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 ⌀ |
|---|---|---|---|---|---|---|
piLocalizationToMaximal_comp_toPiLocalization :
(piLocalizationToMaximal R).comp (toPiLocalization R) = MaximalSpectrum.toPiLocalization R :=
rfl
variable {S} | theorem | RingTheory | [
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.RingTheory.Spectrum.Maximal.Basic",
"Mathlib.RingTheory.Spectrum.Prime.RingHom"
] | Mathlib/RingTheory/Spectrum/Maximal/Localization.lean | piLocalizationToMaximal_comp_toPiLocalization | null |
isMaximal_of_toPiLocalization_surjective (surj : Function.Surjective (toPiLocalization R))
(I : PrimeSpectrum R) : I.1.IsMaximal := by
classical
have ⟨J, max, le⟩ := I.1.exists_le_maximal I.2.ne_top
obtain ⟨r, hr⟩ := surj (Function.update 0 ⟨J, max.isPrime⟩ 1)
by_contra h
have hJ : algebraMap _ _ r = _ := (congr_fun hr _).trans (Function.update_self ..)
have hI : algebraMap _ _ r = _ := congr_fun hr I
rw [← IsLocalization.lift_eq (M := J.primeCompl) (S := Localization J.primeCompl), hJ, map_one,
Function.update_of_ne] at hI
· exact one_ne_zero hI
· intro eq; have : I.1 = J := congr_arg (·.1) eq; exact h (this ▸ max)
· exact fun ⟨s, hs⟩ ↦ IsLocalization.map_units (M := I.1.primeCompl) _ ⟨s, fun h ↦ hs (le h)⟩
variable (f : R →+* S) | theorem | RingTheory | [
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.RingTheory.Spectrum.Maximal.Basic",
"Mathlib.RingTheory.Spectrum.Prime.RingHom"
] | Mathlib/RingTheory/Spectrum/Maximal/Localization.lean | isMaximal_of_toPiLocalization_surjective | null |
noncomputable mapPiLocalization : PiLocalization R →+* PiLocalization S :=
Pi.ringHom fun I ↦ (Localization.localRingHom _ I.1 f rfl).comp (Pi.evalRingHom _ (f.specComap I)) | def | RingTheory | [
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.RingTheory.Spectrum.Maximal.Basic",
"Mathlib.RingTheory.Spectrum.Prime.RingHom"
] | Mathlib/RingTheory/Spectrum/Maximal/Localization.lean | mapPiLocalization | A ring homomorphism induces a homomorphism between the products of localizations at primes. |
mapPiLocalization_naturality :
(mapPiLocalization f).comp (toPiLocalization R) = (toPiLocalization S).comp f := by
ext r I
change Localization.localRingHom _ _ _ rfl (algebraMap _ _ r) = algebraMap _ _ (f r)
simp_rw [← IsLocalization.mk'_one (M := (I.1.comap f).primeCompl), Localization.localRingHom_mk',
← IsLocalization.mk'_one (M := I.1.primeCompl), Submonoid.coe_one, map_one f]
rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.RingTheory.Spectrum.Maximal.Basic",
"Mathlib.RingTheory.Spectrum.Prime.RingHom"
] | Mathlib/RingTheory/Spectrum/Maximal/Localization.lean | mapPiLocalization_naturality | null |
mapPiLocalization_id : mapPiLocalization (.id R) = .id _ := by
ext; exact congr($(Localization.localRingHom_id _) _) | theorem | RingTheory | [
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.RingTheory.Spectrum.Maximal.Basic",
"Mathlib.RingTheory.Spectrum.Prime.RingHom"
] | Mathlib/RingTheory/Spectrum/Maximal/Localization.lean | mapPiLocalization_id | null |
mapPiLocalization_comp (g : S →+* P) :
mapPiLocalization (g.comp f) = (mapPiLocalization g).comp (mapPiLocalization f) := by
ext; exact congr($(Localization.localRingHom_comp _ _ _ _ rfl _ rfl) _) | theorem | RingTheory | [
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.RingTheory.Spectrum.Maximal.Basic",
"Mathlib.RingTheory.Spectrum.Prime.RingHom"
] | Mathlib/RingTheory/Spectrum/Maximal/Localization.lean | mapPiLocalization_comp | null |
mapPiLocalization_bijective (hf : Function.Bijective f) :
Function.Bijective (mapPiLocalization f) := by
let f := RingEquiv.ofBijective f hf
let e := RingEquiv.ofRingHom (mapPiLocalization (f : R →+* S)) (mapPiLocalization f.symm) ?_ ?_
· exact e.bijective
· rw [← mapPiLocalization_comp, RingEquiv.comp_symm, mapPiLocalization_id]
· rw [← mapPiLocalization_comp, RingEquiv.symm_comp, mapPiLocalization_id] | theorem | RingTheory | [
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.RingTheory.Spectrum.Maximal.Basic",
"Mathlib.RingTheory.Spectrum.Prime.RingHom"
] | Mathlib/RingTheory/Spectrum/Maximal/Localization.lean | mapPiLocalization_bijective | null |
toPiLocalization_not_surjective_of_infinite [Infinite ι] :
¬ Function.Surjective (toPiLocalization (Π i, R i)) :=
fun surj ↦ MaximalSpectrum.toPiLocalization_not_surjective_of_infinite R <| by
rw [← piLocalizationToMaximal_comp_toPiLocalization]
exact (piLocalizationToMaximal_surjective _).comp surj
variable {R} | theorem | RingTheory | [
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.RingTheory.Spectrum.Maximal.Basic",
"Mathlib.RingTheory.Spectrum.Prime.RingHom"
] | Mathlib/RingTheory/Spectrum/Maximal/Localization.lean | toPiLocalization_not_surjective_of_infinite | null |
finite_of_toPiLocalization_pi_surjective
(h : Function.Surjective (toPiLocalization (Π i, R i))) :
Finite ι := by
contrapose! h
exact toPiLocalization_not_surjective_of_infinite _ | theorem | RingTheory | [
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.RingTheory.Spectrum.Maximal.Basic",
"Mathlib.RingTheory.Spectrum.Prime.RingHom"
] | Mathlib/RingTheory/Spectrum/Maximal/Localization.lean | finite_of_toPiLocalization_pi_surjective | null |
finite_of_toPiLocalization_surjective
(surj : Function.Surjective (toPiLocalization R)) :
Finite (PrimeSpectrum R) := by
replace surj := (mapPiLocalization_bijective _ ⟨toPiLocalization_injective R, surj⟩).2.comp surj
rw [← RingHom.coe_comp, mapPiLocalization_naturality, RingHom.coe_comp] at surj
exact finite_of_toPiLocalization_pi_surjective surj.of_comp | theorem | RingTheory | [
"Mathlib.RingTheory.Localization.AsSubring",
"Mathlib.RingTheory.Spectrum.Maximal.Basic",
"Mathlib.RingTheory.Spectrum.Prime.RingHom"
] | Mathlib/RingTheory/Spectrum/Maximal/Localization.lean | finite_of_toPiLocalization_surjective | null |
toPrimeSpectrum_range :
Set.range (@toPrimeSpectrum R _) = { x | IsClosed ({x} : Set <| PrimeSpectrum R) } := by
simp only [isClosed_singleton_iff_isMaximal]
ext ⟨x, _⟩
exact ⟨fun ⟨y, hy⟩ => hy ▸ y.isMaximal, fun hx => ⟨⟨x, hx⟩, rfl⟩⟩ | theorem | RingTheory | [
"Mathlib.RingTheory.Spectrum.Maximal.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology"
] | Mathlib/RingTheory/Spectrum/Maximal/Topology.lean | toPrimeSpectrum_range | null |
zariskiTopology : TopologicalSpace <| MaximalSpectrum R :=
PrimeSpectrum.zariskiTopology.induced toPrimeSpectrum | instance | RingTheory | [
"Mathlib.RingTheory.Spectrum.Maximal.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology"
] | Mathlib/RingTheory/Spectrum/Maximal/Topology.lean | zariskiTopology | The Zariski topology on the maximal spectrum of a commutative ring is defined as the subspace
topology induced by the natural inclusion into the prime spectrum. |
toPrimeSpectrum_continuous : Continuous <| @toPrimeSpectrum R _ :=
continuous_induced_dom | theorem | RingTheory | [
"Mathlib.RingTheory.Spectrum.Maximal.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology"
] | Mathlib/RingTheory/Spectrum/Maximal/Topology.lean | toPrimeSpectrum_continuous | null |
nonempty_iff_nontrivial : Nonempty (PrimeSpectrum R) ↔ Nontrivial R := by
refine ⟨fun ⟨p⟩ ↦ ⟨0, 1, fun h ↦ p.2.ne_top ?_⟩, fun h ↦ ?_⟩
· simp [Ideal.eq_top_iff_one p.asIdeal, ← h]
· obtain ⟨I, hI⟩ := Ideal.exists_maximal R
exact ⟨⟨I, hI.isPrime⟩⟩ | lemma | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | nonempty_iff_nontrivial | null |
isEmpty_iff_subsingleton : IsEmpty (PrimeSpectrum R) ↔ Subsingleton R := by
rw [← not_iff_not, not_isEmpty_iff, not_subsingleton_iff_nontrivial, nonempty_iff_nontrivial] | lemma | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | isEmpty_iff_subsingleton | null |
@[simp]
primeSpectrumProdOfSum : PrimeSpectrum R ⊕ PrimeSpectrum S → PrimeSpectrum (R × S)
| Sum.inl ⟨I, _⟩ => ⟨Ideal.prod I ⊤, Ideal.isPrime_ideal_prod_top⟩
| Sum.inr ⟨J, _⟩ => ⟨Ideal.prod ⊤ J, Ideal.isPrime_ideal_prod_top'⟩ | def | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | primeSpectrumProdOfSum | The prime spectrum of the zero ring is empty. -/
instance [Subsingleton R] : IsEmpty (PrimeSpectrum R) :=
isEmpty_iff_subsingleton.mpr inferInstance
lemma nontrivial (p : PrimeSpectrum R) : Nontrivial R :=
nonempty_iff_nontrivial.mp ⟨p⟩
variable (R S)
theorem range_asIdeal : Set.range PrimeSpectrum.asIdeal = {J : Ideal R | J.IsPrime} :=
Set.ext fun J ↦
⟨fun hJ ↦ let ⟨j, hj⟩ := Set.mem_range.mp hJ; Set.mem_setOf.mpr <| hj ▸ j.isPrime,
fun hJ ↦ Set.mem_range.mpr ⟨⟨J, Set.mem_setOf.mp hJ⟩, rfl⟩⟩
/-- The map from the direct sum of prime spectra to the prime spectrum of a direct product. |
noncomputable primeSpectrumProd :
PrimeSpectrum (R × S) ≃ PrimeSpectrum R ⊕ PrimeSpectrum S :=
Equiv.symm <|
Equiv.ofBijective (primeSpectrumProdOfSum R S) (by
constructor
· rintro (⟨I, hI⟩ | ⟨J, hJ⟩) (⟨I', hI'⟩ | ⟨J', hJ'⟩) h <;>
simp only [mk.injEq, Ideal.prod_inj, primeSpectrumProdOfSum] at h
· simp only [h]
· exact False.elim (hI.ne_top h.left)
· exact False.elim (hJ.ne_top h.right)
· simp only [h]
· rintro ⟨I, hI⟩
rcases (Ideal.ideal_prod_prime I).mp hI with (⟨p, ⟨hp, rfl⟩⟩ | ⟨p, ⟨hp, rfl⟩⟩)
· exact ⟨Sum.inl ⟨p, hp⟩, rfl⟩
· exact ⟨Sum.inr ⟨p, hp⟩, rfl⟩)
variable {R S}
@[simp] | def | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | primeSpectrumProd | The prime spectrum of `R × S` is in bijection with the disjoint unions of the prime spectrum of
`R` and the prime spectrum of `S`. |
primeSpectrumProd_symm_inl_asIdeal (x : PrimeSpectrum R) :
((primeSpectrumProd R S).symm <| Sum.inl x).asIdeal = Ideal.prod x.asIdeal ⊤ := by
cases x
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | primeSpectrumProd_symm_inl_asIdeal | null |
primeSpectrumProd_symm_inr_asIdeal (x : PrimeSpectrum S) :
((primeSpectrumProd R S).symm <| Sum.inr x).asIdeal = Ideal.prod ⊤ x.asIdeal := by
cases x
rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | primeSpectrumProd_symm_inr_asIdeal | null |
zeroLocus (s : Set R) : Set (PrimeSpectrum R) :=
{ x | s ⊆ x.asIdeal }
@[simp] | def | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus | The zero locus of a set `s` of elements of a commutative (semi)ring `R` is the set of all
prime ideals of the ring that contain the set `s`.
An element `f` of `R` can be thought of as a dependent function on the prime spectrum of `R`.
At a point `x` (a prime ideal) the function (i.e., element) `f` takes values in the quotient ring
`R` modulo the prime ideal `x`. In this manner, `zeroLocus s` is exactly the subset of
`PrimeSpectrum R` where all "functions" in `s` vanish simultaneously. |
mem_zeroLocus (x : PrimeSpectrum R) (s : Set R) : x ∈ zeroLocus s ↔ s ⊆ x.asIdeal :=
Iff.rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | mem_zeroLocus | null |
zeroLocus_span (s : Set R) : zeroLocus (Ideal.span s : Set R) = zeroLocus s := by
ext x
exact (Submodule.gi R R).gc s x.asIdeal | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_span | null |
vanishingIdeal (t : Set (PrimeSpectrum R)) : Ideal R :=
⨅ x ∈ t, x.asIdeal | def | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | vanishingIdeal | The vanishing ideal of a set `t` of points of the prime spectrum of a commutative ring `R` is
the intersection of all the prime ideals in the set `t`.
An element `f` of `R` can be thought of as a dependent function on the prime spectrum of `R`.
At a point `x` (a prime ideal) the function (i.e., element) `f` takes values in the quotient ring
`R` modulo the prime ideal `x`. In this manner, `vanishingIdeal t` is exactly the ideal of `R`
consisting of all "functions" that vanish on all of `t`. |
coe_vanishingIdeal (t : Set (PrimeSpectrum R)) :
(vanishingIdeal t : Set R) = { f : R | ∀ x ∈ t, f ∈ x.asIdeal } := by
ext f
rw [vanishingIdeal, SetLike.mem_coe, Submodule.mem_iInf]
apply forall_congr'; intro x
rw [Submodule.mem_iInf] | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | coe_vanishingIdeal | null |
mem_vanishingIdeal (t : Set (PrimeSpectrum R)) (f : R) :
f ∈ vanishingIdeal t ↔ ∀ x ∈ t, f ∈ x.asIdeal := by
rw [← SetLike.mem_coe, coe_vanishingIdeal, Set.mem_setOf_eq]
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | mem_vanishingIdeal | null |
vanishingIdeal_singleton (x : PrimeSpectrum R) :
vanishingIdeal ({x} : Set (PrimeSpectrum R)) = x.asIdeal := by simp [vanishingIdeal] | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | vanishingIdeal_singleton | null |
subset_zeroLocus_iff_le_vanishingIdeal (t : Set (PrimeSpectrum R)) (I : Ideal R) :
t ⊆ zeroLocus I ↔ I ≤ vanishingIdeal t :=
⟨fun h _ k => (mem_vanishingIdeal _ _).mpr fun _ j => (mem_zeroLocus _ _).mpr (h j) k, fun h =>
fun x j => (mem_zeroLocus _ _).mpr (le_trans h fun _ h => ((mem_vanishingIdeal _ _).mp h) x j)⟩ | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | subset_zeroLocus_iff_le_vanishingIdeal | null |
gc :
@GaloisConnection (Ideal R) (Set (PrimeSpectrum R))ᵒᵈ _ _ (fun I => zeroLocus I) fun t =>
vanishingIdeal t :=
fun I t => subset_zeroLocus_iff_le_vanishingIdeal t I | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | gc | `zeroLocus` and `vanishingIdeal` form a Galois connection. |
gc_set :
@GaloisConnection (Set R) (Set (PrimeSpectrum R))ᵒᵈ _ _ (fun s => zeroLocus s) fun t =>
vanishingIdeal t := by
have ideal_gc : GaloisConnection Ideal.span _ := (Submodule.gi R R).gc
simpa [zeroLocus_span, Function.comp_def] using ideal_gc.compose (gc R) | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | gc_set | `zeroLocus` and `vanishingIdeal` form a Galois connection. |
subset_zeroLocus_iff_subset_vanishingIdeal (t : Set (PrimeSpectrum R)) (s : Set R) :
t ⊆ zeroLocus s ↔ s ⊆ vanishingIdeal t :=
(gc_set R) s t | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | subset_zeroLocus_iff_subset_vanishingIdeal | null |
subset_vanishingIdeal_zeroLocus (s : Set R) : s ⊆ vanishingIdeal (zeroLocus s) :=
(gc_set R).le_u_l s | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | subset_vanishingIdeal_zeroLocus | null |
le_vanishingIdeal_zeroLocus (I : Ideal R) : I ≤ vanishingIdeal (zeroLocus I) :=
(gc R).le_u_l I
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | le_vanishingIdeal_zeroLocus | null |
vanishingIdeal_zeroLocus_eq_radical (I : Ideal R) :
vanishingIdeal (zeroLocus (I : Set R)) = I.radical :=
Ideal.ext fun f => by
rw [mem_vanishingIdeal, Ideal.radical_eq_sInf, Submodule.mem_sInf]
exact ⟨fun h x hx => h ⟨x, hx.2⟩ hx.1, fun h x hx => h x.1 ⟨hx, x.2⟩⟩ | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | vanishingIdeal_zeroLocus_eq_radical | null |
nilradical_eq_iInf : nilradical R = iInf asIdeal := by
apply range_asIdeal R ▸ nilradical_eq_sInf R
@[simp] theorem vanishingIdeal_univ : vanishingIdeal Set.univ = nilradical R := by
rw [vanishingIdeal, iInf_univ, nilradical_eq_iInf]
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | nilradical_eq_iInf | null |
zeroLocus_radical (I : Ideal R) : zeroLocus (I.radical : Set R) = zeroLocus I :=
vanishingIdeal_zeroLocus_eq_radical I ▸ (gc R).l_u_l_eq_l I | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_radical | null |
subset_zeroLocus_vanishingIdeal (t : Set (PrimeSpectrum R)) :
t ⊆ zeroLocus (vanishingIdeal t) :=
(gc R).l_u_le t | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | subset_zeroLocus_vanishingIdeal | null |
zeroLocus_anti_mono {s t : Set R} (h : s ⊆ t) : zeroLocus t ⊆ zeroLocus s :=
(gc_set R).monotone_l h | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_anti_mono | null |
zeroLocus_anti_mono_ideal {s t : Ideal R} (h : s ≤ t) :
zeroLocus (t : Set R) ⊆ zeroLocus (s : Set R) :=
(gc R).monotone_l h | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_anti_mono_ideal | null |
vanishingIdeal_anti_mono {s t : Set (PrimeSpectrum R)} (h : s ⊆ t) :
vanishingIdeal t ≤ vanishingIdeal s :=
(gc R).monotone_u h | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | vanishingIdeal_anti_mono | null |
zeroLocus_subset_zeroLocus_iff (I J : Ideal R) :
zeroLocus (I : Set R) ⊆ zeroLocus (J : Set R) ↔ J ≤ I.radical := by
rw [subset_zeroLocus_iff_le_vanishingIdeal, vanishingIdeal_zeroLocus_eq_radical] | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_subset_zeroLocus_iff | null |
zeroLocus_subset_zeroLocus_singleton_iff (f g : R) :
zeroLocus ({f} : Set R) ⊆ zeroLocus {g} ↔ g ∈ (Ideal.span ({f} : Set R)).radical := by
rw [← zeroLocus_span {f}, ← zeroLocus_span {g}, zeroLocus_subset_zeroLocus_iff, Ideal.span_le,
Set.singleton_subset_iff, SetLike.mem_coe] | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_subset_zeroLocus_singleton_iff | null |
zeroLocus_bot : zeroLocus ((⊥ : Ideal R) : Set R) = Set.univ :=
(gc R).l_bot
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_bot | null |
zeroLocus_nilradical : zeroLocus (nilradical R : Set R) = Set.univ := by
rw [nilradical, zeroLocus_radical, Ideal.zero_eq_bot, zeroLocus_bot]
@[simp] | lemma | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_nilradical | null |
zeroLocus_singleton_zero : zeroLocus ({0} : Set R) = Set.univ :=
zeroLocus_bot
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_singleton_zero | null |
zeroLocus_empty : zeroLocus (∅ : Set R) = Set.univ :=
(gc_set R).l_bot
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_empty | null |
vanishingIdeal_empty : vanishingIdeal (∅ : Set (PrimeSpectrum R)) = ⊤ := by
simpa using (gc R).u_top | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | vanishingIdeal_empty | null |
zeroLocus_empty_of_one_mem {s : Set R} (h : (1 : R) ∈ s) : zeroLocus s = ∅ := by
rw [Set.eq_empty_iff_forall_notMem]
intro x hx
rw [mem_zeroLocus] at hx
have x_prime : x.asIdeal.IsPrime := by infer_instance
have eq_top : x.asIdeal = ⊤ := by
rw [Ideal.eq_top_iff_one]
exact hx h
apply x_prime.ne_top eq_top
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_empty_of_one_mem | null |
zeroLocus_singleton_one : zeroLocus ({1} : Set R) = ∅ :=
zeroLocus_empty_of_one_mem (Set.mem_singleton (1 : R)) | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_singleton_one | null |
zeroLocus_empty_iff_eq_top {I : Ideal R} : zeroLocus (I : Set R) = ∅ ↔ I = ⊤ := by
constructor
· contrapose!
intro h
rcases Ideal.exists_le_maximal I h with ⟨M, hM, hIM⟩
exact ⟨⟨M, hM.isPrime⟩, hIM⟩
· rintro rfl
apply zeroLocus_empty_of_one_mem
trivial
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_empty_iff_eq_top | null |
zeroLocus_univ : zeroLocus (Set.univ : Set R) = ∅ :=
zeroLocus_empty_of_one_mem (Set.mem_univ 1) | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_univ | null |
vanishingIdeal_eq_top_iff {s : Set (PrimeSpectrum R)} : vanishingIdeal s = ⊤ ↔ s = ∅ := by
rw [← top_le_iff, ← subset_zeroLocus_iff_le_vanishingIdeal, Submodule.top_coe, zeroLocus_univ,
Set.subset_empty_iff] | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | vanishingIdeal_eq_top_iff | null |
zeroLocus_eq_univ_iff (s : Set R) :
zeroLocus s = Set.univ ↔ s ⊆ nilradical R := by
rw [← Set.univ_subset_iff, subset_zeroLocus_iff_subset_vanishingIdeal, vanishingIdeal_univ]
@[deprecated (since := "2025-04-05")] alias zeroLocus_eq_top_iff := zeroLocus_eq_univ_iff | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_eq_univ_iff | null |
zeroLocus_sup (I J : Ideal R) :
zeroLocus ((I ⊔ J : Ideal R) : Set R) = zeroLocus I ∩ zeroLocus J :=
(gc R).l_sup | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_sup | null |
zeroLocus_union (s s' : Set R) : zeroLocus (s ∪ s') = zeroLocus s ∩ zeroLocus s' :=
(gc_set R).l_sup | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_union | null |
vanishingIdeal_union (t t' : Set (PrimeSpectrum R)) :
vanishingIdeal (t ∪ t') = vanishingIdeal t ⊓ vanishingIdeal t' :=
(gc R).u_inf | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | vanishingIdeal_union | null |
zeroLocus_iSup {ι : Sort*} (I : ι → Ideal R) :
zeroLocus ((⨆ i, I i : Ideal R) : Set R) = ⋂ i, zeroLocus (I i) :=
(gc R).l_iSup | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_iSup | null |
zeroLocus_iUnion {ι : Sort*} (s : ι → Set R) :
zeroLocus (⋃ i, s i) = ⋂ i, zeroLocus (s i) :=
(gc_set R).l_iSup | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_iUnion | null |
zeroLocus_iUnion₂ {ι : Sort*} {κ : (i : ι) → Sort*} (s : ∀ i, κ i → Set R) :
zeroLocus (⋃ (i) (j), s i j) = ⋂ (i) (j), zeroLocus (s i j) :=
(gc_set R).l_iSup₂ | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_iUnion₂ | null |
zeroLocus_bUnion (s : Set (Set R)) :
zeroLocus (⋃ s' ∈ s, s' : Set R) = ⋂ s' ∈ s, zeroLocus s' := by simp only [zeroLocus_iUnion] | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_bUnion | null |
vanishingIdeal_iUnion {ι : Sort*} (t : ι → Set (PrimeSpectrum R)) :
vanishingIdeal (⋃ i, t i) = ⨅ i, vanishingIdeal (t i) :=
(gc R).u_iInf | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | vanishingIdeal_iUnion | null |
zeroLocus_inf (I J : Ideal R) :
zeroLocus ((I ⊓ J : Ideal R) : Set R) = zeroLocus I ∪ zeroLocus J :=
Set.ext fun x => x.2.inf_le | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_inf | null |
union_zeroLocus (s s' : Set R) :
zeroLocus s ∪ zeroLocus s' = zeroLocus (Ideal.span s ⊓ Ideal.span s' : Ideal R) := by
rw [zeroLocus_inf]
simp | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | union_zeroLocus | null |
zeroLocus_mul (I J : Ideal R) :
zeroLocus ((I * J : Ideal R) : Set R) = zeroLocus I ∪ zeroLocus J :=
Set.ext fun x => x.2.mul_le | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_mul | null |
zeroLocus_singleton_mul (f g : R) :
zeroLocus ({f * g} : Set R) = zeroLocus {f} ∪ zeroLocus {g} :=
Set.ext fun x => by simpa using x.2.mul_mem_iff_mem_or_mem
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_singleton_mul | null |
zeroLocus_pow (I : Ideal R) {n : ℕ} (hn : n ≠ 0) :
zeroLocus ((I ^ n : Ideal R) : Set R) = zeroLocus I :=
zeroLocus_radical (I ^ n) ▸ (I.radical_pow hn).symm ▸ zeroLocus_radical I
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_pow | null |
zeroLocus_singleton_pow (f : R) (n : ℕ) (hn : 0 < n) :
zeroLocus ({f ^ n} : Set R) = zeroLocus {f} :=
Set.ext fun x => by simpa using x.2.pow_mem_iff_mem n hn | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_singleton_pow | null |
sup_vanishingIdeal_le (t t' : Set (PrimeSpectrum R)) :
vanishingIdeal t ⊔ vanishingIdeal t' ≤ vanishingIdeal (t ∩ t') := by
intro r
rw [Submodule.mem_sup, mem_vanishingIdeal]
rintro ⟨f, hf, g, hg, rfl⟩ x ⟨hxt, hxt'⟩
rw [mem_vanishingIdeal] at hf hg
apply Submodule.add_mem <;> solve_by_elim | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | sup_vanishingIdeal_le | null |
mem_compl_zeroLocus_iff_notMem {f : R} {I : PrimeSpectrum R} :
I ∈ (zeroLocus {f} : Set (PrimeSpectrum R))ᶜ ↔ f ∉ I.asIdeal := by
rw [Set.mem_compl_iff, mem_zeroLocus, Set.singleton_subset_iff]; rfl
@[deprecated (since := "2025-05-23")]
alias mem_compl_zeroLocus_iff_not_mem := mem_compl_zeroLocus_iff_notMem
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | mem_compl_zeroLocus_iff_notMem | null |
zeroLocus_insert_zero (s : Set R) : zeroLocus (insert 0 s) = zeroLocus s := by
rw [← Set.union_singleton, zeroLocus_union, zeroLocus_singleton_zero, Set.inter_univ]
@[simp] | lemma | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_insert_zero | null |
zeroLocus_diff_singleton_zero (s : Set R) : zeroLocus (s \ {0}) = zeroLocus s := by
rw [← zeroLocus_insert_zero, ← zeroLocus_insert_zero (s := s)]; simp | lemma | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_diff_singleton_zero | null |
zeroLocus_smul_of_isUnit {r : R} (hr : IsUnit r) (s : Set R) :
zeroLocus (r • s) = zeroLocus s := by
ext; simp [Set.subset_def, ← Set.image_smul, Ideal.unit_mul_mem_iff_mem _ hr] | lemma | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_smul_of_isUnit | null |
isMax_iff {x : PrimeSpectrum R} :
IsMax x ↔ x.asIdeal.IsMaximal := by
refine ⟨fun hx ↦ ⟨⟨x.2.ne_top, fun I hI ↦ ?_⟩⟩, fun hx y e ↦ (hx.eq_of_le y.2.ne_top e).ge⟩
by_contra e
obtain ⟨m, hm, hm'⟩ := Ideal.exists_le_maximal I e
exact hx.not_lt (show x < ⟨m, hm.isPrime⟩ from hI.trans_le hm') | lemma | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | isMax_iff | Also see `PrimeSpectrum.isClosed_singleton_iff_isMaximal` |
zeroLocus_eq_singleton (m : Ideal R) [m.IsMaximal] :
zeroLocus m = {⟨m, inferInstance⟩} := by
ext I
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· simp only [mem_zeroLocus, SetLike.coe_subset_coe] at h
simpa using PrimeSpectrum.ext_iff.mpr (Ideal.IsMaximal.eq_of_le ‹_› I.2.ne_top h).symm
· simp [Set.mem_singleton_iff.mp h] | lemma | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | zeroLocus_eq_singleton | null |
isMin_iff {x : PrimeSpectrum R} :
IsMin x ↔ x.asIdeal ∈ minimalPrimes R := by
change IsMin _ ↔ Minimal (fun q : Ideal R ↦ q.IsPrime ∧ ⊥ ≤ q) _
simp only [IsMin, Minimal, x.2, bot_le, and_self, and_true, true_and]
exact ⟨fun H y hy e ↦ @H ⟨y, hy⟩ e, fun H y e ↦ H y.2 e⟩ | lemma | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | isMin_iff | null |
exists_primeSpectrum_prod_le (I : Ideal R) :
∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
induction I using IsNoetherian.induction with | hgt M hgt =>
change Ideal R at M
by_cases h_prM : M.IsPrime
· use {⟨M, h_prM⟩}
rw [Multiset.map_singleton, Multiset.prod_singleton]
by_cases htop : M = ⊤
· rw [htop]
exact ⟨0, le_top⟩
have lt_add : ∀ z ∉ M, M < M + span R {z} := by
intro z hz
refine lt_of_le_of_ne le_sup_left fun m_eq => hz ?_
rw [m_eq]
exact Ideal.mem_sup_right (mem_span_singleton_self z)
obtain ⟨x, hx, y, hy, hxy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left htop
obtain ⟨Wx, h_Wx⟩ := hgt (M + span R {x}) (lt_add _ hx)
obtain ⟨Wy, h_Wy⟩ := hgt (M + span R {y}) (lt_add _ hy)
use Wx + Wy
rw [Multiset.map_add, Multiset.prod_add]
apply le_trans (mul_le_mul' h_Wx h_Wy)
rw [add_mul]
apply sup_le (show M * (M + span R {y}) ≤ M from Ideal.mul_le_right)
rw [mul_add]
apply sup_le (show span R {x} * M ≤ M from Ideal.mul_le_left)
rwa [span_mul_span, Set.singleton_mul_singleton, span_singleton_le_iff_mem] | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | exists_primeSpectrum_prod_le | In a Noetherian ring, every ideal contains a product of prime ideals
([samuel1967, § 3.3, Lemma 3]). |
exists_primeSpectrum_prod_le_and_ne_bot_of_domain (h_fA : ¬IsField A) {I : Ideal A}
(h_nzI : I ≠ ⊥) :
∃ Z : Multiset (PrimeSpectrum A),
Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥ := by
induction I using IsNoetherian.induction with | hgt M hgt =>
change Ideal A at M
have hA_nont : Nontrivial A := IsDomain.toNontrivial
by_cases h_topM : M = ⊤
· rcases h_topM with rfl
obtain ⟨p_id, h_nzp, h_pp⟩ : ∃ p : Ideal A, p ≠ ⊥ ∧ p.IsPrime := by
apply Ring.not_isField_iff_exists_prime.mp h_fA
use ({⟨p_id, h_pp⟩} : Multiset (PrimeSpectrum A)), le_top
rwa [Multiset.map_singleton, Multiset.prod_singleton]
by_cases h_prM : M.IsPrime
· use ({⟨M, h_prM⟩} : Multiset (PrimeSpectrum A))
rw [Multiset.map_singleton, Multiset.prod_singleton]
exact ⟨le_rfl, h_nzI⟩
obtain ⟨x, hx, y, hy, h_xy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left h_topM
have lt_add : ∀ z ∉ M, M < M + span A {z} := by
intro z hz
refine lt_of_le_of_ne le_sup_left fun m_eq => hz ?_
rw [m_eq]
exact mem_sup_right (mem_span_singleton_self z)
obtain ⟨Wx, h_Wx_le, h_Wx_ne⟩ := hgt (M + span A {x}) (lt_add _ hx) (ne_bot_of_gt (lt_add _ hx))
obtain ⟨Wy, h_Wy_le, h_Wx_ne⟩ := hgt (M + span A {y}) (lt_add _ hy) (ne_bot_of_gt (lt_add _ hy))
use Wx + Wy
rw [Multiset.map_add, Multiset.prod_add]
refine ⟨le_trans (mul_le_mul' h_Wx_le h_Wy_le) ?_, mt Ideal.mul_eq_bot.mp ?_⟩
· rw [add_mul]
apply sup_le (show M * (M + span A {y}) ≤ M from Ideal.mul_le_right)
rw [mul_add]
apply sup_le (show span A {x} * M ≤ M from Ideal.mul_le_left)
rwa [span_mul_span, Set.singleton_mul_singleton, span_singleton_le_iff_mem]
· rintro (hx | hy) <;> contradiction | theorem | RingTheory | [
"Mathlib.RingTheory.Ideal.MinimalPrime.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas",
"Mathlib.RingTheory.Noetherian.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Defs"
] | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | exists_primeSpectrum_prod_le_and_ne_bot_of_domain | In a Noetherian integral domain which is not a field, every non-zero ideal contains a non-zero
product of prime ideals; in a field, the whole ring is a non-zero ideal containing only 0 as
product or prime ideals ([samuel1967, § 3.3, Lemma 3]) |
isConstructible_comap_C
{s : Set (PrimeSpectrum (Polynomial R))} (hs : IsConstructible s) :
IsConstructible (comap Polynomial.C '' s) := by
obtain ⟨S, rfl⟩ := exists_constructibleSetData_iff.mpr hs
obtain ⟨T, hT, -⟩ := ChevalleyThm.chevalley_polynomialC _ Submodule.mem_top S (by simp)
rw [hT]
exact T.isConstructible_toSet | lemma | RingTheory | [
"Mathlib.RingTheory.Ideal.GoingDown",
"Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity"
] | Mathlib/RingTheory/Spectrum/Prime/Chevalley.lean | isConstructible_comap_C | null |
isConstructible_comap_image
{f : R →+* S} (hf : f.FinitePresentation)
{s : Set (PrimeSpectrum S)} (hs : IsConstructible s) :
IsConstructible (comap f '' s) := by
refine hf.polynomial_induction
(fun _ _ _ _ f ↦ ∀ s, IsConstructible s → IsConstructible (comap f '' s))
(fun _ _ _ _ f ↦ ∀ s, IsConstructible s → IsConstructible (comap f '' s))
(fun _ _ _ ↦ isConstructible_comap_C) ?_ ?_ f s hs
· intro R _ S _ f hf hf' s hs
refine hs.image_of_isClosedEmbedding (isClosedEmbedding_comap_of_surjective _ f hf) ?_
rw [range_comap_of_surjective _ f hf]
exact isRetrocompact_zeroLocus_compl_of_fg hf'
· intro R _ S _ T _ f g H₁ H₂ s hs
simp only [comap_comp, ContinuousMap.coe_comp, Set.image_comp]
exact H₁ _ (H₂ _ hs) | lemma | RingTheory | [
"Mathlib.RingTheory.Ideal.GoingDown",
"Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity"
] | Mathlib/RingTheory/Spectrum/Prime/Chevalley.lean | isConstructible_comap_image | **Chevalley's theorem**: If `f` is of finite presentation,
then the image of a constructible set under `Spec(f)` is constructible. |
isConstructible_range_comap {f : R →+* S} (hf : f.FinitePresentation) :
IsConstructible (Set.range <| comap f) :=
Set.image_univ ▸ isConstructible_comap_image hf .univ
@[stacks 00I1] | lemma | RingTheory | [
"Mathlib.RingTheory.Ideal.GoingDown",
"Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity"
] | Mathlib/RingTheory/Spectrum/Prime/Chevalley.lean | isConstructible_range_comap | null |
isOpenMap_comap_of_hasGoingDown_of_finitePresentation
[Algebra R S] [Algebra.HasGoingDown R S] [Algebra.FinitePresentation R S] :
IsOpenMap (comap (algebraMap R S)) := by
rw [isBasis_basic_opens.isOpenMap_iff]
rintro _ ⟨_, ⟨f, rfl⟩, rfl⟩
exact isOpen_of_stableUnderGeneralization_of_isConstructible
((basicOpen f).2.stableUnderGeneralization.image
(Algebra.HasGoingDown.iff_generalizingMap_primeSpectrumComap.mp ‹_›))
(isConstructible_comap_image (RingHom.finitePresentation_algebraMap.mpr ‹_›)
isConstructible_basicOpen) | lemma | RingTheory | [
"Mathlib.RingTheory.Ideal.GoingDown",
"Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity"
] | Mathlib/RingTheory/Spectrum/Prime/Chevalley.lean | isOpenMap_comap_of_hasGoingDown_of_finitePresentation | null |
private DegreeType := (Fin n → WithBot ℕ) ×ₗ Prop
variable (R n) in | abbrev | RingTheory | [
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Algebra.Polynomial.CoeffMem",
"Mathlib.Data.DFinsupp.WellFounded",
"Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet",
"Mathlib.RingTheory.Spectrum.Prime.Polynomial"
] | Mathlib/RingTheory/Spectrum/Prime/ChevalleyComplexity.lean | DegreeType | The codomain of the measure that will decrease during the induction in the `C : R → R[X]` case
of Chevalley's theorem with complexity bound. |
@[ext]
private InductionObj where
/-- The underlying family of polynomials of an induction object. -/
val : Fin n → R[X] | structure | RingTheory | [
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Algebra.Polynomial.CoeffMem",
"Mathlib.Data.DFinsupp.WellFounded",
"Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet",
"Mathlib.RingTheory.Spectrum.Prime.Polynomial"
] | Mathlib/RingTheory/Spectrum/Prime/ChevalleyComplexity.lean | InductionObj | A type synonym for families of polynomials. This is used in the induction for the case of
`C : R → R[X]` of Chevalley's theorem with complexity bound. |
private coeffSubmodule (e : InductionObj R n) : Submodule R₀ R :=
.span R₀ ({1} ∪ ⋃ i, coeff(e i)) | def | RingTheory | [
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Algebra.Polynomial.CoeffMem",
"Mathlib.Data.DFinsupp.WellFounded",
"Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet",
"Mathlib.RingTheory.Spectrum.Prime.Polynomial"
] | Mathlib/RingTheory/Spectrum/Prime/ChevalleyComplexity.lean | coeffSubmodule | A subgroup containing all coefficients of all polynomials of a given induction object for
Chevalley's theorem with complexity.
Note that we force `1` to lie in that subgroup so that `fun n ↦ e.coeffSubmodule ^ n` is
increasing. |
private coeffSubmodule_mapRingHom_comp (e : InductionObj R n) (f : R →ₐ[R₀] S) :
({ val := mapRingHom f ∘ e } : InductionObj S n).coeffSubmodule R₀
= (e.coeffSubmodule R₀).map f.toLinearMap := by
simp [coeffSubmodule, Submodule.map_span, Set.image_insert_eq, Set.image_iUnion, ← Set.range_comp,
coeff_map_eq_comp]
variable {e T : InductionObj R n} | lemma | RingTheory | [
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Algebra.Polynomial.CoeffMem",
"Mathlib.Data.DFinsupp.WellFounded",
"Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet",
"Mathlib.RingTheory.Spectrum.Prime.Polynomial"
] | Mathlib/RingTheory/Spectrum/Prime/ChevalleyComplexity.lean | coeffSubmodule_mapRingHom_comp | null |
private coeff_mem_coeffSubmodule {i : Fin n} {d : ℕ} :
(e i).coeff d ∈ e.coeffSubmodule R₀ :=
Submodule.subset_span <| .inr <| Set.mem_iUnion.2 ⟨i, Set.mem_range_self _⟩ | lemma | RingTheory | [
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Algebra.Polynomial.CoeffMem",
"Mathlib.Data.DFinsupp.WellFounded",
"Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet",
"Mathlib.RingTheory.Spectrum.Prime.Polynomial"
] | Mathlib/RingTheory/Spectrum/Prime/ChevalleyComplexity.lean | coeff_mem_coeffSubmodule | null |
private one_mem_coeffSubmodule : 1 ∈ e.coeffSubmodule R₀ := Submodule.subset_span (.inl rfl) | lemma | RingTheory | [
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Algebra.Polynomial.CoeffMem",
"Mathlib.Data.DFinsupp.WellFounded",
"Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet",
"Mathlib.RingTheory.Spectrum.Prime.Polynomial"
] | Mathlib/RingTheory/Spectrum/Prime/ChevalleyComplexity.lean | one_mem_coeffSubmodule | null |
private one_le_coeffSubmodule : 1 ≤ e.coeffSubmodule R₀ := by
rw [Submodule.one_eq_span, Submodule.span_le, Set.singleton_subset_iff]
exact one_mem_coeffSubmodule
variable (e) in | lemma | RingTheory | [
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Algebra.Polynomial.CoeffMem",
"Mathlib.Data.DFinsupp.WellFounded",
"Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet",
"Mathlib.RingTheory.Spectrum.Prime.Polynomial"
] | Mathlib/RingTheory/Spectrum/Prime/ChevalleyComplexity.lean | one_le_coeffSubmodule | null |
private degree : DegreeType n :=
toLex (Polynomial.degree ∘ e, ¬ ∃ i, (e i).Monic ∧ ∀ j, e j ≠ 0 → (e i).degree ≤ (e j).degree)
@[simp] private lemma ofLex_degree_fst (i) : (ofLex e.degree).fst i = (e i).degree := rfl | def | RingTheory | [
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Algebra.Polynomial.CoeffMem",
"Mathlib.Data.DFinsupp.WellFounded",
"Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet",
"Mathlib.RingTheory.Spectrum.Prime.Polynomial"
] | Mathlib/RingTheory/Spectrum/Prime/ChevalleyComplexity.lean | degree | The measure that will decrease during the induction in the `C : R → R[X]` case of
Chevalley's theorem with complexity bound. |
private ofLex_degree_snd :
(ofLex e.degree).snd ↔
¬ ∃ i, (e i).Monic ∧ ∀ j, e j ≠ 0 → (e i).degree ≤ (e j).degree := .rfl
variable (e) in | lemma | RingTheory | [
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Algebra.Polynomial.CoeffMem",
"Mathlib.Data.DFinsupp.WellFounded",
"Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet",
"Mathlib.RingTheory.Spectrum.Prime.Polynomial"
] | Mathlib/RingTheory/Spectrum/Prime/ChevalleyComplexity.lean | ofLex_degree_snd | null |
private degBound : ℕ := ∑ i, (e i).degree.succ
variable (e) in | def | RingTheory | [
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Algebra.Polynomial.CoeffMem",
"Mathlib.Data.DFinsupp.WellFounded",
"Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet",
"Mathlib.RingTheory.Spectrum.Prime.Polynomial"
] | Mathlib/RingTheory/Spectrum/Prime/ChevalleyComplexity.lean | degBound | The bound on the degree of the polynomials used to describe the constructible set appearing in
the case of `C : R → R[X]` of Chevalley's theorem with complexity bound. |
private powBound : ℕ := e.degBound ^ e.degBound | def | RingTheory | [
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Algebra.Polynomial.CoeffMem",
"Mathlib.Data.DFinsupp.WellFounded",
"Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet",
"Mathlib.RingTheory.Spectrum.Prime.Polynomial"
] | Mathlib/RingTheory/Spectrum/Prime/ChevalleyComplexity.lean | powBound | The bound on the power of the subgroup generated by the coefficients of the polynomials used to
describe the constructible set appearing in the case of `C : R → R[X]` of Chevalley's theorem with
complexity bound. |
private powBound_ne_zero : e.powBound ≠ 0 := Nat.pow_self_pos.ne'
variable (R₀ R n e) in | lemma | RingTheory | [
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Algebra.Polynomial.CoeffMem",
"Mathlib.Data.DFinsupp.WellFounded",
"Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet",
"Mathlib.RingTheory.Spectrum.Prime.Polynomial"
] | Mathlib/RingTheory/Spectrum/Prime/ChevalleyComplexity.lean | powBound_ne_zero | null |
private Statement [Algebra ℤ R] : Prop :=
∀ f : R[X], ∃ T : ConstructibleSetData R,
comap Polynomial.C '' (zeroLocus (Set.range e) \ zeroLocus {f}) = T.toSet ∧
∀ C ∈ T, C.n ≤ e.degBound ∧ ∀ i, C.g i ∈ e.coeffSubmodule R₀ ^ e.powBound | def | RingTheory | [
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Algebra.Polynomial.CoeffMem",
"Mathlib.Data.DFinsupp.WellFounded",
"Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet",
"Mathlib.RingTheory.Spectrum.Prime.Polynomial"
] | Mathlib/RingTheory/Spectrum/Prime/ChevalleyComplexity.lean | Statement | The statement we induct on in the `C : R → R[X]` case of Chevalley's theorem with complexity
bound. |
private induction_structure (n : ℕ)
(P : ∀ (R : Type u) [CommRing R], (InductionObj R n) → Prop)
(hP₁ : ∀ (R) [CommRing R], P R ⟨0⟩)
(hP₂ : ∀ (R) [CommRing R] (e : InductionObj R n) (i : Fin n),
(e.1 i).Monic → (∀ j ≠ i, e.1 j = 0) → P R e)
(hP₃ : ∀ (R) [CommRing R] (e : InductionObj R n) (i j : Fin n),
(e.1 i).Monic → (e.1 i).degree ≤ (e.1 j).degree → i ≠ j →
P R ⟨update e.1 j (e.1 j %ₘ e.1 i)⟩ → P R e)
(hP₄ : ∀ (R) [CommRing R] (c : R) (i : Fin n) (e : InductionObj R n), c = (e.1 i).leadingCoeff →
c ≠ 0 →
P (Away c) ⟨Polynomial.C (IsLocalization.Away.invSelf (S := Away c) c) •
mapRingHom (algebraMap _ _) ∘ e⟩ →
P (R ⧸ Ideal.span {c}) ⟨mapRingHom (algebraMap _ _) ∘ e⟩ → P R e)
{R} [CommRing R] (e : InductionObj R n) : P R e := by
classical
set v := e.degree with hv
clear_value v
induction v using WellFoundedLT.induction generalizing R with
| ind v H_IH =>
by_cases he0 : e = ⟨0⟩
· exact he0 ▸ hP₁ R
cases subsingleton_or_nontrivial R
· convert hP₁ R; ext; exact Subsingleton.elim _ _
simp only [InductionObj.ext_iff, funext_iff, Pi.zero_apply, not_forall] at he0
by_cases H : (∃ i, (e.1 i).Monic ∧ ∀ j, e.1 j ≠ 0 → (e.1 i).degree ≤ (e.1 j).degree)
· obtain ⟨i, hi, i_min⟩ := H
by_cases H' : ∀ j ≠ i, e.1 j = 0
· exact hP₂ R e i hi H'
· simp only [ne_eq, not_forall] at H'
obtain ⟨j, hj, hj'⟩ := H'
replace i_min := i_min j hj'
apply hP₃ R e i j hi i_min (.symm hj) (H_IH _ ?_ _ rfl)
refine .left _ _ (lt_of_le_of_ne (b := (ofLex v).1) ?_ ?_)
· intro k
simp only [comp_apply, update_apply, hv]
split_ifs with hjk
· rw [hjk]
exact (degree_modByMonic_le _ hi).trans i_min
· exact le_rfl
· simp only [hv, ne_eq, not_forall, funext_iff,
comp_apply]
use j
simp only [update_self]
refine ((degree_modByMonic_lt _ hi).trans_le i_min).ne
obtain ⟨i, hi, i_min⟩ : ∃ i, e.1 i ≠ 0 ∧ ∀ j, e.1 j ≠ 0 → (e.1 i).degree ≤ (e.1 j).degree := by
have : ∃ n : ℕ, ∃ i, (e.1 i).degree = n ∧ (e.1 i) ≠ 0 := by
obtain ⟨i, hi⟩ := he0; exact ⟨(e.1 i).natDegree, i, degree_eq_natDegree hi, hi⟩
let m := Nat.find this
obtain ⟨i, hi, hi'⟩ : ∃ i, (e.1 i).degree = m ∧ (e.1 i) ≠ 0 := Nat.find_spec this
refine ⟨i, hi', fun j hj ↦ ?_⟩
refine hi.le.trans ?_
... | lemma | RingTheory | [
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Algebra.Polynomial.CoeffMem",
"Mathlib.Data.DFinsupp.WellFounded",
"Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet",
"Mathlib.RingTheory.Spectrum.Prime.Polynomial"
] | Mathlib/RingTheory/Spectrum/Prime/ChevalleyComplexity.lean | induction_structure | The structure of the induction in the proof of Chevalley's theorem:
Consider a property on a vector `e` of polynomials. Suppose that it holds for the following cases:
1. The vector contains zeroes only.
2. The vector contains a single monic polynomial (and zero otherwise).
3. Suppose `eᵢ` has the lowest degree among all monic polynomials and `eⱼ` is some other polynomial.
If the property holds when `eⱼ` is replaced by `eⱼ % eᵢ`, then it holds for `e`.
4. Suppose the property holds for both the localization at some leading coefficient of `eᵢ` and
the localization at the leading coefficient of `eᵢ`, then the property holds for `e`.
Then it holds for all vectors `e` over all rings. |
private induction_aux (R : Type*) [CommRing R] [Algebra R₀ R]
(c : R) (i : Fin n) (e : InductionObj R n) (hi : c = (e.1 i).leadingCoeff) (hc : c ≠ 0) :
Statement R₀ (Away c) n
⟨Polynomial.C (IsLocalization.Away.invSelf (S := Away c) c) •
mapRingHom (algebraMap _ _) ∘ e⟩ →
Statement R₀ (R ⧸ Ideal.span {c}) n ⟨mapRingHom (algebraMap _ _) ∘ e⟩ →
Statement R₀ R n e := by
set q₁ := IsScalarTower.toAlgHom R₀ R (Away c)
set q₂ := Ideal.Quotient.mkₐ R₀ (.span {c})
have q₂_surjective : Surjective q₂ := Ideal.Quotient.mk_surjective
set e₁ : InductionObj (Away c) n :=
⟨Polynomial.C (IsLocalization.Away.invSelf (S := Away c) c) • mapRingHom q₁ ∘ e⟩
set e₂ : InductionObj (R ⧸ Ideal.span {c}) n := ⟨mapRingHom q₂ ∘ e⟩
have degBound_e₁_le : e₁.degBound ≤ e.degBound := by
unfold degBound
gcongr with j
exact (degree_mul_le _ _).trans <| (add_le_of_nonpos_left degree_C_le).trans degree_map_le
have degBound_e₂_lt : e₂.degBound < e.degBound := by
unfold degBound
refine Fintype.sum_strictMono <| Pi.lt_def.2 ⟨fun j ↦ ?_, i, ?_⟩
· dsimp
gcongr
exact degree_map_le
· gcongr
exact degree_map_lt (by simp [q₂, ← hi]) (by simpa [hi] using hc)
intro (H₁ : Statement R₀ _ _ e₁) (H₂ : Statement R₀ _ _ e₂) f
obtain ⟨T₁, hT₁⟩ := H₁ (mapRingHom q₁ f)
obtain ⟨T₂, hT₂⟩ := H₂ (mapRingHom q₂ f)
simp only [forall_and] at hT₁ hT₂
obtain ⟨hT₁, hT₁deg, hT₁span⟩ := hT₁
obtain ⟨hT₂, hT₂deg, hT₂span⟩ := hT₂
let _ : Invertible (q₁ c) :=
⟨IsLocalization.Away.invSelf c, by simp [q₁, IsLocalization.Away.invSelf], by
simp [q₁, IsLocalization.Away.invSelf]⟩
have he₁span :
e₁.coeffSubmodule R₀ ^ e₁.powBound = ⅟(q₁ c ^ e₁.powBound) •
(span R₀ ({c} ∪ ⋃ i, coeff(e i)) ^ e₁.powBound).map q₁.toLinearMap := by
unfold coeffSubmodule
rw [Submodule.map_pow, map_span, invOf_pow, ← smul_pow, ← span_smul]
simp [Set.image_insert_eq, Set.smul_set_insert, Set.image_iUnion, Set.smul_set_iUnion, q₁, e₁]
congr! with i
change _ = IsLocalization.Away.invSelf c • _
simp [← Set.range_comp, Set.smul_set_range]
ext
simp
replace hT₁span x hx i :=
smul_mem_pointwise_smul _ (q₁ c ^ e₁.powBound) _ (hT₁span x hx i)
simp only [he₁span, smul_invOf_smul, smul_eq_mul] at hT₁span
choose! g₁ hg₁ hq₁g₁ using hT₁span
choose! n₁ f₁ hf₁ using Away.surj (S := Away c) c
change (∀ _, _ * q₁ _ ^ _ = q₁ _) at hf₁
... | lemma | RingTheory | [
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Algebra.Polynomial.CoeffMem",
"Mathlib.Data.DFinsupp.WellFounded",
"Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet",
"Mathlib.RingTheory.Spectrum.Prime.Polynomial"
] | Mathlib/RingTheory/Spectrum/Prime/ChevalleyComplexity.lean | induction_aux | Part 4 of the induction structure applied to `Statement R₀ R n`. See the docstring of
`induction_structure`. |
private statement : ∀ S : InductionObj R n, Statement R₀ R n S := by
intro S; revert R₀; revert S
classical
apply induction_structure
· intro R _ R₀ _ _ f
refine ⟨(Finset.range (f.natDegree + 2)).image fun j ↦ ⟨f.coeff j, 0, 0⟩, ?_, ?_⟩
· convert image_comap_C_basicOpen f
· simp only [basicOpen_eq_zeroLocus_compl, Set.compl_eq_univ_diff]
congr 1
rw [← Set.univ_subset_iff]
rintro x _ _ ⟨_, rfl⟩
exact zero_mem x.asIdeal
· suffices Set.range f.coeff = ⋃ i < f.natDegree + 2, {f.coeff i} by
simp [BasicConstructibleSetData.toSet, ConstructibleSetData.toSet,
← Set.compl_eq_univ_diff, eq_compl_comm (y := zeroLocus _), ← zeroLocus_iUnion₂, this]
trans f.coeff '' (Set.Iio (f.natDegree + 2))
· refine ((Set.image_subset_range _ _).antisymm ?_).symm
rintro _ ⟨i, rfl⟩
by_cases hi : i ≤ f.natDegree
· exact ⟨i, hi.trans_lt (by simp), rfl⟩
· exact ⟨f.natDegree + 1, by simp,
by simp [f.coeff_eq_zero_of_natDegree_lt (lt_of_not_ge hi)]⟩
· ext; simp [eq_comm]
· simp
· intro R _ g i hi hi_min _ R₀ _ f
let M := R[X] ⧸ Ideal.span {g.1 i}
have : Module.Free R M := .of_basis (AdjoinRoot.powerBasis' hi).basis
have : Module.Finite R M := .of_basis (AdjoinRoot.powerBasis' hi).basis
refine ⟨(Finset.range (Module.finrank R M)).image
fun j ↦ ⟨(Algebra.lmul R M (Ideal.Quotient.mk _ f)).charpoly.coeff j, 0, 0⟩, ?_, ?_⟩
· ext x
have : zeroLocus (Set.range g.val) = zeroLocus {g.1 i} := by
rw [Set.range_eq_iUnion, zeroLocus_iUnion]
refine (Set.iInter_subset _ _).antisymm (Set.subset_iInter fun j ↦ ?_)
by_cases hij : i = j
· subst hij; rfl
· rw [hi_min j (.symm hij), zeroLocus_singleton_zero]; exact Set.subset_univ _
rw [this, ← Polynomial.algebraMap_eq, mem_image_comap_zeroLocus_sdiff,
IsScalarTower.algebraMap_apply R[X] M, isNilpotent_tensor_residueField_iff]
simp [BasicConstructibleSetData.toSet, ConstructibleSetData.toSet, Set.subset_def, M]
· simp
· intro R _ c i j hi hle hne H R₀ _ _ f
cases subsingleton_or_nontrivial R
· use ∅
simp [ConstructibleSetData.toSet, Subsingleton.elim f 0]
obtain ⟨S, hS, hS'⟩ := H (R₀ := R₀) f
refine ⟨S, Eq.trans ?_ hS, ?_⟩
· rw [← zeroLocus_span (Set.range _), ← zeroLocus_span (Set.range _),
idealSpan_range_update_divByMonic hne _ hi]
· intro C hC
let c' : InductionObj _ _ := ⟨update c.val j (c.val j %ₘ c.val i)⟩
... | lemma | RingTheory | [
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Algebra.Polynomial.CoeffMem",
"Mathlib.Data.DFinsupp.WellFounded",
"Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet",
"Mathlib.RingTheory.Spectrum.Prime.Polynomial"
] | Mathlib/RingTheory/Spectrum/Prime/ChevalleyComplexity.lean | statement | The main induction in the proof of Chevalley's theorem for `R →+* R[X]`.
See the docstring of `induction_structure` for the overview. |
chevalley_polynomialC {R : Type*} [CommRing R] (M : Submodule ℤ R) (hM : 1 ∈ M)
(S : ConstructibleSetData R[X]) (hS : ∀ C ∈ S, ∀ j k, (C.g j).coeff k ∈ M) :
∃ T : ConstructibleSetData R,
comap Polynomial.C '' S.toSet = T.toSet ∧ ∀ C ∈ T, C.n ≤ S.degBound ∧
∀ i, C.g i ∈ M ^ S.degBound ^ S.degBound := by
classical
choose f hf₁ hf₂ hf₃ using fun C : BasicConstructibleSetData R[X] ↦ statement (R₀ := ℤ) ⟨C.g⟩ C.f
refine ⟨S.biUnion f, ?_, ?_⟩
· simp only [BasicConstructibleSetData.toSet, ConstructibleSetData.toSet, Set.image_iUnion,
Finset.set_biUnion_biUnion, hf₁]
· simp only [Finset.mem_biUnion, forall_exists_index, and_imp]
intro x y hy hx
have H : degBound ⟨y.g⟩ ≤ S.degBound :=
Finset.le_sup (f := fun e ↦ ∑ i, (e.g i).degree.succ) hy
refine ⟨(hf₂ y x hx).trans H, fun i ↦ SetLike.le_def.mp ?_ (hf₃ y x hx i)⟩
gcongr
· simpa [Submodule.one_eq_span]
· refine Submodule.span_le.mpr ?_
simp [Set.subset_def, hM, forall_comm (α := R), hS y hy]
· delta powBound
by_cases h : S.degBound = 0
· have : degBound ⟨y.g⟩ = 0 := by nlinarith
rw [h, this]
gcongr
rwa [Nat.one_le_iff_ne_zero]
/-! ### The `C : R → R[X₁, ..., Xₘ]` case -/ | lemma | RingTheory | [
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Algebra.Polynomial.CoeffMem",
"Mathlib.Data.DFinsupp.WellFounded",
"Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet",
"Mathlib.RingTheory.Spectrum.Prime.Polynomial"
] | Mathlib/RingTheory/Spectrum/Prime/ChevalleyComplexity.lean | chevalley_polynomialC | The `C : R → R[X]` case of **Chevalley's theorem** with complexity bound. |
numBound (k : ℕ) (D : ℕ → ℕ) : ℕ → ℕ
| 0 => k
| n + 1 => numBound k D n * degBound k D n * D n | def | RingTheory | [
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Algebra.Polynomial.CoeffMem",
"Mathlib.Data.DFinsupp.WellFounded",
"Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet",
"Mathlib.RingTheory.Spectrum.Prime.Polynomial"
] | Mathlib/RingTheory/Spectrum/Prime/ChevalleyComplexity.lean | numBound | The bound on the number of polynomials used to describe the constructible set appearing in the
case of `C : R → R[X₁, ..., Xₘ]` of Chevalley's theorem with complexity bound. |
degBound (k : ℕ) (D : ℕ → ℕ) : ℕ → ℕ
| 0 => 1
| n + 1 => numBound k D (n + 1) ^ numBound k D (n + 1) * degBound k D n | def | RingTheory | [
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Algebra.Polynomial.CoeffMem",
"Mathlib.Data.DFinsupp.WellFounded",
"Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet",
"Mathlib.RingTheory.Spectrum.Prime.Polynomial"
] | Mathlib/RingTheory/Spectrum/Prime/ChevalleyComplexity.lean | degBound | The bound on the degree of the polynomials used to describe the constructible set appearing in
the case of `C : R → R[X₁, ..., Xₘ]` of Chevalley's theorem with complexity bound. |
@[simp] degBound_zero (k : ℕ) (D : ℕ → ℕ) : degBound k D 0 = 1 := by rw [degBound]
@[simp] lemma numBound_zero (k : ℕ) (D : ℕ → ℕ) : numBound k D 0 = k := by rw [numBound]
@[simp] | lemma | RingTheory | [
"Mathlib.Algebra.Order.SuccPred.WithBot",
"Mathlib.Algebra.Polynomial.CoeffMem",
"Mathlib.Data.DFinsupp.WellFounded",
"Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet",
"Mathlib.RingTheory.Spectrum.Prime.Polynomial"
] | Mathlib/RingTheory/Spectrum/Prime/ChevalleyComplexity.lean | degBound_zero | null |
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