fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
(hI : (I * (I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤ 1) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
obtain ⟨J, hJ⟩ : ∃ J : Ideal A, (J : FractionalIdeal A⁰ K) = I * (I : FractionalIdeal A⁰ K)⁻¹ :=
le_one_iff_exists_coeIdeal.mp mul_one_div_le_one
by_cases hJ0 : J = ⊥
· subst hJ0
refine absurd ?_ hI0
rw [eq_bot_iff, ← coeIdeal_le_coeIdeal K, hJ]
exact coe_ideal_le_self_mul_inv K I
by_cases hJ1 : J = ⊤
· rw [← hJ, hJ1, coeIdeal_top]
exact (not_inv_le_one_of_ne_bot (K := K) hJ0 hJ1 (hJ ▸ hI)).elim | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Pointwise",
"Mathlib.RingTheory.DedekindDomain.Basic",
"Mathlib.RingTheory.FractionalIdeal.Inverse",
"Mathlib.RingTheory.Spectrum.Prime.Basic"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean | mul_inv_cancel_of_le_one | null |
coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ ⊥) :
I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
apply mul_inv_cancel_of_le_one hI0
by_cases hJ0 : I * (I : FractionalIdeal A⁰ K)⁻¹ = 0
· rw [hJ0, inv_zero']; exact zero_le _
intro x hx
suffices x ∈ integralClosure A K by
rwa [IsIntegrallyClosed.integralClosure_eq_bot, Algebra.mem_bot, Set.mem_range,
← mem_one_iff] at this
rw [mem_integralClosure_iff_mem_fg]
have x_mul_mem : ∀ b ∈ (I⁻¹ : FractionalIdeal A⁰ K), x * b ∈ (I⁻¹ : FractionalIdeal A⁰ K) := by
intro b hb
rw [mem_inv_iff (coeIdeal_ne_zero.mpr hI0)]
dsimp only at hx
rw [val_eq_coe, mem_coe, mem_inv_iff hJ0] at hx
simp only [mul_assoc, mul_comm b] at hx ⊢
intro y hy
exact hx _ (mul_mem_mul hy hb)
refine ⟨AlgHom.range (Polynomial.aeval x : A[X] →ₐ[A] K),
isNoetherian_submodule.mp (isNoetherian (I : FractionalIdeal A⁰ K)⁻¹) _ fun y hy => ?_,
⟨Polynomial.X, Polynomial.aeval_X x⟩⟩
obtain ⟨p, rfl⟩ := (AlgHom.mem_range _).mp hy
rw [Polynomial.aeval_eq_sum_range]
refine Submodule.sum_mem _ fun i hi => Submodule.smul_mem _ _ ?_
clear hi
induction i with
| zero => rw [pow_zero]; exact one_mem_inv_coe_ideal hI0
| succ i ih => rw [pow_succ']; exact x_mul_mem _ ih | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Pointwise",
"Mathlib.RingTheory.DedekindDomain.Basic",
"Mathlib.RingTheory.FractionalIdeal.Inverse",
"Mathlib.RingTheory.Spectrum.Prime.Basic"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean | coe_ideal_mul_inv | 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 mul_inv_cancel [IsDedekindDomain A] {I : FractionalIdeal A⁰ K} (hne : I ≠ 0) :
I * I⁻¹ = 1 := by
obtain ⟨a, J, ha, hJ⟩ :
∃ (a : A) (aI : Ideal A), a ≠ 0 ∧ I = spanSingleton A⁰ (algebraMap A K a)⁻¹ * aI :=
exists_eq_spanSingleton_mul I
suffices h₂ : I * (spanSingleton A⁰ (algebraMap _ _ a) * (J : FractionalIdeal A⁰ K)⁻¹) = 1 by
rw [mul_inv_cancel_iff]
exact ⟨spanSingleton A⁰ (algebraMap _ _ a) * (J : FractionalIdeal A⁰ K)⁻¹, h₂⟩
subst hJ
rw [mul_assoc, mul_left_comm (J : FractionalIdeal A⁰ K), coe_ideal_mul_inv, mul_one,
spanSingleton_mul_spanSingleton, inv_mul_cancel₀, spanSingleton_one]
· exact mt ((injective_iff_map_eq_zero (algebraMap A K)).mp (IsFractionRing.injective A K) _) ha
· exact coeIdeal_ne_zero.mp (right_ne_zero_of_mul hne) | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Pointwise",
"Mathlib.RingTheory.DedekindDomain.Basic",
"Mathlib.RingTheory.FractionalIdeal.Inverse",
"Mathlib.RingTheory.Spectrum.Prime.Basic"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean | mul_inv_cancel | Nonzero fractional ideals in a Dedekind domain are units.
This is also available as `_root_.mul_inv_cancel`, using the
`Semifield` instance defined below. |
mul_right_le_iff [IsDedekindDomain A] {J : FractionalIdeal A⁰ K} (hJ : J ≠ 0) :
∀ {I I'}, I * J ≤ I' * J ↔ I ≤ I' := by
intro I I'
constructor
· intro h
convert mul_right_mono J⁻¹ h <;> dsimp only <;>
rw [mul_assoc, FractionalIdeal.mul_inv_cancel hJ, mul_one]
· exact fun h => mul_right_mono J h | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Pointwise",
"Mathlib.RingTheory.DedekindDomain.Basic",
"Mathlib.RingTheory.FractionalIdeal.Inverse",
"Mathlib.RingTheory.Spectrum.Prime.Basic"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean | mul_right_le_iff | null |
mul_left_le_iff [IsDedekindDomain A] {J : FractionalIdeal A⁰ K} (hJ : J ≠ 0) {I I'} :
J * I ≤ J * I' ↔ I ≤ I' := by convert mul_right_le_iff hJ using 1; simp only [mul_comm] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Pointwise",
"Mathlib.RingTheory.DedekindDomain.Basic",
"Mathlib.RingTheory.FractionalIdeal.Inverse",
"Mathlib.RingTheory.Spectrum.Prime.Basic"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean | mul_left_le_iff | null |
mul_right_strictMono [IsDedekindDomain A] {I : FractionalIdeal A⁰ K} (hI : I ≠ 0) :
StrictMono (· * I) :=
strictMono_of_le_iff_le fun _ _ => (mul_right_le_iff hI).symm | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Pointwise",
"Mathlib.RingTheory.DedekindDomain.Basic",
"Mathlib.RingTheory.FractionalIdeal.Inverse",
"Mathlib.RingTheory.Spectrum.Prime.Basic"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean | mul_right_strictMono | null |
protected div_eq_mul_inv [IsDedekindDomain A] (I J : FractionalIdeal A⁰ K) :
I / J = I * J⁻¹ := by
by_cases hJ : J = 0
· rw [hJ, div_zero, inv_zero', mul_zero]
refine le_antisymm ((mul_right_le_iff hJ).mp ?_) ((le_div_iff_mul_le hJ).mpr ?_)
· rw [mul_assoc, mul_comm J⁻¹, FractionalIdeal.mul_inv_cancel hJ, mul_one, mul_le]
intro x hx y hy
rw [mem_div_iff_of_nonzero hJ] at hx
exact hx y hy
rw [mul_assoc, mul_comm J⁻¹, FractionalIdeal.mul_inv_cancel hJ, mul_one] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Pointwise",
"Mathlib.RingTheory.DedekindDomain.Basic",
"Mathlib.RingTheory.FractionalIdeal.Inverse",
"Mathlib.RingTheory.Spectrum.Prime.Basic"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean | div_eq_mul_inv | This is also available as `_root_.div_eq_mul_inv`, using the
`Semifield` instance defined below. |
isDedekindDomain_iff_isDedekindDomainInv [IsDomain A] :
IsDedekindDomain A ↔ IsDedekindDomainInv A :=
⟨fun _h _I hI => FractionalIdeal.mul_inv_cancel hI, fun h => h.isDedekindDomain⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Pointwise",
"Mathlib.RingTheory.DedekindDomain.Basic",
"Mathlib.RingTheory.FractionalIdeal.Inverse",
"Mathlib.RingTheory.Spectrum.Prime.Basic"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean | isDedekindDomain_iff_isDedekindDomainInv | `IsDedekindDomain` and `IsDedekindDomainInv` are equivalent ways
to express that an integral domain is a Dedekind domain. |
noncomputable FractionalIdeal.semifield : Semifield (FractionalIdeal A⁰ K) where
__ := coeIdeal_injective.nontrivial
inv_zero := inv_zero' _
div_eq_mul_inv := FractionalIdeal.div_eq_mul_inv
mul_inv_cancel _ := FractionalIdeal.mul_inv_cancel
nnqsmul := _
nnqsmul_def := fun _ _ => rfl | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Pointwise",
"Mathlib.RingTheory.DedekindDomain.Basic",
"Mathlib.RingTheory.FractionalIdeal.Inverse",
"Mathlib.RingTheory.Spectrum.Prime.Basic"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean | FractionalIdeal.semifield | null |
FractionalIdeal.cancelCommMonoidWithZero :
CancelCommMonoidWithZero (FractionalIdeal A⁰ K) where
__ : CommSemiring (FractionalIdeal A⁰ K) := inferInstance
mul_left_cancel_of_ne_zero h _ _ := mul_left_cancel₀ h | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Pointwise",
"Mathlib.RingTheory.DedekindDomain.Basic",
"Mathlib.RingTheory.FractionalIdeal.Inverse",
"Mathlib.RingTheory.Spectrum.Prime.Basic"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean | FractionalIdeal.cancelCommMonoidWithZero | Fractional ideals have cancellative multiplication in a Dedekind domain.
Although this instance is a direct consequence of the instance
`FractionalIdeal.semifield`, we define this instance to provide
a computable alternative. |
noncomputable Ideal.cancelCommMonoidWithZero : CancelCommMonoidWithZero (Ideal A) :=
{ Function.Injective.cancelCommMonoidWithZero (coeIdealHom A⁰ (FractionRing A)) coeIdeal_injective
(RingHom.map_zero _) (RingHom.map_one _) (RingHom.map_mul _) (RingHom.map_pow _) with } | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Pointwise",
"Mathlib.RingTheory.DedekindDomain.Basic",
"Mathlib.RingTheory.FractionalIdeal.Inverse",
"Mathlib.RingTheory.Spectrum.Prime.Basic"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean | Ideal.cancelCommMonoidWithZero | null |
Ideal.isDomain : IsDomain (Ideal A) where | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Pointwise",
"Mathlib.RingTheory.DedekindDomain.Basic",
"Mathlib.RingTheory.FractionalIdeal.Inverse",
"Mathlib.RingTheory.Spectrum.Prime.Basic"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean | Ideal.isDomain | null |
Ideal.dvd_iff_le {I J : Ideal A} : I ∣ J ↔ J ≤ I :=
⟨Ideal.le_of_dvd, fun h => by
by_cases hI : I = ⊥
· have hJ : J = ⊥ := by rwa [hI, ← eq_bot_iff] at h
rw [hI, hJ]
have hI' : (I : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr hI
have : (I : FractionalIdeal A⁰ (FractionRing A))⁻¹ * J ≤ 1 := by
rw [← inv_mul_cancel₀ hI']
exact mul_left_mono _ ((coeIdeal_le_coeIdeal _).mpr h)
obtain ⟨H, hH⟩ := le_one_iff_exists_coeIdeal.mp this
use H
refine coeIdeal_injective (show (J : FractionalIdeal A⁰ (FractionRing A)) = ↑(I * H) from ?_)
rw [coeIdeal_mul, hH, ← mul_assoc, mul_inv_cancel₀ hI', one_mul]⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Pointwise",
"Mathlib.RingTheory.DedekindDomain.Basic",
"Mathlib.RingTheory.FractionalIdeal.Inverse",
"Mathlib.RingTheory.Spectrum.Prime.Basic"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean | Ideal.dvd_iff_le | For ideals in a Dedekind domain, to divide is to contain. |
Ideal.liesOver_iff_dvd_map [Algebra R A] {p : Ideal R} {P : Ideal A} (hP : P ≠ ⊤)
[p.IsMaximal] :
P.LiesOver p ↔ P ∣ Ideal.map (algebraMap R A) p := by
rw [liesOver_iff, dvd_iff_le, under_def, map_le_iff_le_comap,
IsCoatom.le_iff_eq (by rwa [← isMaximal_def]) (comap_ne_top _ hP), eq_comm] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Pointwise",
"Mathlib.RingTheory.DedekindDomain.Basic",
"Mathlib.RingTheory.FractionalIdeal.Inverse",
"Mathlib.RingTheory.Spectrum.Prime.Basic"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean | Ideal.liesOver_iff_dvd_map | null |
Ideal.dvdNotUnit_iff_lt {I J : Ideal A} : DvdNotUnit I J ↔ J < I :=
⟨fun ⟨hI, H, hunit, hmul⟩ =>
lt_of_le_of_ne (Ideal.dvd_iff_le.mp ⟨H, hmul⟩)
(mt
(fun h =>
have : H = 1 := mul_left_cancel₀ hI (by rw [← hmul, h, mul_one])
show IsUnit H from this.symm ▸ isUnit_one)
hunit),
fun h =>
dvdNotUnit_of_dvd_of_not_dvd (Ideal.dvd_iff_le.mpr (le_of_lt h))
(mt Ideal.dvd_iff_le.mp (not_le_of_gt h))⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Pointwise",
"Mathlib.RingTheory.DedekindDomain.Basic",
"Mathlib.RingTheory.FractionalIdeal.Inverse",
"Mathlib.RingTheory.Spectrum.Prime.Basic"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean | Ideal.dvdNotUnit_iff_lt | null |
Ideal.uniqueFactorizationMonoid : UniqueFactorizationMonoid (Ideal A) :=
{ irreducible_iff_prime := by
intro P
exact ⟨fun hirr => ⟨hirr.ne_zero, hirr.not_isUnit, fun I J => by
have : P.IsMaximal := by
refine ⟨⟨mt Ideal.isUnit_iff.mpr hirr.not_isUnit, ?_⟩⟩
intro J hJ
obtain ⟨_J_ne, H, hunit, P_eq⟩ := Ideal.dvdNotUnit_iff_lt.mpr hJ
exact Ideal.isUnit_iff.mp ((hirr.isUnit_or_isUnit P_eq).resolve_right hunit)
rw [Ideal.dvd_iff_le, Ideal.dvd_iff_le, Ideal.dvd_iff_le, SetLike.le_def, SetLike.le_def,
SetLike.le_def]
contrapose!
rintro ⟨⟨x, x_mem, x_notMem⟩, ⟨y, y_mem, y_notMem⟩⟩
exact
⟨x * y, Ideal.mul_mem_mul x_mem y_mem,
mt this.isPrime.mem_or_mem (not_or_intro x_notMem y_notMem)⟩⟩, Prime.irreducible⟩ } | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Pointwise",
"Mathlib.RingTheory.DedekindDomain.Basic",
"Mathlib.RingTheory.FractionalIdeal.Inverse",
"Mathlib.RingTheory.Spectrum.Prime.Basic"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean | Ideal.uniqueFactorizationMonoid | null |
noncomputable Ideal.normalizationMonoid : NormalizationMonoid (Ideal A) := .ofUniqueUnits | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Pointwise",
"Mathlib.RingTheory.DedekindDomain.Basic",
"Mathlib.RingTheory.FractionalIdeal.Inverse",
"Mathlib.RingTheory.Spectrum.Prime.Basic"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean | Ideal.normalizationMonoid | null |
exists_notMem_one_of_ne_bot [IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
(hI1 : I ≠ ⊤) : ∃ x ∈ (I⁻¹ : FractionalIdeal A⁰ K), x ∉ (1 : FractionalIdeal A⁰ K) :=
Set.not_subset.1 <| not_inv_le_one_of_ne_bot hI0 hI1
@[deprecated (since := "2025-05-23")]
alias exists_not_mem_one_of_ne_bot := exists_notMem_one_of_ne_bot | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | exists_notMem_one_of_ne_bot | null |
mul_left_strictMono [IsDedekindDomain A] {I : FractionalIdeal A⁰ K} (hI : I ≠ 0) :
StrictMono (I * ·) :=
strictMono_of_le_iff_le fun _ _ => (mul_left_le_iff hI).symm | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | mul_left_strictMono | null |
@[simp]
Ideal.dvd_span_singleton {I : Ideal A} {x : A} : I ∣ Ideal.span {x} ↔ x ∈ I :=
Ideal.dvd_iff_le.trans (Ideal.span_le.trans Set.singleton_subset_iff) | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.dvd_span_singleton | null |
Ideal.isPrime_of_prime {P : Ideal A} (h : Prime P) : IsPrime P := by
refine ⟨?_, fun hxy => ?_⟩
· rintro rfl
rw [← Ideal.one_eq_top] at h
exact h.not_unit isUnit_one
· simp only [← Ideal.dvd_span_singleton, ← Ideal.span_singleton_mul_span_singleton] at hxy ⊢
exact h.dvd_or_dvd hxy | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.isPrime_of_prime | null |
Ideal.prime_of_isPrime {P : Ideal A} (hP : P ≠ ⊥) (h : IsPrime P) : Prime P := by
refine ⟨hP, mt Ideal.isUnit_iff.mp h.ne_top, fun I J hIJ => ?_⟩
simpa only [Ideal.dvd_iff_le] using h.mul_le.mp (Ideal.le_of_dvd hIJ) | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.prime_of_isPrime | null |
Ideal.prime_iff_isPrime {P : Ideal A} (hP : P ≠ ⊥) : Prime P ↔ IsPrime P :=
⟨Ideal.isPrime_of_prime, Ideal.prime_of_isPrime hP⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.prime_iff_isPrime | In a Dedekind domain, the (nonzero) prime elements of the monoid with zero `Ideal A`
are exactly the prime ideals. |
Ideal.isPrime_iff_bot_or_prime {P : Ideal A} : IsPrime P ↔ P = ⊥ ∨ Prime P :=
⟨fun hp => (eq_or_ne P ⊥).imp_right fun hp0 => Ideal.prime_of_isPrime hp0 hp, fun hp =>
hp.elim (fun h => h.symm ▸ Ideal.bot_prime) Ideal.isPrime_of_prime⟩
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.isPrime_iff_bot_or_prime | In a Dedekind domain, the prime ideals are the zero ideal together with the prime elements
of the monoid with zero `Ideal A`. |
Ideal.prime_span_singleton_iff {a : A} : Prime (Ideal.span {a}) ↔ Prime a := by
rcases eq_or_ne a 0 with rfl | ha
· rw [Set.singleton_zero, span_zero, ← Ideal.zero_eq_bot, ← not_iff_not]
simp only [not_prime_zero, not_false_eq_true]
· have ha' : span {a} ≠ ⊥ := by simpa only [ne_eq, span_singleton_eq_bot] using ha
rw [Ideal.prime_iff_isPrime ha', Ideal.span_singleton_prime ha]
open Submodule.IsPrincipal in | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.prime_span_singleton_iff | null |
Ideal.prime_generator_of_prime {P : Ideal A} (h : Prime P) [P.IsPrincipal] :
Prime (generator P) :=
have : Ideal.IsPrime P := Ideal.isPrime_of_prime h
prime_generator_of_isPrime _ h.ne_zero
open UniqueFactorizationMonoid in
nonrec theorem Ideal.mem_normalizedFactors_iff {p I : Ideal A} (hI : I ≠ ⊥) :
p ∈ normalizedFactors I ↔ p.IsPrime ∧ I ≤ p := by
rw [← Ideal.dvd_iff_le]
by_cases hp : p = 0
· rw [← zero_eq_bot] at hI
simp only [hp, zero_notMem_normalizedFactors, zero_dvd_iff, hI, false_iff, not_and,
not_false_eq_true, implies_true]
· rwa [mem_normalizedFactors_iff hI, prime_iff_isPrime]
variable (A) in
open UniqueFactorizationMonoid in | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.prime_generator_of_prime | null |
Ideal.mem_primesOver_iff_mem_normalizedFactors {p : Ideal R} [h : p.IsMaximal]
[Algebra R A] [NoZeroSMulDivisors R A] (hp : p ≠ ⊥) {P : Ideal A} :
P ∈ p.primesOver A ↔ P ∈ normalizedFactors (Ideal.map (algebraMap R A) p) := by
rw [primesOver, Set.mem_setOf_eq, mem_normalizedFactors_iff (map_ne_bot_of_ne_bot hp),
liesOver_iff, under_def, and_congr_right_iff, map_le_iff_le_comap]
intro hP
refine ⟨fun h ↦ le_of_eq h, fun h' ↦ ((IsCoatom.le_iff_eq (isMaximal_def.mp h) ?_).mp h').symm⟩
exact comap_ne_top (algebraMap R A) (IsPrime.ne_top hP) | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.mem_primesOver_iff_mem_normalizedFactors | null |
Ideal.pow_right_strictAnti (I : Ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
StrictAnti (I ^ · : ℕ → Ideal A) :=
strictAnti_nat_of_succ_lt fun e =>
Ideal.dvdNotUnit_iff_lt.mp ⟨pow_ne_zero _ hI0, I, mt isUnit_iff.mp hI1, pow_succ I e⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.pow_right_strictAnti | null |
Ideal.pow_lt_self (I : Ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) (e : ℕ) (he : 2 ≤ e) :
I ^ e < I := by
convert I.pow_right_strictAnti hI0 hI1 he
dsimp only
rw [pow_one] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.pow_lt_self | null |
Ideal.exists_mem_pow_notMem_pow_succ (I : Ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) (e : ℕ) :
∃ x ∈ I ^ e, x ∉ I ^ (e + 1) :=
SetLike.exists_of_lt (I.pow_right_strictAnti hI0 hI1 e.lt_succ_self)
@[deprecated (since := "2025-05-23")]
alias Ideal.exists_mem_pow_not_mem_pow_succ := Ideal.exists_mem_pow_notMem_pow_succ
open UniqueFactorizationMonoid | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.exists_mem_pow_notMem_pow_succ | null |
Ideal.eq_prime_pow_of_succ_lt_of_le {P I : Ideal A} [P_prime : P.IsPrime] (hP : P ≠ ⊥)
{i : ℕ} (hlt : P ^ (i + 1) < I) (hle : I ≤ P ^ i) : I = P ^ i := by
refine le_antisymm hle ?_
have P_prime' := Ideal.prime_of_isPrime hP P_prime
have h1 : I ≠ ⊥ := (lt_of_le_of_lt bot_le hlt).ne'
have := pow_ne_zero i hP
have h3 := pow_ne_zero (i + 1) hP
rw [← Ideal.dvdNotUnit_iff_lt, dvdNotUnit_iff_normalizedFactors_lt_normalizedFactors h1 h3,
normalizedFactors_pow, normalizedFactors_irreducible P_prime'.irreducible,
Multiset.nsmul_singleton, Multiset.lt_replicate_succ] at hlt
rw [← Ideal.dvd_iff_le, dvd_iff_normalizedFactors_le_normalizedFactors, normalizedFactors_pow,
normalizedFactors_irreducible P_prime'.irreducible, Multiset.nsmul_singleton]
all_goals assumption | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.eq_prime_pow_of_succ_lt_of_le | null |
Ideal.pow_succ_lt_pow {P : Ideal A} [P_prime : P.IsPrime] (hP : P ≠ ⊥) (i : ℕ) :
P ^ (i + 1) < P ^ i :=
lt_of_le_of_ne (Ideal.pow_le_pow_right (Nat.le_succ _))
(mt (pow_inj_of_not_isUnit (mt Ideal.isUnit_iff.mp P_prime.ne_top) hP).mp i.succ_ne_self) | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.pow_succ_lt_pow | null |
Associates.le_singleton_iff (x : A) (n : ℕ) (I : Ideal A) :
Associates.mk I ^ n ≤ Associates.mk (Ideal.span {x}) ↔ x ∈ I ^ n := by
simp_rw [← Associates.dvd_eq_le, ← Associates.mk_pow, Associates.mk_dvd_mk,
Ideal.dvd_span_singleton]
variable {K} | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Associates.le_singleton_iff | null |
FractionalIdeal.le_inv_comm {I J : FractionalIdeal A⁰ K} (hI : I ≠ 0) (hJ : J ≠ 0) :
I ≤ J⁻¹ ↔ J ≤ I⁻¹ := by
rw [inv_eq, inv_eq, le_div_iff_mul_le hI, le_div_iff_mul_le hJ, mul_comm] | lemma | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | FractionalIdeal.le_inv_comm | null |
FractionalIdeal.inv_le_comm {I J : FractionalIdeal A⁰ K} (hI : I ≠ 0) (hJ : J ≠ 0) :
I⁻¹ ≤ J ↔ J⁻¹ ≤ I := by
simpa using le_inv_comm (A := A) (K := K) (inv_ne_zero hI) (inv_ne_zero hJ)
open FractionalIdeal | lemma | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | FractionalIdeal.inv_le_comm | null |
Ideal.exist_integer_multiples_notMem {J : Ideal A} (hJ : J ≠ ⊤) {ι : Type*} (s : Finset ι)
(f : ι → K) {j} (hjs : j ∈ s) (hjf : f j ≠ 0) :
∃ a : K,
(∀ i ∈ s, IsLocalization.IsInteger A (a * f i)) ∧
∃ i ∈ s, a * f i ∉ (J : FractionalIdeal A⁰ K) := by
let I : FractionalIdeal A⁰ K := spanFinset A s f
have hI0 : I ≠ 0 := spanFinset_ne_zero.mpr ⟨j, hjs, hjf⟩
suffices ↑J / I < I⁻¹ by
obtain ⟨_, a, hI, hpI⟩ := SetLike.lt_iff_le_and_exists.mp this
rw [mem_inv_iff hI0] at hI
refine ⟨a, fun i hi => ?_, ?_⟩
· exact (mem_one_iff _).mp (hI (f i) (Submodule.subset_span (Set.mem_image_of_mem f hi)))
· contrapose! hpI
refine (mem_div_iff_of_nonzero hI0).mpr fun y hy => Submodule.span_induction ?_ ?_ ?_ ?_ hy
· rintro _ ⟨i, hi, rfl⟩; exact hpI i hi
· rw [mul_zero]; exact Submodule.zero_mem _
· intro x y _ _ hx hy; rw [mul_add]; exact Submodule.add_mem _ hx hy
· intro b x _ hx; rw [mul_smul_comm]; exact Submodule.smul_mem _ b hx
calc
↑J / I = ↑J * I⁻¹ := div_eq_mul_inv (↑J) I
_ < 1 * I⁻¹ := mul_right_strictMono (inv_ne_zero hI0) ?_
_ = I⁻¹ := one_mul _
rw [← coeIdeal_top]
exact
strictMono_of_le_iff_le (fun _ _ => (coeIdeal_le_coeIdeal K).symm)
(lt_top_iff_ne_top.mpr hJ)
@[deprecated (since := "2025-05-23")]
alias Ideal.exist_integer_multiples_not_mem := Ideal.exist_integer_multiples_notMem | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.exist_integer_multiples_notMem | Strengthening of `IsLocalization.exist_integer_multiples`:
Let `J ≠ ⊤` be an ideal in a Dedekind domain `A`, and `f ≠ 0` a finite collection
of elements of `K = Frac(A)`, then we can multiply the elements of `f` by some `a : K`
to find a collection of elements of `A` that is not completely contained in `J`. |
Ideal.mul_iInf (I : Ideal A) {ι : Type*} [Nonempty ι] (J : ι → Ideal A) :
I * ⨅ i, J i = ⨅ i, I * J i := by
by_cases hI : I = 0
· simp [hI]
refine (le_iInf fun i ↦ Ideal.mul_mono_right (iInf_le _ _)).antisymm ?_
have H : ⨅ i, I * J i ≤ I := (iInf_le _ (Nonempty.some ‹_›)).trans Ideal.mul_le_right
obtain ⟨K, hK⟩ := Ideal.dvd_iff_le.mpr H
rw [hK]
refine mul_le_mul_left' ?_ I
rw [le_iInf_iff]
intro i
rw [← mul_le_mul_iff_of_pos_left (a := I), ← hK]
· exact iInf_le _ _
· exact bot_lt_iff_ne_bot.mpr hI | lemma | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.mul_iInf | null |
Ideal.iInf_mul (I : Ideal A) {ι : Type*} [Nonempty ι] (J : ι → Ideal A) :
(⨅ i, J i) * I = ⨅ i, J i * I := by
simp only [mul_iInf, mul_comm _ I] | lemma | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.iInf_mul | null |
Ideal.mul_inf (I J K : Ideal A) : I * (J ⊓ K) = I * J ⊓ I * K := by
rw [inf_eq_iInf, Ideal.mul_iInf, inf_eq_iInf]
congr! 2 with ⟨⟩ | lemma | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.mul_inf | null |
Ideal.inf_mul (I J K : Ideal A) : (I ⊓ J) * K = I * K ⊓ J * K := by
simp only [Ideal.mul_inf, mul_comm _ K] | lemma | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.inf_mul | null |
FractionalIdeal.mul_inf (I J K : FractionalIdeal A⁰ K) : I * (J ⊓ K) = I * J ⊓ I * K :=
mul_inf₀ (zero_le _) _ _ | lemma | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | FractionalIdeal.mul_inf | null |
FractionalIdeal.inf_mul (I J K : FractionalIdeal A⁰ K) : (I ⊓ J) * K = I * K ⊓ J * K :=
inf_mul₀ (zero_le _) _ _ | lemma | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | FractionalIdeal.inf_mul | null |
@[simp]
sup_mul_inf (I J : Ideal A) : (I ⊔ J) * (I ⊓ J) = I * J := by
letI := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
have hgcd : gcd I J = I ⊔ J := by
rw [gcd_eq_normalize _ _, normalize_eq]
· rw [dvd_iff_le, sup_le_iff, ← dvd_iff_le, ← dvd_iff_le]
exact ⟨gcd_dvd_left _ _, gcd_dvd_right _ _⟩
· rw [dvd_gcd_iff, dvd_iff_le, dvd_iff_le]
simp
have hlcm : lcm I J = I ⊓ J := by
rw [lcm_eq_normalize _ _, normalize_eq]
· rw [lcm_dvd_iff, dvd_iff_le, dvd_iff_le]
simp
· rw [dvd_iff_le, le_inf_iff, ← dvd_iff_le, ← dvd_iff_le]
exact ⟨dvd_lcm_left _ _, dvd_lcm_right _ _⟩
rw [← hgcd, ← hlcm, associated_iff_eq.mp (gcd_mul_lcm _ _)] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | sup_mul_inf | null |
@[ext, nolint unusedArguments]
HeightOneSpectrum where
asIdeal : Ideal R
isPrime : asIdeal.IsPrime
ne_bot : asIdeal ≠ ⊥
attribute [instance] HeightOneSpectrum.isPrime
variable (v : HeightOneSpectrum R) {R} | structure | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | HeightOneSpectrum | Ideals in a Dedekind domain have gcd and lcm operators that (trivially) are compatible with
the normalization operator. -/
noncomputable instance : NormalizedGCDMonoid (Ideal A) :=
{ Ideal.normalizationMonoid with
gcd := (· ⊔ ·)
gcd_dvd_left := fun _ _ => by simpa only [dvd_iff_le] using le_sup_left
gcd_dvd_right := fun _ _ => by simpa only [dvd_iff_le] using le_sup_right
dvd_gcd := by
simp only [dvd_iff_le]
exact fun h1 h2 => @sup_le (Ideal A) _ _ _ _ h1 h2
lcm := (· ⊓ ·)
lcm_zero_left := fun _ => by simp only [zero_eq_bot, bot_inf_eq]
lcm_zero_right := fun _ => by simp only [zero_eq_bot, inf_bot_eq]
gcd_mul_lcm := fun _ _ => by rw [associated_iff_eq, sup_mul_inf]
normalize_gcd := fun _ _ => normalize_eq _
normalize_lcm := fun _ _ => normalize_eq _ }
-- In fact, any lawful gcd and lcm would equal sup and inf respectively.
@[simp]
theorem gcd_eq_sup (I J : Ideal A) : gcd I J = I ⊔ J := rfl
@[simp]
theorem lcm_eq_inf (I J : Ideal A) : lcm I J = I ⊓ J := rfl
theorem isCoprime_iff_gcd {I J : Ideal A} : IsCoprime I J ↔ gcd I J = 1 := by
rw [Ideal.isCoprime_iff_codisjoint, codisjoint_iff, one_eq_top, gcd_eq_sup]
theorem factors_span_eq {p : K[X]} : factors (span {p}) = (factors p).map (fun q ↦ span {q}) := by
rcases eq_or_ne p 0 with rfl | hp; · simpa [Set.singleton_zero] using normalizedFactors_zero
have : ∀ q ∈ (factors p).map (fun q ↦ span {q}), Prime q := fun q hq ↦ by
obtain ⟨r, hr, rfl⟩ := Multiset.mem_map.mp hq
exact prime_span_singleton_iff.mpr <| prime_of_factor r hr
rw [← span_singleton_eq_span_singleton.mpr (factors_prod hp), ← multiset_prod_span_singleton,
factors_eq_normalizedFactors, normalizedFactors_prod_of_prime this]
lemma _root_.FractionalIdeal.sup_mul_inf (I J : FractionalIdeal A⁰ K) :
(I ⊓ J) * (I ⊔ J) = I * J := by
apply mul_left_injective₀ (b := spanSingleton A⁰ (algebraMap A K
(I.den.1 * I.den.1 * J.den.1 * J.den.1))) (by simp [spanSingleton_eq_zero_iff])
have := Ideal.sup_mul_inf (Ideal.span {J.den.1} * I.num) (Ideal.span {I.den.1} * J.num)
simp only [← coeIdeal_inj (K := K), coeIdeal_mul, coeIdeal_sup, coeIdeal_inf,
← den_mul_self_eq_num', coeIdeal_span_singleton] at this
rw [mul_left_comm, ← mul_add, ← mul_add, ← mul_inf₀ (FractionalIdeal.zero_le _),
← mul_inf₀ (FractionalIdeal.zero_le _)] at this
simp only [FractionalIdeal.sup_eq_add, _root_.map_mul, ← spanSingleton_mul_spanSingleton]
convert this using 1 <;> ring
end Ideal
end Gcd
end IsDedekindDomain
section IsDedekindDomain
variable {T : Type*} [CommRing T] [IsDedekindDomain T] {I J : Ideal T}
open Multiset UniqueFactorizationMonoid Ideal
theorem prod_normalizedFactors_eq_self (hI : I ≠ ⊥) : (normalizedFactors I).prod = I :=
associated_iff_eq.1 (prod_normalizedFactors hI)
theorem count_le_of_ideal_ge [DecidableEq (Ideal T)]
{I J : Ideal T} (h : I ≤ J) (hI : I ≠ ⊥) (K : Ideal T) :
count K (normalizedFactors J) ≤ count K (normalizedFactors I) :=
le_iff_count.1 ((dvd_iff_normalizedFactors_le_normalizedFactors (ne_bot_of_le_ne_bot hI h) hI).1
(dvd_iff_le.2 h))
_
theorem sup_eq_prod_inf_factors [DecidableEq (Ideal T)] (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod := by
have H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod =
normalizedFactors I ∩ normalizedFactors J := by
apply normalizedFactors_prod_of_prime
intro p hp
rw [mem_inter] at hp
exact prime_of_normalized_factor p hp.left
have := Multiset.prod_ne_zero_of_prime (normalizedFactors I ∩ normalizedFactors J) fun _ h =>
prime_of_normalized_factor _ (Multiset.mem_inter.1 h).1
apply le_antisymm
· rw [sup_le_iff, ← dvd_iff_le, ← dvd_iff_le]
constructor
· rw [dvd_iff_normalizedFactors_le_normalizedFactors this hI, H]
exact inf_le_left
· rw [dvd_iff_normalizedFactors_le_normalizedFactors this hJ, H]
exact inf_le_right
· rw [← dvd_iff_le, dvd_iff_normalizedFactors_le_normalizedFactors,
normalizedFactors_prod_of_prime, le_iff_count]
· intro a
rw [Multiset.count_inter]
exact le_min (count_le_of_ideal_ge le_sup_left hI a) (count_le_of_ideal_ge le_sup_right hJ a)
· intro p hp
rw [mem_inter] at hp
exact prime_of_normalized_factor p hp.left
· exact ne_bot_of_le_ne_bot hI le_sup_left
· exact this
theorem irreducible_pow_sup [DecidableEq (Ideal T)] (hI : I ≠ ⊥) (hJ : Irreducible J) (n : ℕ) :
J ^ n ⊔ I = J ^ min ((normalizedFactors I).count J) n := by
rw [sup_eq_prod_inf_factors (pow_ne_zero n hJ.ne_zero) hI, min_comm,
normalizedFactors_of_irreducible_pow hJ, normalize_eq J, replicate_inter, prod_replicate]
theorem irreducible_pow_sup_of_le (hJ : Irreducible J) (n : ℕ) (hn : n ≤ emultiplicity J I) :
J ^ n ⊔ I = J ^ n := by
classical
by_cases hI : I = ⊥
· simp_all
rw [irreducible_pow_sup hI hJ, min_eq_right]
rw [emultiplicity_eq_count_normalizedFactors hJ hI, normalize_eq J] at hn
exact_mod_cast hn
theorem irreducible_pow_sup_of_ge (hI : I ≠ ⊥) (hJ : Irreducible J) (n : ℕ)
(hn : emultiplicity J I ≤ n) : J ^ n ⊔ I = J ^ multiplicity J I := by
classical
rw [irreducible_pow_sup hI hJ, min_eq_left]
· congr
rw [← Nat.cast_inj (R := ℕ∞), ← FiniteMultiplicity.emultiplicity_eq_multiplicity,
emultiplicity_eq_count_normalizedFactors hJ hI, normalize_eq J]
rw [← emultiplicity_lt_top]
apply hn.trans_lt
simp
· rw [emultiplicity_eq_count_normalizedFactors hJ hI, normalize_eq J] at hn
exact_mod_cast hn
theorem Ideal.eq_prime_pow_mul_coprime [DecidableEq (Ideal T)] {I : Ideal T} (hI : I ≠ ⊥)
(P : Ideal T) [hpm : P.IsMaximal] :
∃ Q : Ideal T, P ⊔ Q = ⊤ ∧ I = P ^ (Multiset.count P (normalizedFactors I)) * Q := by
use (filter (¬ P = ·) (normalizedFactors I)).prod
constructor
· refine P.sup_multiset_prod_eq_top (fun p hpi ↦ ?_)
have hp : Prime p := prime_of_normalized_factor p (filter_subset _ (normalizedFactors I) hpi)
exact hpm.coprime_of_ne ((isPrime_of_prime hp).isMaximal hp.ne_zero) (of_mem_filter hpi)
· nth_rw 1 [← prod_normalizedFactors_eq_self hI, ← filter_add_not (P = ·) (normalizedFactors I)]
rw [prod_add, pow_count]
end IsDedekindDomain
/-!
### Height one spectrum of a Dedekind domain
If `R` is a Dedekind domain of Krull dimension 1, the maximal ideals of `R` are exactly its nonzero
prime ideals.
We define `HeightOneSpectrum` and provide lemmas to recover the facts that prime ideals of height
one are prime and irreducible.
-/
namespace IsDedekindDomain
variable [IsDedekindDomain R]
/-- The height one prime spectrum of a Dedekind domain `R` is the type of nonzero prime ideals of
`R`. Note that this equals the maximal spectrum if `R` has Krull dimension 1. |
isMaximal : v.asIdeal.IsMaximal := v.isPrime.isMaximal v.ne_bot | instance | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | isMaximal | null |
prime : Prime v.asIdeal := Ideal.prime_of_isPrime v.ne_bot v.isPrime | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | prime | null |
@[simps]
ofPrime {p : Ideal R} (hp : Prime p) : HeightOneSpectrum R :=
⟨p, Ideal.isPrime_of_prime hp, hp.ne_zero⟩ | def | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | ofPrime | The (nonzero) prime elements of the monoid with zero `Ideal R` correspond
to an element of type `HeightOneSpectrum R`.
See `IsDedekindDomain.HeightOneSpectrum.prime` for the inverse direction. |
irreducible : Irreducible v.asIdeal :=
UniqueFactorizationMonoid.irreducible_iff_prime.mpr v.prime | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | irreducible | null |
associates_irreducible : Irreducible <| Associates.mk v.asIdeal :=
Associates.irreducible_mk.mpr v.irreducible | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | associates_irreducible | null |
equivMaximalSpectrum (hR : ¬IsField R) : HeightOneSpectrum R ≃ MaximalSpectrum R where
toFun v := ⟨v.asIdeal, v.isPrime.isMaximal v.ne_bot⟩
invFun v :=
⟨v.asIdeal, v.isMaximal.isPrime, Ring.ne_bot_of_isMaximal_of_not_isField v.isMaximal hR⟩ | def | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | equivMaximalSpectrum | An equivalence between the height one and maximal spectra for rings of Krull dimension 1. |
ideal_ne_top_iff_exists (hR : ¬IsField R) (I : Ideal R) :
I ≠ ⊤ ↔ ∃ P : HeightOneSpectrum R, I ≤ P.asIdeal := by
rw [Ideal.ne_top_iff_exists_maximal]
constructor
· rintro ⟨M, hMmax, hIM⟩
exact ⟨(equivMaximalSpectrum hR).symm ⟨M, hMmax⟩, hIM⟩
· rintro ⟨P, hP⟩
exact ⟨((equivMaximalSpectrum hR) P).asIdeal, ((equivMaximalSpectrum hR) P).isMaximal, hP⟩
variable (R) | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | ideal_ne_top_iff_exists | An ideal of `R` is not the whole ring if and only if it is contained in an element of
`HeightOneSpectrum R` |
iInf_localization_eq_bot [Algebra R K] [hK : IsFractionRing R K] :
(⨅ v : HeightOneSpectrum R,
Localization.subalgebra.ofField K _ v.asIdeal.primeCompl_le_nonZeroDivisors) = ⊥ := by
ext x
rw [Algebra.mem_iInf]
constructor
on_goal 1 => by_cases hR : IsField R
· rcases Function.bijective_iff_has_inverse.mp
(IsField.localization_map_bijective (Rₘ := K) (flip nonZeroDivisors.ne_zero rfl : 0 ∉ R⁰) hR)
with ⟨algebra_map_inv, _, algebra_map_right_inv⟩
exact fun _ => Algebra.mem_bot.mpr ⟨algebra_map_inv x, algebra_map_right_inv x⟩
all_goals rw [← MaximalSpectrum.iInf_localization_eq_bot, Algebra.mem_iInf]
· exact fun hx ⟨v, hv⟩ => hx ((equivMaximalSpectrum hR).symm ⟨v, hv⟩)
· exact fun hx ⟨v, hv, hbot⟩ => hx ⟨v, hv.isMaximal hbot⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | iInf_localization_eq_bot | A Dedekind domain is equal to the intersection of its localizations at all its height one
non-zero prime ideals viewed as subalgebras of its field of fractions. |
@[simps]
idealFactorsFunOfQuotHom {f : R ⧸ I →+* A ⧸ J} (hf : Function.Surjective f) :
{p : Ideal R // p ∣ I} →o {p : Ideal A // p ∣ J} where
toFun X := ⟨comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) X)), by
have : RingHom.ker (Ideal.Quotient.mk J) ≤
comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) X)) :=
ker_le_comap (Ideal.Quotient.mk J)
rw [mk_ker] at this
exact dvd_iff_le.mpr this⟩
monotone' := by
rintro ⟨X, hX⟩ ⟨Y, hY⟩ h
rw [← Subtype.coe_le_coe, Subtype.coe_mk, Subtype.coe_mk] at h ⊢
rw [Subtype.coe_mk, comap_le_comap_iff_of_surjective (Ideal.Quotient.mk J)
Ideal.Quotient.mk_surjective, map_le_iff_le_comap, Subtype.coe_mk,
comap_map_of_surjective _ hf (map (Ideal.Quotient.mk I) Y)]
suffices map (Ideal.Quotient.mk I) X ≤ map (Ideal.Quotient.mk I) Y by
exact le_sup_of_le_left this
rwa [map_le_iff_le_comap, comap_map_of_surjective (Ideal.Quotient.mk I)
Ideal.Quotient.mk_surjective, ← RingHom.ker_eq_comap_bot, mk_ker,
sup_eq_left.mpr <| le_of_dvd hY]
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | idealFactorsFunOfQuotHom | The map from ideals of `R` dividing `I` to the ideals of `A` dividing `J` induced by
a homomorphism `f : R/I →+* A/J` |
idealFactorsFunOfQuotHom_id :
idealFactorsFunOfQuotHom (RingHom.id (A ⧸ J)).surjective = OrderHom.id :=
OrderHom.ext _ _
(funext fun X => by
simp only [idealFactorsFunOfQuotHom, map_id, OrderHom.coe_mk, OrderHom.id_coe, id,
comap_map_of_surjective (Ideal.Quotient.mk J) Ideal.Quotient.mk_surjective, ←
RingHom.ker_eq_comap_bot (Ideal.Quotient.mk J), mk_ker,
sup_eq_left.mpr (dvd_iff_le.mp X.prop), Subtype.coe_eta])
variable {B : Type*} [CommRing B] [IsDedekindDomain B] {L : Ideal B} | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | idealFactorsFunOfQuotHom_id | null |
idealFactorsFunOfQuotHom_comp {f : R ⧸ I →+* A ⧸ J} {g : A ⧸ J →+* B ⧸ L}
(hf : Function.Surjective f) (hg : Function.Surjective g) :
(idealFactorsFunOfQuotHom hg).comp (idealFactorsFunOfQuotHom hf) =
idealFactorsFunOfQuotHom (show Function.Surjective (g.comp f) from hg.comp hf) := by
refine OrderHom.ext _ _ (funext fun x => ?_)
rw [idealFactorsFunOfQuotHom, idealFactorsFunOfQuotHom, OrderHom.comp_coe, OrderHom.coe_mk,
OrderHom.coe_mk, Function.comp_apply, idealFactorsFunOfQuotHom, OrderHom.coe_mk,
Subtype.mk_eq_mk, Subtype.coe_mk, map_comap_of_surjective (Ideal.Quotient.mk J)
Ideal.Quotient.mk_surjective, map_map]
variable [IsDedekindDomain R] (f : R ⧸ I ≃+* A ⧸ J) | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | idealFactorsFunOfQuotHom_comp | null |
idealFactorsEquivOfQuotEquiv : { p : Ideal R | p ∣ I } ≃o { p : Ideal A | p ∣ J } := by
have f_surj : Function.Surjective (f : R ⧸ I →+* A ⧸ J) := f.surjective
have fsym_surj : Function.Surjective (f.symm : A ⧸ J →+* R ⧸ I) := f.symm.surjective
refine OrderIso.ofHomInv (idealFactorsFunOfQuotHom f_surj) (idealFactorsFunOfQuotHom fsym_surj)
?_ ?_
· have := idealFactorsFunOfQuotHom_comp fsym_surj f_surj
simp only [RingEquiv.comp_symm, idealFactorsFunOfQuotHom_id] at this
rw [← this, OrderHom.coe_eq, OrderHom.coe_eq]
· have := idealFactorsFunOfQuotHom_comp f_surj fsym_surj
simp only [RingEquiv.symm_comp, idealFactorsFunOfQuotHom_id] at this
rw [← this, OrderHom.coe_eq, OrderHom.coe_eq] | def | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | idealFactorsEquivOfQuotEquiv | The bijection between ideals of `R` dividing `I` and the ideals of `A` dividing `J` induced by
an isomorphism `f : R/I ≅ A/J`. |
idealFactorsEquivOfQuotEquiv_symm :
(idealFactorsEquivOfQuotEquiv f).symm = idealFactorsEquivOfQuotEquiv f.symm := rfl | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | idealFactorsEquivOfQuotEquiv_symm | null |
idealFactorsEquivOfQuotEquiv_is_dvd_iso {L M : Ideal R} (hL : L ∣ I) (hM : M ∣ I) :
(idealFactorsEquivOfQuotEquiv f ⟨L, hL⟩ : Ideal A) ∣ idealFactorsEquivOfQuotEquiv f ⟨M, hM⟩ ↔
L ∣ M := by
suffices
idealFactorsEquivOfQuotEquiv f ⟨M, hM⟩ ≤ idealFactorsEquivOfQuotEquiv f ⟨L, hL⟩ ↔
(⟨M, hM⟩ : { p : Ideal R | p ∣ I }) ≤ ⟨L, hL⟩
by rw [dvd_iff_le, dvd_iff_le, Subtype.coe_le_coe, this, Subtype.mk_le_mk]
exact (idealFactorsEquivOfQuotEquiv f).le_iff_le
open UniqueFactorizationMonoid | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | idealFactorsEquivOfQuotEquiv_is_dvd_iso | null |
idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors (hJ : J ≠ ⊥)
{L : Ideal R} (hL : L ∈ normalizedFactors I) :
↑(idealFactorsEquivOfQuotEquiv f ⟨L, dvd_of_mem_normalizedFactors hL⟩)
∈ normalizedFactors J := by
have hI : I ≠ ⊥ := by
intro hI
rw [hI, bot_eq_zero, normalizedFactors_zero, ← Multiset.empty_eq_zero] at hL
exact Finset.notMem_empty _ hL
refine mem_normalizedFactors_factor_dvd_iso_of_mem_normalizedFactors hI hJ hL
(d := (idealFactorsEquivOfQuotEquiv f).toEquiv) ?_
rintro ⟨l, hl⟩ ⟨l', hl'⟩
rw [Subtype.coe_mk, Subtype.coe_mk]
apply idealFactorsEquivOfQuotEquiv_is_dvd_iso f | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors | null |
normalizedFactorsEquivOfQuotEquiv (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
{ L : Ideal R | L ∈ normalizedFactors I } ≃ { M : Ideal A | M ∈ normalizedFactors J } where
toFun j :=
⟨idealFactorsEquivOfQuotEquiv f ⟨↑j, dvd_of_mem_normalizedFactors j.prop⟩,
idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors f hJ j.prop⟩
invFun j :=
⟨(idealFactorsEquivOfQuotEquiv f).symm ⟨↑j, dvd_of_mem_normalizedFactors j.prop⟩, by
rw [idealFactorsEquivOfQuotEquiv_symm]
exact
idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors f.symm hI
j.prop⟩
left_inv := fun ⟨j, hj⟩ => by simp
right_inv := fun ⟨j, hj⟩ => by simp [-Set.coe_setOf]
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | normalizedFactorsEquivOfQuotEquiv | The bijection between the sets of normalized factors of I and J induced by a ring
isomorphism `f : R/I ≅ A/J`. |
normalizedFactorsEquivOfQuotEquiv_symm (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
(normalizedFactorsEquivOfQuotEquiv f hI hJ).symm =
normalizedFactorsEquivOfQuotEquiv f.symm hJ hI := rfl | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | normalizedFactorsEquivOfQuotEquiv_symm | null |
normalizedFactorsEquivOfQuotEquiv_emultiplicity_eq_emultiplicity (hI : I ≠ ⊥) (hJ : J ≠ ⊥)
(L : Ideal R) (hL : L ∈ normalizedFactors I) :
emultiplicity (↑(normalizedFactorsEquivOfQuotEquiv f hI hJ ⟨L, hL⟩)) J = emultiplicity L I := by
rw [normalizedFactorsEquivOfQuotEquiv, Equiv.coe_fn_mk, Subtype.coe_mk]
refine emultiplicity_factor_dvd_iso_eq_emultiplicity_of_mem_normalizedFactors hI hJ hL
(d := (idealFactorsEquivOfQuotEquiv f).toEquiv) ?_
exact fun ⟨l, hl⟩ ⟨l', hl'⟩ => idealFactorsEquivOfQuotEquiv_is_dvd_iso f hl hl' | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | normalizedFactorsEquivOfQuotEquiv_emultiplicity_eq_emultiplicity | The map `normalizedFactorsEquivOfQuotEquiv` preserves multiplicities. |
Ring.DimensionLeOne.prime_le_prime_iff_eq [Ring.DimensionLEOne R] {P Q : Ideal R}
[hP : P.IsPrime] [hQ : Q.IsPrime] (hP0 : P ≠ ⊥) : P ≤ Q ↔ P = Q :=
⟨(hP.isMaximal hP0).eq_of_le hQ.ne_top, Eq.le⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ring.DimensionLeOne.prime_le_prime_iff_eq | null |
Ideal.IsPrime.mul_mem_pow (I : Ideal R) [hI : I.IsPrime] {a b : R} {n : ℕ}
(h : a * b ∈ I ^ n) : a ∈ I ∨ b ∈ I ^ n := by
cases n; · simp
by_cases hI0 : I = ⊥; · simpa [pow_succ, hI0] using h
simp only [← Submodule.span_singleton_le_iff_mem, Ideal.submodule_span_eq, ← Ideal.dvd_iff_le, ←
Ideal.span_singleton_mul_span_singleton] at h ⊢
by_cases ha : I ∣ span {a}
· exact Or.inl ha
rw [mul_comm] at h
exact Or.inr (Prime.pow_dvd_of_dvd_mul_right ((Ideal.prime_iff_isPrime hI0).mpr hI) _ ha h) | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.IsPrime.mul_mem_pow | null |
Ideal.IsPrime.mem_pow_mul (I : Ideal R) [hI : I.IsPrime] {a b : R} {n : ℕ}
(h : a * b ∈ I ^ n) : a ∈ I ^ n ∨ b ∈ I := by
rw [mul_comm] at h
rw [or_comm]
exact Ideal.IsPrime.mul_mem_pow _ h | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.IsPrime.mem_pow_mul | null |
Ideal.count_normalizedFactors_eq {p x : Ideal R} [hp : p.IsPrime] {n : ℕ} (hle : x ≤ p ^ n)
[DecidableEq (Ideal R)] (hlt : ¬x ≤ p ^ (n + 1)) : (normalizedFactors x).count p = n :=
count_normalizedFactors_eq' ((Ideal.isPrime_iff_bot_or_prime.mp hp).imp_right Prime.irreducible)
(normalize_eq _) (Ideal.dvd_iff_le.mpr hle) (mt Ideal.le_of_dvd hlt) | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.count_normalizedFactors_eq | null |
count_associates_factors_eq [DecidableEq (Ideal R)] [DecidableEq <| Associates (Ideal R)]
[∀ (p : Associates <| Ideal R), Decidable (Irreducible p)]
{I J : Ideal R} (hI : I ≠ 0) (hJ : J.IsPrime) (hJ₀ : J ≠ ⊥) :
(Associates.mk J).count (Associates.mk I).factors = Multiset.count J (normalizedFactors I) := by
replace hI : Associates.mk I ≠ 0 := Associates.mk_ne_zero.mpr hI
have hJ' : Irreducible (Associates.mk J) := by
simpa only [Associates.irreducible_mk] using (Ideal.prime_of_isPrime hJ₀ hJ).irreducible
apply (Ideal.count_normalizedFactors_eq (p := J) (x := I) _ _).symm
all_goals
rw [← Ideal.dvd_iff_le, ← Associates.mk_dvd_mk, Associates.mk_pow]
simp only [Associates.dvd_eq_le]
rw [Associates.prime_pow_dvd_iff_le hI hJ']
cutsat | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | count_associates_factors_eq | The number of times an ideal `I` occurs as normalized factor of another ideal `J` is stable
when regarding these ideals as associated elements of the monoid of ideals. |
Ideal.count_associates_eq [DecidableEq (Associates (Ideal R))]
[∀ (p : Associates <| Ideal R), Decidable (Irreducible p)]
{a a₀ x : R} {n : ℕ} (hx : Prime x) (ha : ¬x ∣ a) (heq : a₀ = x ^ n * a) :
(Associates.mk (span {x})).count (Associates.mk (span {a₀})).factors = n := by
have hx0 : x ≠ 0 := Prime.ne_zero hx
classical
rw [count_associates_factors_eq, UniqueFactorizationMonoid.count_normalizedFactors_eq]
· exact (prime_span_singleton_iff.mpr hx).irreducible
· exact normalize_eq _
· simp only [span_singleton_pow, heq, dvd_span_singleton]
exact Ideal.mul_mem_right _ _ (mem_span_singleton_self (x ^ n))
· simp only [span_singleton_pow, heq, dvd_span_singleton, mem_span_singleton]
rw [pow_add, pow_one, mul_dvd_mul_iff_left (pow_ne_zero n hx0)]
exact ha
· simp only [Submodule.zero_eq_bot, ne_eq, span_singleton_eq_bot]
aesop
· exact (span_singleton_prime hx0).mpr hx
· simp only [ne_eq, span_singleton_eq_bot]; exact hx0 | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.count_associates_eq | Variant of `UniqueFactorizationMonoid.count_normalizedFactors_eq` for associated Ideals. |
Ideal.count_associates_eq' [DecidableEq (Associates (Ideal R))]
[∀ (p : Associates <| Ideal R), Decidable (Irreducible p)]
{a x : R} (hx : Prime x) {n : ℕ} (hle : x ^ n ∣ a) (hlt : ¬x ^ (n + 1) ∣ a) :
(Associates.mk (span {x})).count (Associates.mk (span {a})).factors = n := by
obtain ⟨q, hq⟩ := hle
apply Ideal.count_associates_eq hx _ hq
contrapose! hlt with hdvd
obtain ⟨q', hq'⟩ := hdvd
use q'
rw [hq, hq']
ring | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.count_associates_eq' | Variant of `UniqueFactorizationMonoid.count_normalizedFactors_eq` for associated Ideals. |
Ideal.le_mul_of_no_prime_factors {I J K : Ideal R}
(coprime : ∀ P, J ≤ P → K ≤ P → ¬IsPrime P) (hJ : I ≤ J) (hK : I ≤ K) : I ≤ J * K := by
simp only [← Ideal.dvd_iff_le] at coprime hJ hK ⊢
by_cases hJ0 : J = 0
· simpa only [hJ0, zero_mul] using hJ
obtain ⟨I', rfl⟩ := hK
rw [mul_comm]
refine mul_dvd_mul_left K
(UniqueFactorizationMonoid.dvd_of_dvd_mul_right_of_no_prime_factors (b := K) hJ0 ?_ hJ)
exact fun hPJ hPK => mt Ideal.isPrime_of_prime (coprime _ hPJ hPK) | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.le_mul_of_no_prime_factors | null |
IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type*} (s : Finset ι) (f : ι → Ideal R)
(e : ι → ℕ) (prime : ∀ i ∈ s, Prime (f i))
(coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → f i ≠ f j) :
(s.inf fun i => f i ^ e i) = ∏ i ∈ s, f i ^ e i := by
letI := Classical.decEq ι
revert prime coprime
refine s.induction ?_ ?_
· simp
intro a s ha ih prime coprime
specialize
ih (fun i hi => prime i (Finset.mem_insert_of_mem hi)) fun i hi j hj =>
coprime i (Finset.mem_insert_of_mem hi) j (Finset.mem_insert_of_mem hj)
rw [Finset.inf_insert, Finset.prod_insert ha, ih]
refine le_antisymm (Ideal.le_mul_of_no_prime_factors ?_ inf_le_left inf_le_right) Ideal.mul_le_inf
intro P hPa hPs hPp
obtain ⟨b, hb, hPb⟩ := hPp.prod_le.mp hPs
haveI := Ideal.isPrime_of_prime (prime a (Finset.mem_insert_self a s))
haveI := Ideal.isPrime_of_prime (prime b (Finset.mem_insert_of_mem hb))
refine coprime a (Finset.mem_insert_self a s) b (Finset.mem_insert_of_mem hb) ?_ ?_
· exact (ne_of_mem_of_not_mem hb ha).symm
· refine ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp (hPp.le_of_pow_le hPa)).trans
((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp (hPp.le_of_pow_le hPb)).symm
· exact (prime a (Finset.mem_insert_self a s)).ne_zero
· exact (prime b (Finset.mem_insert_of_mem hb)).ne_zero | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | IsDedekindDomain.inf_prime_pow_eq_prod | The intersection of distinct prime powers in a Dedekind domain is the product of these
prime powers. |
noncomputable IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type*} [Fintype ι] (I : Ideal R)
(P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i, Prime (P i))
(coprime : Pairwise fun i j => P i ≠ P j)
(prod_eq : ∏ i, P i ^ e i = I) : R ⧸ I ≃+* ∀ i, R ⧸ P i ^ e i :=
(Ideal.quotEquivOfEq
(by
simp only [← prod_eq, Finset.inf_eq_iInf, Finset.mem_univ, ciInf_pos,
← IsDedekindDomain.inf_prime_pow_eq_prod _ _ _ (fun i _ => prime i)
(coprime.set_pairwise _)])).trans <|
Ideal.quotientInfRingEquivPiQuotient _ fun i j hij => Ideal.coprime_of_no_prime_ge <| by
intro P hPi hPj hPp
haveI := Ideal.isPrime_of_prime (prime i)
haveI := Ideal.isPrime_of_prime (prime j)
exact coprime hij <| ((Ring.DimensionLeOne.prime_le_prime_iff_eq (prime i).ne_zero).mp
(hPp.le_of_pow_le hPi)).trans <| Eq.symm <|
(Ring.DimensionLeOne.prime_le_prime_iff_eq (prime j).ne_zero).mp (hPp.le_of_pow_le hPj)
open scoped Classical in | def | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | IsDedekindDomain.quotientEquivPiOfProdEq | **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as
`∏ i, P i ^ e i`, then `R ⧸ I` factors as `Π i, R ⧸ (P i ^ e i)`. |
noncomputable IsDedekindDomain.quotientEquivPiFactors {I : Ideal R} (hI : I ≠ ⊥) :
R ⧸ I ≃+* ∀ P : (factors I).toFinset, R ⧸ (P : Ideal R) ^ (Multiset.count ↑P (factors I)) :=
IsDedekindDomain.quotientEquivPiOfProdEq _ _ _
(fun P : (factors I).toFinset => prime_of_factor _ (Multiset.mem_toFinset.mp P.prop))
(fun _ _ hij => Subtype.coe_injective.ne hij)
(calc
(∏ P : (factors I).toFinset, (P : Ideal R) ^ (factors I).count (P : Ideal R)) =
∏ P ∈ (factors I).toFinset, P ^ (factors I).count P :=
(factors I).toFinset.prod_coe_sort fun P => P ^ (factors I).count P
_ = ((factors I).map fun P => P).prod := (Finset.prod_multiset_map_count (factors I) id).symm
_ = (factors I).prod := by rw [Multiset.map_id']
_ = I := associated_iff_eq.mp (factors_prod hI)
)
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | IsDedekindDomain.quotientEquivPiFactors | **Chinese remainder theorem** for a Dedekind domain: `R ⧸ I` factors as `Π i, R ⧸ (P i ^ e i)`,
where `P i` ranges over the prime factors of `I` and `e i` over the multiplicities. |
IsDedekindDomain.quotientEquivPiFactors_mk {I : Ideal R} (hI : I ≠ ⊥) (x : R) :
IsDedekindDomain.quotientEquivPiFactors hI (Ideal.Quotient.mk I x) = fun _P =>
Ideal.Quotient.mk _ x := rfl | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | IsDedekindDomain.quotientEquivPiFactors_mk | null |
noncomputable IsDedekindDomain.quotientEquivPiOfFinsetProdEq {ι : Type*} {s : Finset ι}
(I : Ideal R) (P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (P i))
(coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → P i ≠ P j)
(prod_eq : ∏ i ∈ s, P i ^ e i = I) : R ⧸ I ≃+* ∀ i : s, R ⧸ P i ^ e i :=
IsDedekindDomain.quotientEquivPiOfProdEq I (fun i : s => P i) (fun i : s => e i)
(fun i => prime i i.2) (fun i j h => coprime i i.2 j j.2 (Subtype.coe_injective.ne h))
(_root_.trans (Finset.prod_coe_sort s fun i => P i ^ e i) prod_eq) | def | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | IsDedekindDomain.quotientEquivPiOfFinsetProdEq | **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as
`∏ i ∈ s, P i ^ e i`, then `R ⧸ I` factors as `Π (i : s), R ⧸ (P i ^ e i)`.
This is a version of `IsDedekindDomain.quotientEquivPiOfProdEq` where we restrict
the product to a finite subset `s` of a potentially infinite indexing type `ι`. |
IsDedekindDomain.exists_representative_mod_finset {ι : Type*} {s : Finset ι}
(P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (P i))
(coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → P i ≠ P j) (x : ∀ i : s, R ⧸ P i ^ e i) :
∃ y, ∀ (i) (hi : i ∈ s), Ideal.Quotient.mk (P i ^ e i) y = x ⟨i, hi⟩ := by
let f := IsDedekindDomain.quotientEquivPiOfFinsetProdEq _ P e prime coprime rfl
obtain ⟨y, rfl⟩ := f.surjective x
obtain ⟨z, rfl⟩ := Ideal.Quotient.mk_surjective y
exact ⟨z, fun i _hi => rfl⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | IsDedekindDomain.exists_representative_mod_finset | Corollary of the Chinese remainder theorem: given elements `x i : R / P i ^ e i`,
we can choose a representative `y : R` such that `y ≡ x i (mod P i ^ e i)`. |
IsDedekindDomain.exists_forall_sub_mem_ideal {ι : Type*} {s : Finset ι} (P : ι → Ideal R)
(e : ι → ℕ) (prime : ∀ i ∈ s, Prime (P i))
(coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → P i ≠ P j) (x : s → R) :
∃ y, ∀ (i) (hi : i ∈ s), y - x ⟨i, hi⟩ ∈ P i ^ e i := by
obtain ⟨y, hy⟩ :=
IsDedekindDomain.exists_representative_mod_finset P e prime coprime fun i =>
Ideal.Quotient.mk _ (x i)
exact ⟨y, fun i hi => Ideal.Quotient.eq.mp (hy i hi)⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | IsDedekindDomain.exists_forall_sub_mem_ideal | Corollary of the Chinese remainder theorem: given elements `x i : R`,
we can choose a representative `y : R` such that `y - x i ∈ P i ^ e i`. |
span_singleton_dvd_span_singleton_iff_dvd {a b : R} :
Ideal.span {a} ∣ Ideal.span ({b} : Set R) ↔ a ∣ b :=
⟨fun h => mem_span_singleton.mp (dvd_iff_le.mp h (mem_span_singleton.mpr (dvd_refl b))), fun h =>
dvd_iff_le.mpr fun _d hd => mem_span_singleton.mpr (dvd_trans h (mem_span_singleton.mp hd))⟩
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | span_singleton_dvd_span_singleton_iff_dvd | null |
Ideal.squarefree_span_singleton {a : R} :
Squarefree (span {a}) ↔ Squarefree a := by
refine ⟨fun h x hx ↦ ?_, fun h I hI ↦ ?_⟩
· rw [← span_singleton_dvd_span_singleton_iff_dvd, ← span_singleton_mul_span_singleton] at hx
simpa using h _ hx
· rw [← span_singleton_generator I, span_singleton_mul_span_singleton,
span_singleton_dvd_span_singleton_iff_dvd] at hI
exact isUnit_iff.mpr <| eq_top_of_isUnit_mem _ (Submodule.IsPrincipal.generator_mem I) (h _ hI) | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | Ideal.squarefree_span_singleton | null |
singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [NormalizationMonoid R]
{a b : R} (ha : a ∈ normalizedFactors b) :
Ideal.span ({a} : Set R) ∈ normalizedFactors (Ideal.span ({b} : Set R)) := by
by_cases hb : b = 0
· rw [Ideal.span_singleton_eq_bot.mpr hb, bot_eq_zero, normalizedFactors_zero]
rw [hb, normalizedFactors_zero] at ha
exact absurd ha (Multiset.notMem_zero a)
· suffices Prime (Ideal.span ({a} : Set R)) by
obtain ⟨c, hc, hc'⟩ := exists_mem_normalizedFactors_of_dvd ?_ this.irreducible
(dvd_iff_le.mpr (span_singleton_le_span_singleton.mpr (dvd_of_mem_normalizedFactors ha)))
rwa [associated_iff_eq.mp hc']
· by_contra h
exact hb (span_singleton_eq_bot.mp h)
rw [prime_iff_isPrime]
· exact (span_singleton_prime (prime_of_normalized_factor a ha).ne_zero).mpr
(prime_of_normalized_factor a ha)
· by_contra h
exact (prime_of_normalized_factor a ha).ne_zero (span_singleton_eq_bot.mp h) | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | singleton_span_mem_normalizedFactors_of_mem_normalizedFactors | null |
emultiplicity_eq_emultiplicity_span {a b : R} :
emultiplicity (Ideal.span {a}) (Ideal.span ({b} : Set R)) = emultiplicity a b := by
by_cases h : FiniteMultiplicity a b
· rw [h.emultiplicity_eq_multiplicity]
apply emultiplicity_eq_of_dvd_of_not_dvd <;>
rw [Ideal.span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd]
· exact pow_multiplicity_dvd a b
· apply h.not_pow_dvd_of_multiplicity_lt
apply lt_add_one
· suffices ¬FiniteMultiplicity (Ideal.span ({a} : Set R)) (Ideal.span ({b} : Set R)) by
rw [emultiplicity_eq_top.2 h, emultiplicity_eq_top.2 this]
exact FiniteMultiplicity.not_iff_forall.mpr fun n => by
rw [Ideal.span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd]
exact FiniteMultiplicity.not_iff_forall.mp h n | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | emultiplicity_eq_emultiplicity_span | null |
noncomputable normalizedFactorsEquivSpanNormalizedFactors {r : R} (hr : r ≠ 0) :
{ d : R | d ∈ normalizedFactors r } ≃
{ I : Ideal R | I ∈ normalizedFactors (Ideal.span ({r} : Set R)) } := by
refine Equiv.ofBijective ?_ ?_
· exact fun d =>
⟨Ideal.span {↑d}, singleton_span_mem_normalizedFactors_of_mem_normalizedFactors d.prop⟩
· refine ⟨?_, ?_⟩
· rintro ⟨a, ha⟩ ⟨b, hb⟩ h
rw [Subtype.mk_eq_mk, Ideal.span_singleton_eq_span_singleton, Subtype.coe_mk,
Subtype.coe_mk] at h
exact Subtype.mk_eq_mk.mpr (mem_normalizedFactors_eq_of_associated ha hb h)
· rintro ⟨i, hi⟩
have : i.IsPrime := isPrime_of_prime (prime_of_normalized_factor i hi)
have := exists_mem_normalizedFactors_of_dvd hr
(Submodule.IsPrincipal.prime_generator_of_isPrime i
(prime_of_normalized_factor i hi).ne_zero).irreducible ?_
· obtain ⟨a, ha, ha'⟩ := this
use ⟨a, ha⟩
simp only [← span_singleton_eq_span_singleton.mpr ha',
Ideal.span_singleton_generator]
· exact (Submodule.IsPrincipal.mem_iff_generator_dvd i).mp
((show Ideal.span {r} ≤ i from dvd_iff_le.mp (dvd_of_mem_normalizedFactors hi))
(mem_span_singleton.mpr (dvd_refl r))) | def | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | normalizedFactorsEquivSpanNormalizedFactors | The bijection between the (normalized) prime factors of `r` and the (normalized) prime factors
of `span {r}` |
emultiplicity_normalizedFactorsEquivSpanNormalizedFactors_eq_emultiplicity {r d : R}
(hr : r ≠ 0) (hd : d ∈ normalizedFactors r) :
emultiplicity d r =
emultiplicity (normalizedFactorsEquivSpanNormalizedFactors hr ⟨d, hd⟩ : Ideal R)
(Ideal.span {r}) := by
simp only [normalizedFactorsEquivSpanNormalizedFactors, emultiplicity_eq_emultiplicity_span,
Subtype.coe_mk, Equiv.ofBijective_apply] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | emultiplicity_normalizedFactorsEquivSpanNormalizedFactors_eq_emultiplicity | The bijection `normalizedFactorsEquivSpanNormalizedFactors` between the set of prime
factors of `r` and the set of prime factors of the ideal `⟨r⟩` preserves multiplicities. See
`count_normalizedFactorsSpan_eq_count` for the version stated in terms of multisets `count`. |
emultiplicity_normalizedFactorsEquivSpanNormalizedFactors_symm_eq_emultiplicity {r : R}
(hr : r ≠ 0) (I : { I : Ideal R | I ∈ normalizedFactors (Ideal.span ({r} : Set R)) }) :
emultiplicity ((normalizedFactorsEquivSpanNormalizedFactors hr).symm I : R) r =
emultiplicity (I : Ideal R) (Ideal.span {r}) := by
obtain ⟨x, hx⟩ := (normalizedFactorsEquivSpanNormalizedFactors hr).surjective I
obtain ⟨a, ha⟩ := x
rw [hx.symm, Equiv.symm_apply_apply, Subtype.coe_mk,
emultiplicity_normalizedFactorsEquivSpanNormalizedFactors_eq_emultiplicity hr ha]
variable [DecidableEq R] [DecidableEq (Ideal R)] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | emultiplicity_normalizedFactorsEquivSpanNormalizedFactors_symm_eq_emultiplicity | The bijection `normalized_factors_equiv_span_normalized_factors.symm` between the set of prime
factors of the ideal `⟨r⟩` and the set of prime factors of `r` preserves multiplicities. |
count_span_normalizedFactors_eq {r X : R} (hr : r ≠ 0) (hX : Prime X) :
Multiset.count (Ideal.span {X} : Ideal R) (normalizedFactors (Ideal.span {r})) =
Multiset.count (normalize X) (normalizedFactors r) := by
have := emultiplicity_eq_emultiplicity_span (R := R) (a := X) (b := r)
rw [emultiplicity_eq_count_normalizedFactors (Prime.irreducible hX) hr,
emultiplicity_eq_count_normalizedFactors (Prime.irreducible ?_), normalize_apply,
normUnit_eq_one, Units.val_one, one_eq_top, mul_top, Nat.cast_inj] at this
· simp only [normalize_apply, this]
· simp only [Submodule.zero_eq_bot, ne_eq, span_singleton_eq_bot, hr, not_false_eq_true]
· simpa only [prime_span_singleton_iff] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | count_span_normalizedFactors_eq | The bijection between the set of prime factors of the ideal `⟨r⟩` and the set of prime factors
of `r` preserves `count` of the corresponding multisets. See
`multiplicity_normalizedFactorsEquivSpanNormalizedFactors_eq_multiplicity` for the version
stated in terms of multiplicity. |
count_span_normalizedFactors_eq_of_normUnit {r X : R}
(hr : r ≠ 0) (hX₁ : normUnit X = 1) (hX : Prime X) :
Multiset.count (Ideal.span {X} : Ideal R) (normalizedFactors (Ideal.span {r})) =
Multiset.count X (normalizedFactors r) := by
simpa [hX₁, normalize_apply] using count_span_normalizedFactors_eq hr hX | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | count_span_normalizedFactors_eq_of_normUnit | null |
noncomputable primesOverFinset {A : Type*} [CommRing A] (p : Ideal A) (B : Type*)
[CommRing B] [IsDedekindDomain B] [Algebra A B] : Finset (Ideal B) :=
(factors (p.map (algebraMap A B))).toFinset
variable {A : Type*} [CommRing A] {p : Ideal A} (hpb : p ≠ ⊥) [hpm : p.IsMaximal]
(B : Type*) [CommRing B] [IsDedekindDomain B] [Algebra A B] [NoZeroSMulDivisors A B]
include hpb in | abbrev | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | primesOverFinset | The finite set of all prime factors of the pushforward of `p`. |
coe_primesOverFinset : primesOverFinset p B = primesOver p B := by
classical
ext P
rw [primesOverFinset, factors_eq_normalizedFactors, Finset.mem_coe, Multiset.mem_toFinset]
exact (P.mem_normalizedFactors_iff (map_ne_bot_of_ne_bot hpb)).trans <| Iff.intro
(fun ⟨hPp, h⟩ => ⟨hPp, ⟨hpm.eq_of_le (comap_ne_top _ hPp.ne_top) (le_comap_of_map_le h)⟩⟩)
(fun ⟨hPp, h⟩ => ⟨hPp, map_le_of_le_comap h.1.le⟩) | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | coe_primesOverFinset | null |
noncomputable equivPrimesOver (hp : p ≠ 0) :
{v : HeightOneSpectrum B // v.asIdeal ∣ map (algebraMap A B) p} ≃ p.primesOver B :=
Set.BijOn.equiv HeightOneSpectrum.asIdeal
⟨fun v hv ↦ ⟨v.isPrime, by rwa [liesOver_iff_dvd_map v.isPrime.ne_top]⟩,
fun _ _ _ _ h ↦ HeightOneSpectrum.ext_iff.mpr h,
fun Q hQ ↦ ⟨⟨Q, hQ.1, ne_bot_of_mem_primesOver hp hQ⟩,
(liesOver_iff_dvd_map hQ.1.ne_top).mp hQ.2, rfl⟩⟩
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | equivPrimesOver | The bijection between the elements of the height one prime spectrum of `B` that divide the lift
of the maximal ideal `p` in `B` and the primes over `p` in `B`. |
equivPrimesOver_apply (hp : p ≠ 0)
(v : {v : HeightOneSpectrum B // v.asIdeal ∣ map (algebraMap A B) p}) :
equivPrimesOver B hp v = v.1.asIdeal := rfl | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | equivPrimesOver_apply | null |
primesOver_finite : (primesOver p B).Finite := by
by_cases hpb : p = ⊥
· rw [hpb] at hpm ⊢
haveI : IsDomain A := IsDomain.of_bot_isPrime A
rw [primesOver_bot A B]
exact Set.finite_singleton ⊥
· rw [← coe_primesOverFinset hpb B]
exact (primesOverFinset p B).finite_toSet | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | primesOver_finite | null |
primesOver_ncard_ne_zero : (primesOver p B).ncard ≠ 0 := by
rcases exists_ideal_liesOver_maximal_of_isIntegral p B with ⟨P, hPm, hp⟩
exact Set.ncard_ne_zero_of_mem ⟨hPm.isPrime, hp⟩ (primesOver_finite p B) | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | primesOver_ncard_ne_zero | null |
one_le_primesOver_ncard : 1 ≤ (primesOver p B).ncard :=
Nat.one_le_iff_ne_zero.mpr (primesOver_ncard_ne_zero p B) | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.Algebra.Squarefree.Basic",
"Mathlib.RingTheory.ChainOfDivisors",
"Mathlib.RingTheory.DedekindDomain.Ideal.Basic",
"Mathlib.RingTheory.Spectrum.Maximal.Localization",
"Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso"
] | Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean | one_le_primesOver_ncard | null |
CotangentSpace : Type _ := S ⊗[P.Ring] Ω[P.Ring⁄R] | abbrev | RingTheory | [
"Mathlib.RingTheory.Kaehler.Polynomial",
"Mathlib.Algebra.Module.FinitePresentation",
"Mathlib.RingTheory.Extension.Presentation.Basic"
] | Mathlib/RingTheory/Extension/Cotangent/Basic.lean | CotangentSpace | The cotangent space on `P = R[X]`.
This is isomorphic to `Sⁿ` with `n` being the number of variables of `P`. |
noncomputable
cotangentComplex : P.Cotangent →ₗ[S] P.CotangentSpace :=
letI f : P.Cotangent ≃ₗ[P.Ring] P.ker.Cotangent :=
{ __ := AddEquiv.refl _, map_smul' := Cotangent.val_smul' }
(kerCotangentToTensor R P.Ring S ∘ₗ f).extendScalarsOfSurjective P.algebraMap_surjective
@[simp] | def | RingTheory | [
"Mathlib.RingTheory.Kaehler.Polynomial",
"Mathlib.Algebra.Module.FinitePresentation",
"Mathlib.RingTheory.Extension.Presentation.Basic"
] | Mathlib/RingTheory/Extension/Cotangent/Basic.lean | cotangentComplex | The cotangent complex given by a presentation `R[X] → S` (i.e. a closed embedding `S ↪ Aⁿ`). |
cotangentComplex_mk (x) : P.cotangentComplex (.mk x) = 1 ⊗ₜ .D _ _ x :=
rfl
universe w' u' v'
variable {R' : Type u'} {S' : Type v'} [CommRing R'] [CommRing S'] [Algebra R' S']
variable (P' : Extension.{w'} R' S')
variable [Algebra R R'] [Algebra S S'] [Algebra R S'] [IsScalarTower R R' S']
attribute [local instance] SMulCommClass.of_commMonoid
variable {P P'}
universe w'' u'' v''
variable {R'' : Type u''} {S'' : Type v''} [CommRing R''] [CommRing S''] [Algebra R'' S'']
variable {P'' : Extension.{w''} R'' S''}
variable [Algebra R R''] [Algebra S S''] [Algebra R S'']
[IsScalarTower R R'' S'']
variable [Algebra R' R''] [Algebra S' S''] [Algebra R' S'']
[IsScalarTower R' R'' S'']
variable [IsScalarTower R R' R''] [IsScalarTower S S' S''] | lemma | RingTheory | [
"Mathlib.RingTheory.Kaehler.Polynomial",
"Mathlib.Algebra.Module.FinitePresentation",
"Mathlib.RingTheory.Extension.Presentation.Basic"
] | Mathlib/RingTheory/Extension/Cotangent/Basic.lean | cotangentComplex_mk | null |
protected noncomputable
map (f : Hom P P') : P.CotangentSpace →ₗ[S] P'.CotangentSpace := by
letI := ((algebraMap S S').comp (algebraMap P.Ring S)).toAlgebra
haveI : IsScalarTower P.Ring S S' := IsScalarTower.of_algebraMap_eq' rfl
letI := f.toAlgHom.toAlgebra
haveI : IsScalarTower P.Ring P'.Ring S' :=
IsScalarTower.of_algebraMap_eq (fun x ↦ (f.algebraMap_toRingHom x).symm)
apply LinearMap.liftBaseChange
refine (TensorProduct.mk _ _ _ 1).restrictScalars _ ∘ₗ KaehlerDifferential.map R R' P.Ring P'.Ring
@[simp] | def | RingTheory | [
"Mathlib.RingTheory.Kaehler.Polynomial",
"Mathlib.Algebra.Module.FinitePresentation",
"Mathlib.RingTheory.Extension.Presentation.Basic"
] | Mathlib/RingTheory/Extension/Cotangent/Basic.lean | map | This is the map on the cotangent space associated to a map of presentation.
The matrix associated to this map is the Jacobian matrix. See `CotangentSpace.repr_map`. |
map_tmul (f : Hom P P') (x y) :
CotangentSpace.map f (x ⊗ₜ .D _ _ y) = (algebraMap _ _ x) ⊗ₜ .D _ _ (f.toAlgHom y) := by
simp only [CotangentSpace.map, AlgHom.toRingHom_eq_coe, LinearMap.liftBaseChange_tmul,
LinearMap.coe_comp, LinearMap.coe_restrictScalars, Function.comp_apply, map_D, mk_apply]
rw [smul_tmul', ← Algebra.algebraMap_eq_smul_one]
rfl
@[simp] | lemma | RingTheory | [
"Mathlib.RingTheory.Kaehler.Polynomial",
"Mathlib.Algebra.Module.FinitePresentation",
"Mathlib.RingTheory.Extension.Presentation.Basic"
] | Mathlib/RingTheory/Extension/Cotangent/Basic.lean | map_tmul | null |
map_id :
CotangentSpace.map (.id P) = LinearMap.id := by ext; simp | lemma | RingTheory | [
"Mathlib.RingTheory.Kaehler.Polynomial",
"Mathlib.Algebra.Module.FinitePresentation",
"Mathlib.RingTheory.Extension.Presentation.Basic"
] | Mathlib/RingTheory/Extension/Cotangent/Basic.lean | map_id | null |
map_comp (f : Hom P P') (g : Hom P' P'') :
CotangentSpace.map (g.comp f) =
(CotangentSpace.map g).restrictScalars S ∘ₗ CotangentSpace.map f := by
ext x
induction x using TensorProduct.induction_on with
| zero =>
simp only [map_zero]
| add =>
simp only [map_add, LinearMap.coe_comp, LinearMap.coe_restrictScalars, Function.comp_apply, *]
| tmul x y =>
obtain ⟨y, rfl⟩ := KaehlerDifferential.tensorProductTo_surjective _ _ y
induction y with
| zero => simp only [map_zero, tmul_zero]
| add => simp only [map_add, tmul_add, LinearMap.coe_comp, LinearMap.coe_restrictScalars,
Function.comp_apply, *]
| tmul => simp only [Derivation.tensorProductTo_tmul, tmul_smul, smul_tmul', map_tmul,
Hom.toAlgHom_apply, Hom.comp_toRingHom, RingHom.coe_comp, Function.comp_apply,
LinearMap.coe_comp, LinearMap.coe_restrictScalars,
← IsScalarTower.algebraMap_apply S S' S''] | lemma | RingTheory | [
"Mathlib.RingTheory.Kaehler.Polynomial",
"Mathlib.Algebra.Module.FinitePresentation",
"Mathlib.RingTheory.Extension.Presentation.Basic"
] | Mathlib/RingTheory/Extension/Cotangent/Basic.lean | map_comp | null |
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