 {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_presentation) (hf : f.finite_presentation) :

(g.comp f).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

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theorem add_monoid_algebra.mem_adjoin_support {R : Type u_1} {M : Type u_2} [comm_semiring R] [add_monoid M] (f : add_monoid_algebra R M) :

f ∈ algebra.adjoin R (add_monoid_algebra.of' R M '' ↑(f.support))

An element of add_monoid_algebra R M is in the subalgebra generated by its support.

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theorem add_monoid_algebra.support_gen_of_gen {R : Type u_1} {M : Type u_2} [comm_semiring R] [add_monoid M] {S : set (add_monoid_algebra R M)} (hS : algebra.adjoin R S = ⊤) :

algebra.adjoin R (⋃ (f : add_monoid_algebra R M) (H : f ∈ S), add_monoid_algebra.of' R M '' ↑(f.support)) = ⊤

If a set S generates, as algebra, add_monoid_algebra R M, then the set of supports of elements of S generates add_monoid_algebra R M.

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theorem add_monoid_algebra.support_gen_of_gen' {R : Type u_1} {M : Type u_2} [comm_semiring R] [add_monoid M] {S : set (add_monoid_algebra R M)} (hS : algebra.adjoin R S = ⊤) :

algebra.adjoin R (add_monoid_algebra.of' R M '' ⋃ (f : add_monoid_algebra R M) (H : f ∈ S), ↑(f.support)) = ⊤

If a set S generates, as algebra, add_monoid_algebra R M, then the image of the union of the supports of elements of S generates add_monoid_algebra R M.

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theorem add_monoid_algebra.exists_finset_adjoin_eq_top {R : Type u_1} {M : Type u_2} [comm_ring R] [add_comm_monoid M] [h : algebra.finite_type R (add_monoid_algebra R M)] :

∃ (G : finset M), algebra.adjoin R (add_monoid_algebra.of' R M '' ↑G) = ⊤

If add_monoid_algebra R M is of finite type, there there is a G : finset M such that its image generates, as algera, add_monoid_algebra R M.

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theorem add_monoid_algebra.of'_mem_span {R : Type u_1} {M : Type u_2} [comm_ring R] [add_comm_monoid M] [nontrivial R] {m : M} {S : set M} :

add_monoid_algebra.of' R M m ∈ submodule.span R (add_monoid_algebra.of' R M '' S) ↔ m ∈ S

The image of an element m : M in add_monoid_algebra R M belongs the submodule generated by S : set M if and only if m ∈ S.

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theorem add_monoid_algebra.mem_closure_of_mem_span_closure {R : Type u_1} {M : Type u_2} [comm_ring R] [add_comm_monoid M] [nontrivial R] {m : M} {S : set M} (h : add_monoid_algebra.of' R M m ∈ submodule.span R ↑(submonoid.closure (add_monoid_algebra.of' R M '' S))) :

m ∈ add_submonoid.closure S

If the image of an element m : M in add_monoid_algebra R M belongs the submodule generated by the closure of some S : set M then m ∈ closure S.

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theorem add_monoid_algebra.mv_polynomial_aeval_of_surjective_of_closure {R : Type u_1} {M : Type u_2} [add_comm_monoid M] [comm_semiring R] {S : set M} (hS : add_submonoid.closure S = ⊤) :

function.surjective ⇑(mv_polynomial.aeval (λ (s : ↥S), add_monoid_algebra.of' R M ↑s))

If a set S generates an additive monoid M, then the image of M generates, as algebra, add_monoid_algebra R M.

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@[instance]

def add_monoid_algebra.finite_type_of_fg {R : Type u_1} {M : Type u_2} [add_comm_monoid M] [comm_ring R] [h : add_monoid.fg M] :

algebra.finite_type R (add_monoid_algebra R M)

If an additive monoid M is finitely generated then add_monoid_algebra R M is of finite type.

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theorem add_monoid_algebra.finite_type_iff_fg {R : Type u_1} {M : Type u_2} [add_comm_monoid M] [comm_ring R] [nontrivial R] :

add_monoid.fg M ↔ algebra.finite_type R (add_monoid_algebra R M)

An additive monoid M is finitely generated if and only if add_monoid_algebra R M is of finite type.

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theorem add_monoid_algebra.fg_of_finite_type {R : Type u_1} {M : Type u_2} [add_comm_monoid M] [comm_ring R] [nontrivial R] [h : algebra.finite_type R (add_monoid_algebra R M)] :

add_monoid.fg M

If add_monoid_algebra R M is of finite type then M is finitely generated.

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theorem monoid_algebra.mem_adjoint_support {R : Type u_1} {M : Type u_2} [comm_semiring R] [monoid M] (f : monoid_algebra R M) :

f ∈ algebra.adjoin R (⇑(monoid_algebra.of R M) '' ↑(f.support))

An element of monoid_algebra R M is in the subalgebra generated by its support.

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theorem monoid_algebra.support_gen_of_gen {R : Type u_1} {M : Type u_2} [comm_semiring R] [monoid M] {S : set (monoid_algebra R M)} (hS : algebra.adjoin R S = ⊤) :

algebra.adjoin R (⋃ (f : monoid_algebra R M) (H : f ∈ S), ⇑(monoid_algebra.of R M) '' ↑(f.support)) = ⊤

If a set S generates, as algebra, monoid_algebra R M, then the set of supports of elements of S generates monoid_algebra R M.

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theorem monoid_algebra.support_gen_of_gen' {R : Type u_1} {M : Type u_2} [comm_semiring R] [monoid M] {S : set (monoid_algebra R M)} (hS : algebra.adjoin R S = ⊤) :

algebra.adjoin R (⇑(monoid_algebra.of R M) '' ⋃ (f : monoid_algebra R M) (H : f ∈ S), ↑(f.support)) = ⊤

If a set S generates, as algebra, monoid_algebra R M, then the image of the union of the supports of elements of S generates monoid_algebra R M.

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theorem monoid_algebra.exists_finset_adjoin_eq_top {R : Type u_1} {M : Type u_2} [comm_ring R] [comm_monoid M] [h : algebra.finite_type R (monoid_algebra R M)] :

∃ (G : finset M), algebra.adjoin R (⇑(monoid_algebra.of R M) '' ↑G) = ⊤

If monoid_algebra R M is of finite type, there there is a G : finset M such that its image generates, as algera, monoid_algebra R M.

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theorem monoid_algebra.of_mem_span_of_iff {R : Type u_1} {M : Type u_2} [comm_ring R] [comm_monoid M] [nontrivial R] {m : M} {S : set M} :

⇑(monoid_algebra.of R M) m ∈ submodule.span R (⇑(monoid_algebra.of R M) '' S) ↔ m ∈ S

The image of an element m : M in monoid_algebra R M belongs the submodule generated by S : set M if and only if m ∈ S.

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theorem monoid_algebra.mem_closure_of_mem_span_closure {R : Type u_1} {M : Type u_2} [comm_ring R] [comm_monoid M] [nontrivial R] {m : M} {S : set M} (h : ⇑(monoid_algebra.of R M) m ∈ submodule.span R ↑(submonoid.closure (⇑(monoid_algebra.of R M) '' S))) :

m ∈ submonoid.closure S

If the image of an element m : M in monoid_algebra R M belongs the submodule generated by the closure of some S : set M then m ∈ closure S.

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theorem monoid_algebra.mv_polynomial_aeval_of_surjective_of_closure {R : Type u_1} {M : Type u_2} [comm_monoid M] [comm_semiring R] {S : set M} (hS : submonoid.closure S = ⊤) :

function.surjective ⇑(mv_polynomial.aeval (λ (s : ↥S), ⇑(monoid_algebra.of R M) ↑s))

If a set S generates a monoid M, then the image of M generates, as algebra, monoid_algebra R M.

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@[instance]

def monoid_algebra.finite_type_of_fg {R : Type u_1} {M : Type u_2} [comm_monoid M] [comm_ring R] [monoid.fg M] :

algebra.finite_type R (monoid_algebra R M)

If a monoid M is finitely generated then monoid_algebra R M is of finite type.

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theorem monoid_algebra.finite_type_iff_fg {R : Type u_1} {M : Type u_2} [comm_monoid M] [comm_ring R] [nontrivial R] :

monoid.fg M ↔ algebra.finite_type R (monoid_algebra R M)

A monoid M is finitely generated if and only if monoid_algebra R M is of finite type.

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theorem monoid_algebra.fg_of_finite_type {R : Type u_1} {M : Type u_2} [comm_monoid M] [comm_ring R] [nontrivial R] [h : algebra.finite_type R (monoid_algebra R M)] :

monoid.fg M

If monoid_algebra R M is of finite type then M is finitely generated.

mathlib documentation

Finiteness conditions in commutative algebra #

Main declarations #