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ring_theory.dedekind_domain

Dedekind domains #

This file defines the notion of a Dedekind domain (or Dedekind ring), giving three equivalent definitions (TODO: and shows that they are equivalent).

Main definitions #

is_dedekind_domain defines a Dedekind domain as a commutative ring that is Noetherian, integrally closed in its field of fractions and has Krull dimension at most one. is_dedekind_domain_iff shows that this does not depend on the choice of field of fractions.
is_dedekind_domain_dvr alternatively defines a Dedekind domain as an integral domain that is Noetherian, and the localization at every nonzero prime ideal is a DVR.
is_dedekind_domain_inv alternatively defines a Dedekind domain as an integral domain where every nonzero fractional ideal is invertible.
is_dedekind_domain_inv_iff shows that this does note depend on the choice of field of fractions.

Implementation notes #

The definitions that involve a field of fractions choose a canonical field of fractions, but are independent of that choice. The ..._iff lemmas express this independence.

Often, definitions assume that Dedekind domains are not fields. We found it more practical to add a (h : ¬ is_field A) assumption whenever this is explicitly needed.

References #

Tags #

dedekind domain, dedekind ring

def ring.dimension_le_one (R : Type u_1) [comm_ring R] :

Prop

A ring R has Krull dimension at most one if all nonzero prime ideals are maximal.

Equations

ring.dimension_le_one R = ∀ (p : ideal R), p ≠ ⊥ → p.is_prime → p.is_maximal

theorem ring.dimension_le_one.principal_ideal_ring (A : Type u_2) [comm_ring A] [is_domain A] [is_principal_ideal_ring A] :

ring.dimension_le_one A

theorem ring.dimension_le_one.is_integral_closure (R : Type u_1) (A : Type u_2) [comm_ring R] [comm_ring A] (B : Type u_3) [comm_ring B] [is_domain B] [nontrivial R] [algebra R A] [algebra R B] [algebra B A] [is_scalar_tower R B A] [is_integral_closure B R A] (h : ring.dimension_le_one R) :

ring.dimension_le_one B

theorem ring.dimension_le_one.integral_closure (R : Type u_1) (A : Type u_2) [comm_ring R] [comm_ring A] [nontrivial R] [is_domain A] [algebra R A] (h : ring.dimension_le_one R) :

ring.dimension_le_one ↥(integral_closure R A)

@[class]

structure is_dedekind_domain (A : Type u_2) [comm_ring A] [is_domain A] :

Prop

is_noetherian_ring : is_noetherian_ring A
dimension_le_one : ring.dimension_le_one A
is_integrally_closed : is_integrally_closed A

A Dedekind domain is an integral domain that is Noetherian, integrally closed, and has Krull dimension at most one.

This is definition 3.2 of [Neu99].

The integral closure condition is independent of the choice of field of fractions: use is_dedekind_domain_iff to prove is_dedekind_domain for a given fraction_map.

This is the default implementation, but there are equivalent definitions, is_dedekind_domain_dvr and is_dedekind_domain_inv. TODO: Prove that these are actually equivalent definitions.

Instances

theorem is_dedekind_domain_iff (A : Type u_2) [comm_ring A] [is_domain A] (K : Type u_1) [field K] [algebra A K] [is_fraction_ring A K] :

is_dedekind_domain A ↔ is_noetherian_ring A ∧ ring.dimension_le_one A ∧ ∀ {x : K}, is_integral A x → (∃ (y : A), ⇑(algebra_map A K) y = x)

An integral domain is a Dedekind domain iff and only if it is Noetherian, has dimension ≤ 1, and is integrally closed in a given fraction field. In particular, this definition does not depend on the choice of this fraction field.

@[protected, instance]

def is_principal_ideal_ring.is_dedekind_domain (A : Type u_2) [comm_ring A] [is_domain A] [is_principal_ideal_ring A] :

is_dedekind_domain A

structure is_dedekind_domain_dvr (A : Type u_2) [comm_ring A] [is_domain A] :

Prop

is_noetherian_ring : is_noetherian_ring A
is_dvr_at_nonzero_prime : ∀ (P : ideal A), P ≠ ⊥ → ∀ (ᾰ : P.is_prime), discrete_valuation_ring (localization.at_prime P)

A Dedekind domain is an integral domain that is Noetherian, and the localization at every nonzero prime is a discrete valuation ring.

This is equivalent to is_dedekind_domain. TODO: prove the equivalence.

@[protected, instance]

noncomputable def fractional_ideal.has_inv (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] :

has_inv (fractional_ideal R₁⁰ K)

Equations

fractional_ideal.has_inv K = {inv := λ (I : fractional_ideal R₁⁰ K), 1 / I}

theorem fractional_ideal.inv_eq (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] {I : fractional_ideal R₁⁰ K} :

I⁻¹ = 1 / I

theorem fractional_ideal.inv_zero' (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] :

theorem fractional_ideal.inv_nonzero (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] {J : fractional_ideal R₁⁰ K} (h : J ≠ 0) :

J⁻¹ = ⟨↑1 / ↑J, _⟩

theorem fractional_ideal.coe_inv_of_nonzero (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] {J : fractional_ideal R₁⁰ K} (h : J ≠ 0) :

↑J⁻¹ = is_localization.coe_submodule K ⊤ / ↑J

theorem fractional_ideal.mem_inv_iff {K : Type u_3} [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] {I : fractional_ideal R₁⁰ K} (hI : I ≠ 0) {x : K} :

x ∈ I⁻¹ ↔ ∀ (y : K), y ∈ I → x * y ∈ 1

theorem fractional_ideal.inv_anti_mono {K : Type u_3} [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] {I J : fractional_ideal R₁⁰ K} (hI : I ≠ 0) (hJ : J ≠ 0) (hIJ : I ≤ J) :

J⁻¹ ≤ I⁻¹

theorem fractional_ideal.le_self_mul_inv {K : Type u_3} [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] {I : fractional_ideal R₁⁰ K} (hI : I ≤ 1) :

I ≤ I * I⁻¹

theorem fractional_ideal.coe_ideal_le_self_mul_inv (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] (I : ideal R₁) :

↑I ≤ (↑I) * (↑I)⁻¹

theorem fractional_ideal.right_inverse_eq (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] (I J : fractional_ideal R₁⁰ K) (h : I * J = 1) :

I⁻¹ is the inverse of I if I has an inverse.

theorem fractional_ideal.mul_inv_cancel_iff (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] {I : fractional_ideal R₁⁰ K} :

I * I⁻¹ = 1 ↔ ∃ (J : fractional_ideal R₁⁰ K), I * J = 1

theorem fractional_ideal.mul_inv_cancel_iff_is_unit (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] {I : fractional_ideal R₁⁰ K} :

I * I⁻¹ = 1 ↔ is_unit I

@[simp]

theorem fractional_ideal.map_inv (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] {K' : Type u_5} [field K'] [algebra R₁ K'] [is_fraction_ring R₁ K'] (I : fractional_ideal R₁⁰ K) (h : K ≃ₐ[R₁] K') :

fractional_ideal.map ↑h I⁻¹ = (fractional_ideal.map ↑h I)⁻¹

@[simp]

theorem fractional_ideal.span_singleton_inv (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] (x : K) :

(fractional_ideal.span_singleton R₁⁰ x)⁻¹ = fractional_ideal.span_singleton R₁⁰ x⁻¹

theorem fractional_ideal.mul_generator_self_inv (K : Type u_3) [field K] {R₁ : Type u_1} [comm_ring R₁] [algebra R₁ K] [is_localization R₁⁰ K] (I : fractional_ideal R₁⁰ K) [↑I.is_principal] (h : I ≠ 0) :

I * fractional_ideal.span_singleton R₁⁰ (submodule.is_principal.generator ↑I)⁻¹ = 1

theorem fractional_ideal.invertible_of_principal (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] (I : fractional_ideal R₁⁰ K) [↑I.is_principal] (h : I ≠ 0) :

I * I⁻¹ = 1

theorem fractional_ideal.invertible_iff_generator_nonzero (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] (I : fractional_ideal R₁⁰ K) [↑I.is_principal] :

I * I⁻¹ = 1 ↔ submodule.is_principal.generator ↑I ≠ 0

theorem fractional_ideal.is_principal_inv (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] (I : fractional_ideal R₁⁰ K) [↑I.is_principal] (h : I ≠ 0) :

I⁻¹.val.is_principal

@[simp]

theorem fractional_ideal.one_inv (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] :

def is_dedekind_domain_inv (A : Type u_2) [comm_ring A] [is_domain A] :

Prop

A Dedekind domain is an integral domain such that every fractional ideal has an inverse.

This is equivalent to is_dedekind_domain. In particular we provide a fractional_ideal.comm_group_with_zero instance, assuming is_dedekind_domain A, which implies is_dedekind_domain_inv. For integral ideals, is_dedekind_domain(_inv) implies only ideal.cancel_comm_monoid_with_zero.

Equations

is_dedekind_domain_inv A = ∀ (I : fractional_ideal A⁰ (fraction_ring A)), I ≠ ⊥ → I * I⁻¹ = 1

theorem is_dedekind_domain_inv_iff {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] :

is_dedekind_domain_inv A ↔ ∀ (I : fractional_ideal A⁰ K), I ≠ ⊥ → I * I⁻¹ = 1

theorem fractional_ideal.adjoin_integral_eq_one_of_is_unit {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] (x : K) (hx : is_integral A x) (hI : is_unit (fractional_ideal.adjoin_integral A⁰ x hx)) :

fractional_ideal.adjoin_integral A⁰ x hx = 1

theorem is_dedekind_domain_inv.mul_inv_eq_one {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] (h : is_dedekind_domain_inv A) {I : fractional_ideal A⁰ K} (hI : I ≠ 0) :

I * I⁻¹ = 1

theorem is_dedekind_domain_inv.inv_mul_eq_one {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] (h : is_dedekind_domain_inv A) {I : fractional_ideal A⁰ K} (hI : I ≠ 0) :

I⁻¹ * I = 1

@[protected]

theorem is_dedekind_domain_inv.is_unit {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] (h : is_dedekind_domain_inv A) {I : fractional_ideal A⁰ K} (hI : I ≠ 0) :

theorem is_dedekind_domain_inv.is_noetherian_ring {A : Type u_2} [comm_ring A] [is_domain A] (h : is_dedekind_domain_inv A) :

is_noetherian_ring A

theorem is_dedekind_domain_inv.integrally_closed {A : Type u_2} [comm_ring A] [is_domain A] (h : is_dedekind_domain_inv A) :

is_integrally_closed A

theorem is_dedekind_domain_inv.dimension_le_one {A : Type u_2} [comm_ring A] [is_domain A] (h : is_dedekind_domain_inv A) :

ring.dimension_le_one A

theorem is_dedekind_domain_inv.is_dedekind_domain {A : Type u_2} [comm_ring A] [is_domain A] (h : is_dedekind_domain_inv A) :

is_dedekind_domain A

Showing one side of the equivalence between the definitions is_dedekind_domain_inv and is_dedekind_domain of Dedekind domains.

theorem exists_multiset_prod_cons_le_and_prod_not_le {A : Type u_2} [comm_ring A] [is_domain A] [is_dedekind_domain A] (hNF : ¬is_field A) {I M : ideal A} (hI0 : I ≠ ⊥) (hIM : I ≤ M) [hM : M.is_maximal] :

∃ (Z : multiset (prime_spectrum A)), (M ::ₘ multiset.map prime_spectrum.as_ideal Z).prod ≤ I ∧ ¬(multiset.map prime_spectrum.as_ideal Z).prod ≤ I

Specialization of exists_prime_spectrum_prod_le_and_ne_bot_of_domain to Dedekind domains: Let I : ideal A be a nonzero ideal, where A is a Dedekind domain that is not a field. Then exists_prime_spectrum_prod_le_and_ne_bot_of_domain states we can find a product of prime ideals that is contained within I. This lemma extends that result by making the product minimal: let M be a maximal ideal that contains I, then the product including M is contained within I and the product excluding M is not contained within I.

theorem fractional_ideal.exists_not_mem_one_of_ne_bot {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] [is_dedekind_domain A] (hNF : ¬is_field A) {I : ideal A} (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :

∃ (x : K), x ∈ (↑I)⁻¹ ∧ x ∉ 1

theorem fractional_ideal.one_mem_inv_coe_ideal {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] {I : ideal A} (hI : I ≠ ⊥) :

1 ∈ (↑I)⁻¹

theorem fractional_ideal.mul_inv_cancel_of_le_one {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] [h : is_dedekind_domain A] {I : ideal A} (hI0 : I ≠ ⊥) (hI : ((↑I) * (↑I)⁻¹)⁻¹ ≤ 1) :

(↑I) * (↑I)⁻¹ = 1

theorem fractional_ideal.coe_ideal_mul_inv {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] [h : is_dedekind_domain A] (I : ideal A) (hI0 : I ≠ ⊥) :

(↑I) * (↑I)⁻¹ = 1

Nonzero integral ideals in a Dedekind domain are invertible.

We will use this to show that nonzero fractional ideals are invertible, and finally conclude that fractional ideals in a Dedekind domain form a group with zero.

@[protected]

theorem fractional_ideal.mul_inv_cancel {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] [is_dedekind_domain A] {I : fractional_ideal A⁰ K} (hne : I ≠ 0) :

I * I⁻¹ = 1

Nonzero fractional ideals in a Dedekind domain are units.

This is also available as _root_.mul_inv_cancel, using the comm_group_with_zero instance defined below.

theorem fractional_ideal.mul_right_le_iff {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] [is_dedekind_domain A] {J : fractional_ideal A⁰ K} (hJ : J ≠ 0) {I I' : fractional_ideal A⁰ K} :

I * J ≤ I' * J ↔ I ≤ I'

theorem fractional_ideal.mul_left_le_iff {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] [is_dedekind_domain A] {J : fractional_ideal A⁰ K} (hJ : J ≠ 0) {I I' : fractional_ideal A⁰ K} :

J * I ≤ J * I' ↔ I ≤ I'

theorem fractional_ideal.mul_right_strict_mono {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] [is_dedekind_domain A] {I : fractional_ideal A⁰ K} (hI : I ≠ 0) :

strict_mono (λ (_x : fractional_ideal A⁰ K), _x * I)

theorem fractional_ideal.mul_left_strict_mono {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] [is_dedekind_domain A] {I : fractional_ideal A⁰ K} (hI : I ≠ 0) :

strict_mono (has_mul.mul I)

@[protected]

theorem fractional_ideal.div_eq_mul_inv {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] [is_dedekind_domain A] (I J : fractional_ideal A⁰ K) :

I / J = I * J⁻¹

This is also available as _root_.div_eq_mul_inv, using the comm_group_with_zero instance defined below.

theorem is_dedekind_domain_iff_is_dedekind_domain_inv {A : Type u_2} [comm_ring A] [is_domain A] :

is_dedekind_domain A ↔ is_dedekind_domain_inv A

is_dedekind_domain and is_dedekind_domain_inv are equivalent ways to express that an integral domain is a Dedekind domain.

@[protected, instance]

noncomputable def fractional_ideal.comm_group_with_zero {A : Type u_2} (K : Type u_3) [comm_ring A] [field K] [is_domain A] [is_dedekind_domain A] [algebra A K] [is_fraction_ring A K] :

comm_group_with_zero (fractional_ideal A⁰ K)

Equations

fractional_ideal.comm_group_with_zero K = {mul := comm_semiring.mul fractional_ideal.comm_semiring, mul_assoc := _, one := comm_semiring.one fractional_ideal.comm_semiring, one_mul := _, mul_one := _, npow := comm_semiring.npow fractional_ideal.comm_semiring, npow_zero' := _, npow_succ' := _, mul_comm := _, zero := comm_semiring.zero fractional_ideal.comm_semiring, zero_mul := _, mul_zero := _, inv := λ (I : fractional_ideal A⁰ K), I⁻¹, div := has_div.div fractional_ideal.fractional_ideal_has_div, div_eq_mul_inv := _, zpow := group_with_zero.zpow._default comm_semiring.mul _ comm_semiring.one _ _ comm_semiring.npow _ _ (λ (I : fractional_ideal A⁰ K), I⁻¹), zpow_zero' := _, zpow_succ' := _, zpow_neg' := _, exists_pair_ne := _, inv_zero := _, mul_inv_cancel := _}

@[protected, instance]

noncomputable def ideal.cancel_comm_monoid_with_zero {A : Type u_2} [comm_ring A] [is_domain A] [is_dedekind_domain A] :

cancel_comm_monoid_with_zero (ideal A)

Equations

ideal.cancel_comm_monoid_with_zero = function.injective.cancel_comm_monoid_with_zero ⇑(fractional_ideal.coe_ideal_hom A⁰ (fraction_ring A)) ideal.cancel_comm_monoid_with_zero._proof_2 ideal.cancel_comm_monoid_with_zero._proof_3 ideal.cancel_comm_monoid_with_zero._proof_4 ideal.cancel_comm_monoid_with_zero._proof_5

theorem ideal.dvd_iff_le {A : Type u_2} [comm_ring A] [is_domain A] [is_dedekind_domain A] {I J : ideal A} :

I ∣ J ↔ J ≤ I

For ideals in a Dedekind domain, to divide is to contain.

theorem ideal.dvd_not_unit_iff_lt {A : Type u_2} [comm_ring A] [is_domain A] [is_dedekind_domain A] {I J : ideal A} :

dvd_not_unit I J ↔ J < I

@[protected, instance]

def ideal.wf_dvd_monoid {A : Type u_2} [comm_ring A] [is_domain A] [is_dedekind_domain A] :

wf_dvd_monoid (ideal A)

@[protected, instance]

def ideal.unique_factorization_monoid {A : Type u_2} [comm_ring A] [is_domain A] [is_dedekind_domain A] :

unique_factorization_monoid (ideal A)

@[protected, instance]

noncomputable def ideal.normalization_monoid {A : Type u_2} [comm_ring A] [is_domain A] [is_dedekind_domain A] :

normalization_monoid (ideal A)

Equations

ideal.normalization_monoid = normalization_monoid_of_unique_units

@[simp]

theorem ideal.dvd_span_singleton {A : Type u_2} [comm_ring A] [is_domain A] [is_dedekind_domain A] {I : ideal A} {x : A} :

I ∣ ideal.span {x} ↔ x ∈ I

theorem ideal.is_prime_of_prime {A : Type u_2} [comm_ring A] [is_domain A] [is_dedekind_domain A] {P : ideal A} (h : prime P) :

theorem ideal.prime_of_is_prime {A : Type u_2} [comm_ring A] [is_domain A] [is_dedekind_domain A] {P : ideal A} (hP : P ≠ ⊥) (h : P.is_prime) :

theorem ideal.prime_iff_is_prime {A : Type u_2} [comm_ring A] [is_domain A] [is_dedekind_domain A] {P : ideal A} (hP : P ≠ ⊥) :

prime P ↔ P.is_prime

In a Dedekind domain, the (nonzero) prime elements of the monoid with zero ideal A are exactly the prime ideals.

`is_integral_closure` section #

We show that an integral closure of a Dedekind domain in a finite separable field extension is again a Dedekind domain. This implies the ring of integers of a number field is a Dedekind domain.

theorem is_integral_closure.range_le_span_dual_basis {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] {L : Type u_4} [field L] (C : Type u_5) [comm_ring C] [algebra K L] [finite_dimensional K L] [algebra A L] [is_scalar_tower A K L] [algebra C L] [is_integral_closure C A L] [algebra A C] [is_scalar_tower A C L] [is_separable K L] {ι : Type u_1} [fintype ι] [decidable_eq ι] (b : basis ι K L) (hb_int : ∀ (i : ι), is_integral A (⇑b i)) [is_integrally_closed A] :

(linear_map.restrict_scalars A (algebra.linear_map C L)).range ≤ submodule.span A (set.range ⇑((algebra.trace_form K L).dual_basis _ b))

theorem integral_closure_le_span_dual_basis {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] {L : Type u_4} [field L] [algebra K L] [finite_dimensional K L] [algebra A L] [is_scalar_tower A K L] [is_separable K L] {ι : Type u_1} [fintype ι] [decidable_eq ι] (b : basis ι K L) (hb_int : ∀ (i : ι), is_integral A (⇑b i)) [is_integrally_closed A] :

(integral_closure A L).to_submodule ≤ submodule.span A (set.range ⇑((algebra.trace_form K L).dual_basis _ b))

theorem exists_integral_multiples (A : Type u_2) (K : Type u_3) [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] {L : Type u_4} [field L] [algebra K L] [finite_dimensional K L] [algebra A L] [is_scalar_tower A K L] (s : finset L) :

∃ (y : A) (H : y ≠ 0), ∀ (x : L), x ∈ s → is_integral A (y • x)

Send a set of x'es in a finite extension L of the fraction field of R to (y : R) • x ∈ integral_closure R L.

theorem finite_dimensional.exists_is_basis_integral (A : Type u_2) (K : Type u_3) [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] (L : Type u_4) [field L] [algebra K L] [finite_dimensional K L] [algebra A L] [is_scalar_tower A K L] :

∃ (s : finset L) (b : basis ↥s K L), ∀ (x : ↥s), is_integral A (⇑b x)

If L is a finite extension of K = Frac(A), then L has a basis over A consisting of integral elements.

theorem is_integral_closure.is_noetherian_ring (A : Type u_2) (K : Type u_3) [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] (L : Type u_4) [field L] (C : Type u_5) [comm_ring C] [algebra K L] [finite_dimensional K L] [algebra A L] [is_scalar_tower A K L] [algebra C L] [is_integral_closure C A L] [algebra A C] [is_scalar_tower A C L] [is_separable K L] [is_integrally_closed A] [is_noetherian_ring A] :

is_noetherian_ring C

theorem integral_closure.is_noetherian_ring {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] (L : Type u_4) [field L] [algebra K L] [finite_dimensional K L] [algebra A L] [is_scalar_tower A K L] [is_separable K L] [is_integrally_closed A] [is_noetherian_ring A] :

is_noetherian_ring ↥(integral_closure A L)

theorem is_integral_closure.is_dedekind_domain (A : Type u_2) (K : Type u_3) [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] (L : Type u_4) [field L] (C : Type u_5) [comm_ring C] [algebra K L] [finite_dimensional K L] [algebra A L] [is_scalar_tower A K L] [algebra C L] [is_integral_closure C A L] [algebra A C] [is_scalar_tower A C L] [is_separable K L] [is_domain C] [h : is_dedekind_domain A] :

is_dedekind_domain C

theorem integral_closure.is_dedekind_domain (A : Type u_2) (K : Type u_3) [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] (L : Type u_4) [field L] [algebra K L] [finite_dimensional K L] [algebra A L] [is_scalar_tower A K L] [is_separable K L] [h : is_dedekind_domain A] :

is_dedekind_domain ↥(integral_closure A L)

@[protected, instance]

def integral_closure.is_dedekind_domain_fraction_ring (A : Type u_2) [comm_ring A] [is_domain A] (L : Type u_4) [field L] [algebra A L] [algebra (fraction_ring A) L] [is_scalar_tower A (fraction_ring A) L] [finite_dimensional (fraction_ring A) L] [is_separable (fraction_ring A) L] [is_dedekind_domain A] :

is_dedekind_domain ↥(integral_closure A L)

theorem prod_normalized_factors_eq_self {T : Type u_4} [comm_ring T] [is_domain T] [is_dedekind_domain T] {I : ideal T} (hI : I ≠ ⊥) :

(unique_factorization_monoid.normalized_factors I).prod = I

theorem normalized_factors_prod {T : Type u_4} [comm_ring T] [is_domain T] [is_dedekind_domain T] {α : multiset (ideal T)} (h : ∀ (p : ideal T), p ∈ α → prime p) :

unique_factorization_monoid.normalized_factors α.prod = α

theorem count_le_of_ideal_ge {T : Type u_4} [comm_ring T] [is_domain T] [is_dedekind_domain T] {I J : ideal T} (h : I ≤ J) (hI : I ≠ ⊥) (K : ideal T) :

multiset.count K (unique_factorization_monoid.normalized_factors J) ≤ multiset.count K (unique_factorization_monoid.normalized_factors I)

theorem sup_eq_prod_inf_factors {T : Type u_4} [comm_ring T] [is_domain T] [is_dedekind_domain T] (I J : ideal T) (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :

I ⊔ J = (unique_factorization_monoid.normalized_factors I ∩ unique_factorization_monoid.normalized_factors J).prod