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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 #

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

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

Instances
@[class]
structure 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.

Instances
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
theorem module.finite_def {R : Type u_1} {M : Type u_2} [comm_semiring R] [add_comm_monoid M] [module R M] :
@[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] :
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 : ) (s : fin n → M), submodule.span R (set.range s) =
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) :
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) :
@[instance]
def module.finite.self (R : Type u_1) [comm_ring R] :
@[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] :
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) :
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] :
@[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] :
theorem algebra.finite_type.self (R : Type u_1) [comm_ring R] :
theorem algebra.finite_type.mv_polynomial (R : Type u_1) [comm_ring R] (ι : Type u_2) [fintype ι] :
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) :
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) :
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) :
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] :

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

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.

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] :

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

A finitely presented algebra is of finite type.

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

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) :

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

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

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

R is finitely presented as R-algebra.

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) :

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

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) :

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

theorem algebra.finite_presentation.iff {R : Type u_1} {A : Type u_2} [comm_ring R] [comm_ring A] [algebra R A] :
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] :

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.

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

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) :

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

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
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
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
theorem ring_hom.finite.id (A : Type u_1) [comm_ring A] :
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) :
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
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) :
theorem ring_hom.finite_type.id (A : Type u_1) [comm_ring A] :
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) :
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) :
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) :
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) :
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) :
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) :
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) :
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
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
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
theorem alg_hom.finite.id (R : Type u_1) (A : Type u_2) [comm_ring R] [comm_ring A] [algebra R A] :
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
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) :
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) :
theorem alg_hom.finite_type.id (R : Type u_1) (A : Type u_2) [comm_ring R] [comm_ring A] [algebra R A] :
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) :
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) :
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) :
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) :
theorem alg_hom.finite_presentation.id (R : Type u_1) (A : Type u_2) [comm_ring R] [comm_ring A] [algebra R A] :
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) :
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) :
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) :
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} :

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

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 = ) :

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.

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 = ) :

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.

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.

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} :

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.

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.

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

@[instance]

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

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

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

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

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

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.

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.

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)] :

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.

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} :

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.

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.

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

@[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] :

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

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

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

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)] :

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