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 ⌀ |
|---|---|---|---|---|---|---|
isBasis_basic_opens : TopologicalSpace.Opens.IsBasis (Set.range (@basicOpen R _)) := by
unfold TopologicalSpace.Opens.IsBasis
convert isTopologicalBasis_basic_opens (R := R)
rw [← Set.range_comp]
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isBasis_basic_opens | null |
basicOpen_eq_bot_iff (f : R) : basicOpen f = ⊥ ↔ IsNilpotent f := by
rw [← TopologicalSpace.Opens.coe_inj, basicOpen_eq_zeroLocus_compl]
simp only [Set.eq_univ_iff_forall, Set.singleton_subset_iff, TopologicalSpace.Opens.coe_bot,
nilpotent_iff_mem_prime, Set.compl_empty_iff, mem_zeroLocus, SetLike.mem_coe]
exact ⟨fun h I hI => h ⟨I, hI⟩, fun h ⟨I, hI⟩ => h I hI⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | basicOpen_eq_bot_iff | null |
localization_away_comap_range (S : Type v) [CommSemiring S] [Algebra R S] (r : R)
[IsLocalization.Away r S] : Set.range (comap (algebraMap R S)) = basicOpen r := by
rw [localization_comap_range S (Submonoid.powers r)]
ext x
simp only [mem_zeroLocus, basicOpen_eq_zeroLocus_compl, SetLike.mem_coe, Set.mem_setOf_eq,
Set.singleton_subset_iff, Set.mem_compl_iff, disjoint_iff_inf_le]
constructor
· intro h₁ h₂
exact h₁ ⟨Submonoid.mem_powers r, h₂⟩
· rintro h₁ _ ⟨⟨n, rfl⟩, h₃⟩
exact h₁ (x.2.mem_of_pow_mem _ h₃) | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | localization_away_comap_range | null |
localization_away_isOpenEmbedding (S : Type v) [CommSemiring S] [Algebra R S] (r : R)
[IsLocalization.Away r S] : IsOpenEmbedding (comap (algebraMap R S)) where
toIsEmbedding := localization_comap_isEmbedding S (Submonoid.powers r)
isOpen_range := by
rw [localization_away_comap_range S r]
exact isOpen_basicOpen | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | localization_away_isOpenEmbedding | null |
isCompact_basicOpen (f : R) : IsCompact (basicOpen f : Set (PrimeSpectrum R)) := by
rw [← localization_away_comap_range (Localization (Submonoid.powers f))]
exact isCompact_range (map_continuous _) | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isCompact_basicOpen | null |
comap_basicOpen (f : R →+* S) (x : R) :
TopologicalSpace.Opens.comap (comap f) (basicOpen x) = basicOpen (f x) :=
rfl
open TopologicalSpace in | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | comap_basicOpen | null |
iSup_basicOpen_eq_top_iff {ι : Type*} {f : ι → R} :
(⨆ i : ι, PrimeSpectrum.basicOpen (f i)) = ⊤ ↔ Ideal.span (Set.range f) = ⊤ := by
rw [SetLike.ext'_iff, Opens.coe_iSup]
simp only [PrimeSpectrum.basicOpen_eq_zeroLocus_compl, Opens.coe_top, ← Set.compl_iInter,
← PrimeSpectrum.zeroLocus_iUnion]
rw [← PrimeSpectrum.zeroLocus_empty_iff_eq_top, compl_involutive.eq_iff]
simp only [Set.iUnion_singleton_eq_range, Set.compl_univ, PrimeSpectrum.zeroLocus_span] | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | iSup_basicOpen_eq_top_iff | null |
iSup_basicOpen_eq_top_iff' {s : Set R} :
(⨆ i ∈ s, PrimeSpectrum.basicOpen i) = ⊤ ↔ Ideal.span s = ⊤ := by
conv_rhs => rw [← Subtype.range_val (s := s), ← iSup_basicOpen_eq_top_iff]
simp | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | iSup_basicOpen_eq_top_iff' | null |
isLocalization_away_iff_atPrime_of_basicOpen_eq_singleton [Algebra R S]
{f : R} {p : PrimeSpectrum R} (h : (basicOpen f).1 = {p}) :
IsLocalization.Away f S ↔ IsLocalization.AtPrime S p.1 :=
have : IsLocalization.AtPrime (Localization.Away f) p.1 := by
refine .of_le_of_exists_dvd (.powers f) _
(Submonoid.powers_le.mpr <| by apply h ▸ Set.mem_singleton p) fun r hr ↦ ?_
contrapose! hr
simp_rw [← Ideal.mem_span_singleton] at hr
have ⟨q, prime, le, disj⟩ := Ideal.exists_le_prime_disjoint (Ideal.span {r})
(.powers f) (Set.disjoint_right.mpr hr)
have : ⟨q, prime⟩ ∈ (basicOpen f).1 := Set.disjoint_right.mp disj (Submonoid.mem_powers f)
rw [h, Set.mem_singleton_iff] at this
rw [← this]
exact not_not.mpr (q.span_singleton_le_iff_mem.mp le)
IsLocalization.isLocalization_iff_of_isLocalization _ _ (Localization.Away f)
open Localization Polynomial Set in | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isLocalization_away_iff_atPrime_of_basicOpen_eq_singleton | null |
range_comap_algebraMap_localization_compl_eq_range_comap_quotientMk
{R : Type*} [CommRing R] (c : R) :
letI := (mapRingHom (algebraMap R (Away c))).toAlgebra
(range (comap (algebraMap R[X] (Away c)[X])))ᶜ
= range (comap (mapRingHom (Ideal.Quotient.mk (.span {c})))) := by
letI := (mapRingHom (algebraMap R (Away c))).toAlgebra
have := Polynomial.isLocalization (.powers c) (Away c)
rw [Submonoid.map_powers] at this
have surj : Function.Surjective (mapRingHom (Ideal.Quotient.mk (.span {c}))) :=
Polynomial.map_surjective _ Ideal.Quotient.mk_surjective
rw [range_comap_of_surjective _ _ surj, localization_away_comap_range _ (C c)]
simp [Polynomial.ker_mapRingHom, Ideal.map_span] | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | range_comap_algebraMap_localization_compl_eq_range_comap_quotientMk | null |
toPiLocalization_surjective_of_discreteTopology :
Function.Surjective (toPiLocalization R) := fun x ↦ by
have (p : PrimeSpectrum R) : ∃ f, (basicOpen f : Set _) = {p} :=
have ⟨_, ⟨f, rfl⟩, hpf, hfp⟩ := isTopologicalBasis_basic_opens.isOpen_iff.mp
(isOpen_discrete {p}) p rfl
⟨f, hfp.antisymm <| Set.singleton_subset_iff.mpr hpf⟩
choose f hf using this
let e := Equiv.ofInjective f fun p q eq ↦ Set.singleton_injective (hf p ▸ eq ▸ hf q)
have loc a : IsLocalization.AtPrime (Localization.Away a.1) (e.symm a).1 :=
(isLocalization_away_iff_atPrime_of_basicOpen_eq_singleton <| hf _).mp <| by
simp_rw [e, Equiv.apply_ofInjective_symm]; infer_instance
let algE a := IsLocalization.algEquiv (e.symm a).1.primeCompl
(Localization.AtPrime (e.symm a).1) (Localization.Away a.1)
have span_eq : Ideal.span (Set.range f) = ⊤ := iSup_basicOpen_eq_top_iff.mp <| top_unique
fun p _ ↦ TopologicalSpace.Opens.mem_iSup.mpr ⟨p, (hf p).ge rfl⟩
replace hf a : (basicOpen a.1 : Set _) = {e.symm a} := by
simp_rw [e, ← hf, Equiv.apply_ofInjective_symm]
obtain ⟨r, eq, -⟩ := Localization.existsUnique_algebraMap_eq_of_span_eq_top _ span_eq
(fun a ↦ algE a (x _)) fun a b ↦ by
obtain rfl | ne := eq_or_ne a b; · rfl
have nil : IsNilpotent (a * b : R) := (basicOpen_eq_bot_iff _).mp <| by
simp_rw [basicOpen_mul, SetLike.ext'_iff, TopologicalSpace.Opens.coe_inf, hf]
exact bot_unique (fun _ ⟨ha, hb⟩ ↦ ne <| e.symm.injective (ha.symm.trans hb))
apply (IsLocalization.subsingleton (M := .powers (a * b : R)) nil).elim
refine ⟨r, funext fun I ↦ ?_⟩
have := eq (e I)
rwa [← AlgEquiv.symm_apply_eq, AlgEquiv.commutes, e.symm_apply_apply] at this | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | toPiLocalization_surjective_of_discreteTopology | null |
maximalSpectrumToPiLocalization_surjective_of_discreteTopology :
Function.Surjective (MaximalSpectrum.toPiLocalization R) := by
rw [← piLocalizationToMaximal_comp_toPiLocalization]
exact (piLocalizationToMaximal_surjective R).comp
(toPiLocalization_surjective_of_discreteTopology R) | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | maximalSpectrumToPiLocalization_surjective_of_discreteTopology | null |
@[stacks 00JA
"See also `PrimeSpectrum.discreteTopology_iff_finite_isMaximal_and_sInf_le_nilradical`."]
_root_.MaximalSpectrum.toPiLocalizationEquiv :
R ≃+* MaximalSpectrum.PiLocalization R :=
.ofBijective _ ⟨MaximalSpectrum.toPiLocalization_injective R,
maximalSpectrumToPiLocalization_surjective_of_discreteTopology R⟩ | def | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | _root_.MaximalSpectrum.toPiLocalizationEquiv | If the prime spectrum of a commutative semiring R has discrete Zariski topology, then R is
canonically isomorphic to the product of its localizations at the (finitely many) maximal ideals. |
discreteTopology_iff_toPiLocalization_surjective {R} [CommSemiring R] :
DiscreteTopology (PrimeSpectrum R) ↔ Function.Surjective (toPiLocalization R) :=
⟨fun _ ↦ toPiLocalization_surjective_of_discreteTopology _,
discreteTopology_of_toLocalization_surjective⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | discreteTopology_iff_toPiLocalization_surjective | null |
discreteTopology_iff_toPiLocalization_bijective {R} [CommSemiring R] :
DiscreteTopology (PrimeSpectrum R) ↔ Function.Bijective (toPiLocalization R) :=
discreteTopology_iff_toPiLocalization_surjective.trans
(and_iff_right <| toPiLocalization_injective _).symm | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | discreteTopology_iff_toPiLocalization_bijective | null |
le_iff_mem_closure (x y : PrimeSpectrum R) :
x ≤ y ↔ y ∈ closure ({x} : Set (PrimeSpectrum R)) := by
rw [← asIdeal_le_asIdeal, ← zeroLocus_vanishingIdeal_eq_closure, mem_zeroLocus,
vanishingIdeal_singleton, SetLike.coe_subset_coe] | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | le_iff_mem_closure | null |
le_iff_specializes (x y : PrimeSpectrum R) : x ≤ y ↔ x ⤳ y :=
(le_iff_mem_closure x y).trans specializes_iff_mem_closure.symm | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | le_iff_specializes | null |
@[simps!]
nhdsOrderEmbedding : PrimeSpectrum R ↪o Filter (PrimeSpectrum R) :=
OrderEmbedding.ofMapLEIff nhds fun a b => (le_iff_specializes a b).symm | def | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | nhdsOrderEmbedding | `nhds` as an order embedding. |
localizationMapOfSpecializes {x y : PrimeSpectrum R} (h : x ⤳ y) :
Localization.AtPrime y.asIdeal →+* Localization.AtPrime x.asIdeal :=
@IsLocalization.lift _ _ _ _ _ _ _ _ Localization.isLocalization
(algebraMap R (Localization.AtPrime x.asIdeal))
(by
rintro ⟨a, ha⟩
rw [← PrimeSpectrum.le_iff_specializes, ← asIdeal_le_asIdeal, ← SetLike.coe_subset_coe, ←
Set.compl_subset_compl] at h
exact (IsLocalization.map_units (Localization.AtPrime x.asIdeal)
⟨a, show a ∈ x.asIdeal.primeCompl from h ha⟩ :)) | def | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | localizationMapOfSpecializes | If `x` specializes to `y`, then there is a natural map from the localization of `y` to the
localization of `x`. |
isClosed_image_of_stableUnderSpecialization
(Z : Set (PrimeSpectrum S)) (hZ : IsClosed Z)
(hf : StableUnderSpecialization (comap f '' Z)) :
IsClosed (comap f '' Z) := by
obtain ⟨I, rfl⟩ := (PrimeSpectrum.isClosed_iff_zeroLocus_ideal Z).mp hZ
refine (isClosed_iff_zeroLocus _).mpr ⟨I.comap f, le_antisymm ?_ fun p hp ↦ ?_⟩
· rintro _ ⟨q, hq, rfl⟩
exact Ideal.comap_mono hq
· obtain ⟨q, hqI, hq, hqle⟩ := p.asIdeal.exists_ideal_comap_le_prime I hp
exact hf ((le_iff_specializes ⟨q.comap f, inferInstance⟩ p).mp hqle) ⟨⟨q, hq⟩, hqI, rfl⟩
@[stacks 00HY] | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isClosed_image_of_stableUnderSpecialization | null |
isClosed_range_of_stableUnderSpecialization
(hf : StableUnderSpecialization (Set.range (comap f))) :
IsClosed (Set.range (comap f)) := by
rw [← Set.image_univ] at hf ⊢
exact isClosed_image_of_stableUnderSpecialization _ _ isClosed_univ hf
variable {f} in
@[stacks 00HY] | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isClosed_range_of_stableUnderSpecialization | null |
stableUnderSpecialization_range_iff :
StableUnderSpecialization (Set.range (comap f)) ↔ IsClosed (Set.range (comap f)) :=
⟨isClosed_range_of_stableUnderSpecialization f, fun h ↦ h.stableUnderSpecialization⟩ | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | stableUnderSpecialization_range_iff | null |
stableUnderSpecialization_image_iff
(Z : Set (PrimeSpectrum S)) (hZ : IsClosed Z) :
StableUnderSpecialization (comap f '' Z) ↔ IsClosed (comap f '' Z) :=
⟨isClosed_image_of_stableUnderSpecialization f Z hZ, fun h ↦ h.stableUnderSpecialization⟩ | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | stableUnderSpecialization_image_iff | null |
isQuotientMap_of_specializingMap (h₂ : SpecializingMap (comap f)) :
Topology.IsQuotientMap (comap f) := by
rw [Topology.isQuotientMap_iff_isClosed]
exact ⟨h₁, fun s ↦ ⟨fun hs ↦ hs.preimage (comap f).continuous,
fun hsc ↦ Set.image_preimage_eq s h₁ ▸ isClosed_image_of_stableUnderSpecialization _ _ hsc
(h₂.stableUnderSpecialization_image hsc.stableUnderSpecialization)⟩⟩ | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isQuotientMap_of_specializingMap | If `f : Spec S → Spec R` is specializing and surjective, the topology on `Spec R` is the
quotient topology induced by `f`. |
isQuotientMap_of_generalizingMap (h₂ : GeneralizingMap (comap f)) :
Topology.IsQuotientMap (comap f) := by
rw [Topology.isQuotientMap_iff_isClosed]
refine ⟨h₁, fun s ↦ ⟨fun hs ↦ hs.preimage (comap f).continuous,
fun hsc ↦ Set.image_preimage_eq s h₁ ▸ ?_⟩⟩
apply isClosed_image_of_stableUnderSpecialization _ _ hsc
rw [Set.image_preimage_eq s h₁, ← stableUnderGeneralization_compl_iff]
convert h₂.stableUnderGeneralization_image hsc.isOpen_compl.stableUnderGeneralization
rw [← Set.preimage_compl, Set.image_preimage_eq _ h₁] | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isQuotientMap_of_generalizingMap | If `f : Spec S → Spec R` is generalizing and surjective, the topology on `Spec R` is the
quotient topology induced by `f`. |
vanishingIdeal_range_comap :
vanishingIdeal (Set.range (comap f)) = (RingHom.ker f).radical := by
ext x
rw [RingHom.ker_eq_comap_bot, ← Ideal.comap_radical, Ideal.radical_eq_sInf]
simp only [mem_vanishingIdeal, Set.mem_range, forall_exists_index, forall_apply_eq_imp_iff,
comap_asIdeal, Ideal.mem_comap, bot_le, true_and, Submodule.mem_sInf, Set.mem_setOf_eq]
exact ⟨fun H I hI ↦ H ⟨I, hI⟩, fun H I ↦ H I.1 I.2⟩ | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | vanishingIdeal_range_comap | null |
closure_range_comap :
closure (Set.range (comap f)) = zeroLocus (RingHom.ker f) := by
rw [← zeroLocus_vanishingIdeal_eq_closure, vanishingIdeal_range_comap, zeroLocus_radical] | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | closure_range_comap | null |
denseRange_comap_iff_ker_le_nilRadical :
DenseRange (comap f) ↔ RingHom.ker f ≤ nilradical R := by
rw [denseRange_iff_closure_range, closure_range_comap, zeroLocus_eq_univ_iff,
SetLike.coe_subset_coe]
@[stacks 00FL] | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | denseRange_comap_iff_ker_le_nilRadical | null |
denseRange_comap_iff_minimalPrimes :
DenseRange (comap f) ↔ ∀ I (h : I ∈ minimalPrimes R), ⟨I, h.1.1⟩ ∈ Set.range (comap f) := by
constructor
· intro H I hI
have : I ∈ (RingHom.ker f).minimalPrimes := by
rw [denseRange_comap_iff_ker_le_nilRadical] at H
simp only [minimalPrimes, Ideal.minimalPrimes, Set.mem_setOf] at hI ⊢
convert hI using 2 with p
exact ⟨fun h ↦ ⟨h.1, bot_le⟩, fun h ↦ ⟨h.1, H.trans (h.1.radical_le_iff.mpr bot_le)⟩⟩
obtain ⟨p, hp, _, rfl⟩ := Ideal.exists_comap_eq_of_mem_minimalPrimes f (I := ⊥) I this
exact ⟨⟨p, hp⟩, rfl⟩
· intro H p
obtain ⟨q, hq, hq'⟩ := Ideal.exists_minimalPrimes_le (J := p.asIdeal) bot_le
exact ((le_iff_specializes ⟨q, hq.1.1⟩ p).mp hq').mem_closed isClosed_closure
(subset_closure (H q hq)) | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | denseRange_comap_iff_minimalPrimes | null |
protected pointsEquivIrreducibleCloseds :
PrimeSpectrum R ≃o (TopologicalSpace.IrreducibleCloseds (PrimeSpectrum R))ᵒᵈ where
__ := irreducibleSetEquivPoints.toEquiv.symm.trans OrderDual.toDual
map_rel_iff' {p q} :=
(RelIso.symm irreducibleSetEquivPoints).map_rel_iff.trans (le_iff_specializes p q).symm | def | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | pointsEquivIrreducibleCloseds | Zero loci of prime ideals are closed irreducible sets in the Zariski topology and any closed
irreducible set is a zero locus of some prime ideal. |
stableUnderSpecialization_singleton {x : PrimeSpectrum R} :
StableUnderSpecialization {x} ↔ x.asIdeal.IsMaximal := by
simp_rw [← isMax_iff, StableUnderSpecialization, ← le_iff_specializes, Set.mem_singleton_iff,
@forall_comm _ (_ = _), forall_eq]
exact ⟨fun H a h ↦ (H a h).le, fun H a h ↦ le_antisymm (H h) h⟩ | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | stableUnderSpecialization_singleton | null |
stableUnderGeneralization_singleton {x : PrimeSpectrum R} :
StableUnderGeneralization {x} ↔ x.asIdeal ∈ minimalPrimes R := by
simp_rw [← isMin_iff, StableUnderGeneralization, ← le_iff_specializes, Set.mem_singleton_iff,
@forall_comm _ (_ = _), forall_eq]
exact ⟨fun H a h ↦ (H a h).ge, fun H a h ↦ le_antisymm h (H h)⟩ | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | stableUnderGeneralization_singleton | null |
isCompact_isOpen_iff {s : Set (PrimeSpectrum R)} :
IsCompact s ∧ IsOpen s ↔ ∃ t : Finset R, (zeroLocus t)ᶜ = s := by
rw [isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis _
isTopologicalBasis_basic_opens isCompact_basicOpen]
simp only [basicOpen_eq_zeroLocus_compl, ← Set.compl_iInter₂, ← zeroLocus_iUnion₂,
Set.biUnion_of_singleton]
exact ⟨fun ⟨s, hs, e⟩ ↦ ⟨hs.toFinset, by simpa using e.symm⟩,
fun ⟨s, e⟩ ↦ ⟨s, s.finite_toSet, by simpa using e.symm⟩⟩ | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isCompact_isOpen_iff | null |
isCompact_isOpen_iff_ideal {s : Set (PrimeSpectrum R)} :
IsCompact s ∧ IsOpen s ↔ ∃ I : Ideal R, I.FG ∧ (zeroLocus I)ᶜ = s := by
rw [isCompact_isOpen_iff]
exact ⟨fun ⟨s, e⟩ ↦ ⟨.span s, ⟨s, rfl⟩, by simpa using e⟩,
fun ⟨I, ⟨s, hs⟩, e⟩ ↦ ⟨s, by simpa [hs.symm] using e⟩⟩ | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isCompact_isOpen_iff_ideal | null |
basicOpen_eq_zeroLocus_of_mul_add (e f : R) (mul : e * f = 0) (add : e + f = 1) :
basicOpen e = zeroLocus {f} := by
ext p
suffices e ∉ p.asIdeal ↔ f ∈ p.asIdeal by simpa
refine ⟨(p.2.mem_or_mem_of_mul_eq_zero mul).resolve_left, fun h₁ h₂ ↦ p.2.1 ?_⟩
rw [Ideal.eq_top_iff_one, ← add]
exact add_mem h₂ h₁ | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | basicOpen_eq_zeroLocus_of_mul_add | null |
zeroLocus_eq_basicOpen_of_mul_add (e f : R) (mul : e * f = 0) (add : e + f = 1) :
zeroLocus {e} = basicOpen f := by
rw [basicOpen_eq_zeroLocus_of_mul_add f e] <;> simp only [mul, add, mul_comm, add_comm] | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | zeroLocus_eq_basicOpen_of_mul_add | null |
isClopen_basicOpen_of_mul_add (e f : R) (mul : e * f = 0) (add : e + f = 1) :
IsClopen (basicOpen e : Set (PrimeSpectrum R)) :=
⟨basicOpen_eq_zeroLocus_of_mul_add e f mul add ▸ isClosed_zeroLocus _, (basicOpen e).2⟩ | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isClopen_basicOpen_of_mul_add | null |
basicOpen_injOn_isIdempotentElem :
{e : R | IsIdempotentElem e}.InjOn basicOpen := fun x hx y hy eq ↦ by
by_contra! ne
wlog ne' : x * y ≠ x generalizing x y
· apply this y hy x hx eq.symm ne.symm
rwa [mul_comm, of_not_not ne']
have : x ∉ Ideal.span {y} := fun mem ↦ ne' <| by
obtain ⟨r, rfl⟩ := Ideal.mem_span_singleton'.mp mem
rw [mul_assoc, hy]
have ⟨p, prime, le, notMem⟩ := Ideal.exists_le_prime_notMem_of_isIdempotentElem _ x hx this
exact ne_of_mem_of_not_mem' (a := ⟨p, prime⟩) notMem
(not_not.mpr <| p.span_singleton_le_iff_mem.mp le) eq | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | basicOpen_injOn_isIdempotentElem | null |
exists_mul_eq_zero_add_eq_one_basicOpen_eq_of_isClopen {s : Set (PrimeSpectrum R)}
(hs : IsClopen s) : ∃ e f : R, e * f = 0 ∧ e + f = 1 ∧ s = basicOpen e ∧ sᶜ = basicOpen f := by
cases subsingleton_or_nontrivial R
· refine ⟨0, 0, ?_, ?_, ?_, ?_⟩ <;> apply Subsingleton.elim
obtain ⟨I, hI, hI'⟩ := isCompact_isOpen_iff_ideal.mp ⟨hs.1.isCompact, hs.2⟩
obtain ⟨J, hJ, hJ'⟩ := isCompact_isOpen_iff_ideal.mp
⟨hs.2.isClosed_compl.isCompact, hs.1.isOpen_compl⟩
simp only [compl_eq_iff_isCompl, ← eq_compl_iff_isCompl, compl_compl] at hI' hJ'
have : I * J ≤ nilradical R := by
refine Ideal.radical_le_radical_iff.mp (le_of_eq ?_)
rw [← zeroLocus_eq_iff, Ideal.zero_eq_bot, zeroLocus_bot,
zeroLocus_mul, hI', hJ', Set.compl_union_self]
obtain ⟨n, hn⟩ := Ideal.exists_pow_le_of_le_radical_of_fg this (Submodule.FG.mul hI hJ)
have hnz : n ≠ 0 := by rintro rfl; simp at hn
rw [mul_pow, Ideal.zero_eq_bot] at hn
have : I ^ n ⊔ J ^ n = ⊤ := by
rw [eq_top_iff, ← Ideal.span_pow_eq_top (I ∪ J : Set R) _ n, Ideal.span_le, Set.image_union,
Set.union_subset_iff]
constructor
· rintro _ ⟨x, hx, rfl⟩; exact Ideal.mem_sup_left (Ideal.pow_mem_pow hx n)
· rintro _ ⟨x, hx, rfl⟩; exact Ideal.mem_sup_right (Ideal.pow_mem_pow hx n)
· rw [Ideal.span_union, Ideal.span_eq, Ideal.span_eq, ← zeroLocus_empty_iff_eq_top,
zeroLocus_sup, hI', hJ', Set.compl_inter_self]
rw [Ideal.eq_top_iff_one, Submodule.mem_sup] at this
obtain ⟨x, hx, y, hy, add⟩ := this
have mul : x * y = 0 := hn (Ideal.mul_mem_mul hx hy)
have : s = basicOpen x := by
refine subset_antisymm ?_ ?_
· rw [← hJ', basicOpen_eq_zeroLocus_of_mul_add _ _ mul add]
exact zeroLocus_anti_mono (Set.singleton_subset_iff.mpr <| Ideal.pow_le_self hnz hy)
· rw [basicOpen_eq_zeroLocus_compl, Set.compl_subset_comm, ← hI']
exact zeroLocus_anti_mono (Set.singleton_subset_iff.mpr <| Ideal.pow_le_self hnz hx)
refine ⟨x, y, mul, add, this, ?_⟩
rw [this, basicOpen_eq_zeroLocus_of_mul_add _ _ mul add, basicOpen_eq_zeroLocus_compl] | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | exists_mul_eq_zero_add_eq_one_basicOpen_eq_of_isClopen | null |
exists_idempotent_basicOpen_eq_of_isClopen {s : Set (PrimeSpectrum R)}
(hs : IsClopen s) : ∃ e : R, IsIdempotentElem e ∧ s = basicOpen e :=
have ⟨e, _, mul, add, eq, _⟩ := exists_mul_eq_zero_add_eq_one_basicOpen_eq_of_isClopen hs
⟨e, (IsIdempotentElem.of_mul_add mul add).1, eq⟩
@[stacks 00EE] | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | exists_idempotent_basicOpen_eq_of_isClopen | null |
existsUnique_idempotent_basicOpen_eq_of_isClopen {s : Set (PrimeSpectrum R)}
(hs : IsClopen s) : ∃! e : R, IsIdempotentElem e ∧ s = basicOpen e := by
refine existsUnique_of_exists_of_unique (exists_idempotent_basicOpen_eq_of_isClopen hs) ?_
rintro x y ⟨hx, rfl⟩ ⟨hy, eq⟩
exact basicOpen_injOn_isIdempotentElem hx hy (SetLike.ext' eq)
open TopologicalSpace.Opens in | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | existsUnique_idempotent_basicOpen_eq_of_isClopen | null |
isClopen_iff_mul_add {s : Set (PrimeSpectrum R)} :
IsClopen s ↔ ∃ e f : R, e * f = 0 ∧ e + f = 1 ∧ s = basicOpen e := by
refine ⟨fun h ↦ ?_, ?_⟩
· have ⟨e, f, h⟩ := exists_mul_eq_zero_add_eq_one_basicOpen_eq_of_isClopen h
exact ⟨e, f, by simp only [h, and_self]⟩
rintro ⟨e, f, mul, add, rfl⟩
exact isClopen_basicOpen_of_mul_add e f mul add | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isClopen_iff_mul_add | null |
isClopen_iff_mul_add_zeroLocus {s : Set (PrimeSpectrum R)} :
IsClopen s ↔ ∃ e f : R, e * f = 0 ∧ e + f = 1 ∧ s = zeroLocus {e} := by
rw [isClopen_iff_mul_add, exists_swap]
refine exists₂_congr fun e f ↦ ?_
rw [mul_comm, add_comm, ← and_assoc, ← and_assoc, and_congr_right]
intro ⟨mul, add⟩
rw [zeroLocus_eq_basicOpen_of_mul_add e f mul add]
open TopologicalSpace (Clopens) | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isClopen_iff_mul_add_zeroLocus | null |
mulZeroAddOneEquivClopens :
{e : R × R // e.1 * e.2 = 0 ∧ e.1 + e.2 = 1} ≃o Clopens (PrimeSpectrum R) where
toEquiv := .ofBijective
(fun e ↦ ⟨basicOpen e.1.1, isClopen_iff_mul_add.mpr ⟨_, _, e.2.1, e.2.2, rfl⟩⟩) <| by
refine ⟨fun ⟨x, hx⟩ ⟨y, hy⟩ eq ↦ mul_eq_zero_add_eq_one_ext_left ?_, fun s ↦ ?_⟩
· exact basicOpen_injOn_isIdempotentElem (IsIdempotentElem.of_mul_add hx.1 hx.2).1
(IsIdempotentElem.of_mul_add hy.1 hy.2).1 <| SetLike.ext' (congr_arg (·.1) eq)
· have ⟨e, f, mul, add, eq⟩ := isClopen_iff_mul_add.mp s.2
exact ⟨⟨(e, f), mul, add⟩, SetLike.ext' eq.symm⟩
map_rel_iff' {a b} := show basicOpen _ ≤ basicOpen _ ↔ _ by
rw [← inf_eq_left, ← basicOpen_mul]
refine ⟨fun h ↦ ?_, (by rw [·])⟩
rw [← inf_eq_left]
have := (IsIdempotentElem.of_mul_add a.2.1 a.2.2).1
exact mul_eq_zero_add_eq_one_ext_left (basicOpen_injOn_isIdempotentElem
(this.mul (IsIdempotentElem.of_mul_add b.2.1 b.2.2).1) this h) | def | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | mulZeroAddOneEquivClopens | Clopen subsets in the prime spectrum of a commutative semiring are in order-preserving
bijection with pairs of elements with product 0 and sum 1. (By definition, `(e₁, f₁) ≤ (e₂, f₂)`
iff `e₁ * e₂ = e₁`.) Both elements in such pairs must be idempotents, but there may exists
idempotents that do not form such pairs (does not have a "complement"). For example, in the
semiring {0, 0.5, 1} with ⊔ as + and ⊓ as *, 0.5 has no complement. |
isRetrocompact_zeroLocus_compl {s : Set R} (hs : s.Finite) :
IsRetrocompact (zeroLocus s)ᶜ :=
(QuasiSeparatedSpace.isRetrocompact_iff_isCompact (isClosed_zeroLocus _).isOpen_compl).mpr
(isCompact_isOpen_iff.mpr ⟨hs.toFinset, by simp⟩).1 | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isRetrocompact_zeroLocus_compl | null |
isRetrocompact_zeroLocus_compl_of_fg {I : Ideal R} (hI : I.FG) :
IsRetrocompact (zeroLocus (I : Set R))ᶜ := by
obtain ⟨s, rfl⟩ := hI
rw [zeroLocus_span]
exact isRetrocompact_zeroLocus_compl s.finite_toSet | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isRetrocompact_zeroLocus_compl_of_fg | null |
isRetrocompact_basicOpen {f : R} :
IsRetrocompact (basicOpen f : Set (PrimeSpectrum R)) := by
simpa using isRetrocompact_zeroLocus_compl (Set.finite_singleton f) | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isRetrocompact_basicOpen | null |
isConstructible_basicOpen {f : R} :
IsConstructible (basicOpen f : Set (PrimeSpectrum R)) :=
isRetrocompact_basicOpen.isConstructible (basicOpen f).2 | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isConstructible_basicOpen | null |
isClosedMap_comap_of_isIntegral (hf : f.IsIntegral) :
IsClosedMap (comap f) := by
refine fun s hs ↦ isClosed_image_of_stableUnderSpecialization _ _ hs ?_
rintro _ y e ⟨x, hx, rfl⟩
algebraize [f]
obtain ⟨q, hq₁, hq₂, hq₃⟩ := Ideal.exists_ideal_over_prime_of_isIntegral y.asIdeal x.asIdeal
((le_iff_specializes _ _).mpr e)
refine ⟨⟨q, hq₂⟩, ((le_iff_specializes _ ⟨q, hq₂⟩).mp hq₁).mem_closed hs hx,
PrimeSpectrum.ext hq₃⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isClosedMap_comap_of_isIntegral | null |
isClosed_comap_singleton_of_isIntegral (hf : f.IsIntegral)
(x : PrimeSpectrum S) (hx : IsClosed ({x} : Set (PrimeSpectrum S))) :
IsClosed ({comap f x} : Set (PrimeSpectrum R)) := by
simpa using isClosedMap_comap_of_isIntegral f hf _ hx | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isClosed_comap_singleton_of_isIntegral | null |
closure_image_comap_zeroLocus (I : Ideal S) :
closure (comap f '' zeroLocus I) = zeroLocus (I.comap f) := by
apply subset_antisymm
· rw [(isClosed_zeroLocus _).closure_subset_iff, Set.image_subset_iff, preimage_comap_zeroLocus]
exact zeroLocus_anti_mono (Set.image_preimage_subset _ _)
· rintro x (hx : I.comap f ≤ x.asIdeal)
obtain ⟨q, hq₁, hq₂⟩ := Ideal.exists_minimalPrimes_le hx
obtain ⟨p', hp', hp'', rfl⟩ := Ideal.exists_comap_eq_of_mem_minimalPrimes f _ hq₁
let p'' : PrimeSpectrum S := ⟨p', hp'⟩
apply isClosed_closure.stableUnderSpecialization ((le_iff_specializes
(comap f ⟨p', hp'⟩) x).mp hq₂) (subset_closure (by exact ⟨_, hp'', rfl⟩)) | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | closure_image_comap_zeroLocus | null |
isIntegral_of_isClosedMap_comap_mapRingHom (h : IsClosedMap (comap (mapRingHom f))) :
f.IsIntegral := by
algebraize [f]
suffices Algebra.IsIntegral R S by rwa [Algebra.isIntegral_def] at this
nontriviality R
nontriviality S
constructor
intro r
let p : S[X] := C r * X - 1
have : (1 : R[X]) ∈ Ideal.span {X} ⊔ (Ideal.span {p}).comap (mapRingHom f) := by
have H := h _ (isClosed_zeroLocus {p})
rw [← zeroLocus_span, ← closure_eq_iff_isClosed, closure_image_comap_zeroLocus] at H
rw [← Ideal.eq_top_iff_one, sup_comm, ← zeroLocus_empty_iff_eq_top, zeroLocus_sup, H]
suffices ∀ (a : PrimeSpectrum S[X]), p ∈ a.asIdeal → X ∉ a.asIdeal by
simpa [Set.eq_empty_iff_forall_notMem]
intro q hpq hXq
have : 1 ∈ q.asIdeal := by simpa [p] using (sub_mem (q.asIdeal.mul_mem_left (C r) hXq) hpq)
exact q.2.ne_top (q.asIdeal.eq_top_iff_one.mpr this)
obtain ⟨a, b, hb, e⟩ := Ideal.mem_span_singleton_sup.mp this
obtain ⟨c, hc : b.map (algebraMap R S) = _⟩ := Ideal.mem_span_singleton.mp hb
refine ⟨b.reverse * X ^ (1 + c.natDegree), ?_, ?_⟩
· refine Monic.mul ?_ (by simp)
have h : b.coeff 0 = 1 := by simpa using congr(($e).coeff 0)
have : b.natTrailingDegree = 0 := by simp [h]
rw [Monic.def, reverse_leadingCoeff, trailingCoeff, this, h]
· have : p.natDegree ≤ 1 := by simpa using natDegree_linear_le (a := r) (b := -1)
rw [eval₂_eq_eval_map, reverse, Polynomial.map_mul, ← reflect_map, Polynomial.map_pow,
map_X, ← revAt_zero (1 + _), ← reflect_monomial,
← reflect_mul _ _ natDegree_map_le (by simp), pow_zero, mul_one, hc,
← add_assoc, reflect_mul _ _ (this.trans (by simp)) le_rfl,
eval_mul, reflect_sub, reflect_mul _ _ (by simp) (by simp)]
simp [← pow_succ'] | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isIntegral_of_isClosedMap_comap_mapRingHom | null |
_root_.RingHom.IsIntegral.specComap_surjective {f : R →+* S} (hf : f.IsIntegral)
(hinj : Function.Injective f) : Function.Surjective f.specComap := by
algebraize [f]
intro ⟨p, hp⟩
obtain ⟨Q, _, hQ, rfl⟩ := Ideal.exists_ideal_over_prime_of_isIntegral p (⊥ : Ideal S)
(by simp [Ideal.comap_bot_of_injective (algebraMap R S) hinj])
exact ⟨⟨Q, hQ⟩, rfl⟩ | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | _root_.RingHom.IsIntegral.specComap_surjective | null |
@[stacks 00ES]
protected _root_.minimalPrimes.equivIrreducibleComponents :
minimalPrimes R ≃o (irreducibleComponents <| PrimeSpectrum R)ᵒᵈ := by
let e : {p : Ideal R | p.IsPrime ∧ ⊥ ≤ p} ≃o PrimeSpectrum R :=
⟨⟨fun x ↦ ⟨x.1, x.2.1⟩, fun x ↦ ⟨x.1, x.2, bot_le⟩, fun _ ↦ rfl, fun _ ↦ rfl⟩, Iff.rfl⟩
rw [irreducibleComponents_eq_maximals_closed]
exact OrderIso.setOfMinimalIsoSetOfMaximal
(e.trans ((PrimeSpectrum.pointsEquivIrreducibleCloseds R).trans
(TopologicalSpace.IrreducibleCloseds.orderIsoSubtype' (PrimeSpectrum R)).dual)) | def | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | _root_.minimalPrimes.equivIrreducibleComponents | Zero loci of minimal prime ideals of `R` are irreducible components in `Spec R` and any
irreducible component is a zero locus of some minimal prime ideal. |
vanishingIdeal_irreducibleComponents :
vanishingIdeal '' (irreducibleComponents <| PrimeSpectrum R) = minimalPrimes R := by
rw [irreducibleComponents_eq_maximals_closed, minimalPrimes_eq_minimals,
image_antitone_setOf_maximal (fun s t hs _ ↦ (vanishingIdeal_anti_mono_iff hs.1).symm),
← funext (@Set.mem_setOf_eq _ · Ideal.IsPrime), ← vanishingIdeal_isClosed_isIrreducible]
rfl | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | vanishingIdeal_irreducibleComponents | null |
zeroLocus_minimalPrimes :
zeroLocus ∘ (↑) '' minimalPrimes R = irreducibleComponents (PrimeSpectrum R) := by
rw [← vanishingIdeal_irreducibleComponents, ← Set.image_comp, Set.EqOn.image_eq_self]
intro s hs
simpa [zeroLocus_vanishingIdeal_eq_closure, closure_eq_iff_isClosed]
using isClosed_of_mem_irreducibleComponents s hs
variable {R} | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | zeroLocus_minimalPrimes | null |
vanishingIdeal_mem_minimalPrimes {s : Set (PrimeSpectrum R)} :
vanishingIdeal s ∈ minimalPrimes R ↔ closure s ∈ irreducibleComponents (PrimeSpectrum R) := by
constructor
· rw [← zeroLocus_minimalPrimes, ← zeroLocus_vanishingIdeal_eq_closure]
exact Set.mem_image_of_mem _
· rw [← vanishingIdeal_irreducibleComponents, ← vanishingIdeal_closure]
exact Set.mem_image_of_mem _ | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | vanishingIdeal_mem_minimalPrimes | null |
zeroLocus_ideal_mem_irreducibleComponents {I : Ideal R} :
zeroLocus I ∈ irreducibleComponents (PrimeSpectrum R) ↔ I.radical ∈ minimalPrimes R := by
rw [← vanishingIdeal_zeroLocus_eq_radical]
conv_lhs => rw [← (isClosed_zeroLocus _).closure_eq]
exact vanishingIdeal_mem_minimalPrimes.symm | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | zeroLocus_ideal_mem_irreducibleComponents | null |
closedPoint : PrimeSpectrum R :=
⟨maximalIdeal R, (maximalIdeal.isMaximal R).isPrime⟩ | def | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | closedPoint | The closed point in the prime spectrum of a local ring. |
isLocalHom_iff_comap_closedPoint {S : Type v} [CommSemiring S] [IsLocalRing S]
(f : R →+* S) : IsLocalHom f ↔ PrimeSpectrum.comap f (closedPoint S) = closedPoint R := by
have := (local_hom_TFAE f).out 0 4
rw [this, PrimeSpectrum.ext_iff]
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isLocalHom_iff_comap_closedPoint | null |
comap_closedPoint {S : Type v} [CommSemiring S] [IsLocalRing S] (f : R →+* S)
[IsLocalHom f] : PrimeSpectrum.comap f (closedPoint S) = closedPoint R :=
(isLocalHom_iff_comap_closedPoint f).mp inferInstance | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | comap_closedPoint | null |
specializes_closedPoint (x : PrimeSpectrum R) : x ⤳ closedPoint R :=
(PrimeSpectrum.le_iff_specializes _ _).mp (IsLocalRing.le_maximalIdeal x.2.1) | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | specializes_closedPoint | null |
closedPoint_mem_iff (U : TopologicalSpace.Opens <| PrimeSpectrum R) :
closedPoint R ∈ U ↔ U = ⊤ := by
constructor
· rw [eq_top_iff]
exact fun h x _ => (specializes_closedPoint x).mem_open U.2 h
· rintro rfl
exact TopologicalSpace.Opens.mem_top _ | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | closedPoint_mem_iff | null |
closed_point_mem_iff {U : TopologicalSpace.Opens (PrimeSpectrum R)} :
closedPoint R ∈ U ↔ U = ⊤ :=
⟨(eq_top_iff.mpr fun x _ ↦ (specializes_closedPoint x).mem_open U.2 ·), (· ▸ trivial)⟩
@[simp] | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | closed_point_mem_iff | null |
PrimeSpectrum.comap_residue (T : Type u) [CommRing T] [IsLocalRing T]
(x : PrimeSpectrum (ResidueField T)) : PrimeSpectrum.comap (residue T) x = closedPoint T := by
rw [Subsingleton.elim x ⊥]
ext1
exact Ideal.mk_ker
variable (R) in | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | PrimeSpectrum.comap_residue | null |
isClosed_singleton_closedPoint : IsClosed {closedPoint R} := by
rw [PrimeSpectrum.isClosed_singleton_iff_isMaximal, closedPoint]
infer_instance | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isClosed_singleton_closedPoint | null |
PrimeSpectrum.topologicalKrullDim_eq_ringKrullDim [CommSemiring R] :
topologicalKrullDim (PrimeSpectrum R) = ringKrullDim R :=
Order.krullDim_orderDual.symm.trans <| Order.krullDim_eq_of_orderIso
(PrimeSpectrum.pointsEquivIrreducibleCloseds R).symm | theorem | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | PrimeSpectrum.topologicalKrullDim_eq_ringKrullDim | null |
@[stacks 00EC]
basicOpen_eq_zeroLocus_of_isIdempotentElem
(e : R) (he : IsIdempotentElem e) :
basicOpen e = zeroLocus {1 - e} :=
basicOpen_eq_zeroLocus_of_mul_add _ _ (by simp [mul_sub, he.eq]) (by simp)
@[stacks 00EC] | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | basicOpen_eq_zeroLocus_of_isIdempotentElem | null |
zeroLocus_eq_basicOpen_of_isIdempotentElem
(e : R) (he : IsIdempotentElem e) :
zeroLocus {e} = basicOpen (1 - e) := by
rw [basicOpen_eq_zeroLocus_of_isIdempotentElem _ he.one_sub, sub_sub_cancel] | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | zeroLocus_eq_basicOpen_of_isIdempotentElem | null |
isClopen_iff {s : Set (PrimeSpectrum R)} :
IsClopen s ↔ ∃ e : R, IsIdempotentElem e ∧ s = basicOpen e := by
refine ⟨exists_idempotent_basicOpen_eq_of_isClopen, ?_⟩
rintro ⟨e, he, rfl⟩
refine ⟨?_, (basicOpen e).2⟩
rw [PrimeSpectrum.basicOpen_eq_zeroLocus_of_isIdempotentElem e he]
exact isClosed_zeroLocus _ | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isClopen_iff | null |
isClopen_iff_zeroLocus {s : Set (PrimeSpectrum R)} :
IsClopen s ↔ ∃ e : R, IsIdempotentElem e ∧ s = zeroLocus {e} :=
isClopen_iff.trans <| ⟨fun ⟨e, he, h⟩ ↦ ⟨1 - e, he.one_sub,
h.trans (basicOpen_eq_zeroLocus_of_isIdempotentElem e he)⟩,
fun ⟨e, he, h⟩ ↦ ⟨1 - e, he.one_sub, h.trans (zeroLocus_eq_basicOpen_of_isIdempotentElem e he)⟩⟩
open TopologicalSpace (Clopens Opens) | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isClopen_iff_zeroLocus | null |
@[stacks 00EE]
isIdempotentElemEquivClopens :
{e : R // IsIdempotentElem e} ≃o Clopens (PrimeSpectrum R) :=
.trans .isIdempotentElemMulZeroAddOne mulZeroAddOneEquivClopens | def | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isIdempotentElemEquivClopens | Clopen subsets in the prime spectrum of a commutative ring are in 1-1 correspondence
with idempotent elements in the ring. |
basicOpen_isIdempotentElemEquivClopens_symm (s) :
basicOpen (isIdempotentElemEquivClopens (R := R).symm s).1 = s.toOpens :=
Opens.ext <| congr_arg (·.1) (isIdempotentElemEquivClopens.apply_symm_apply s) | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | basicOpen_isIdempotentElemEquivClopens_symm | null |
coe_isIdempotentElemEquivClopens_apply (e) :
(isIdempotentElemEquivClopens e : Set (PrimeSpectrum R)) = basicOpen (e.1 : R) := rfl | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | coe_isIdempotentElemEquivClopens_apply | null |
isIdempotentElemEquivClopens_apply_toOpens (e) :
(isIdempotentElemEquivClopens e).toOpens = basicOpen (e.1 : R) := rfl | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isIdempotentElemEquivClopens_apply_toOpens | null |
isIdempotentElemEquivClopens_mul (e₁ e₂ : {e : R | IsIdempotentElem e}) :
isIdempotentElemEquivClopens ⟨_, e₁.2.mul e₂.2⟩ =
isIdempotentElemEquivClopens e₁ ⊓ isIdempotentElemEquivClopens e₂ :=
map_inf .. | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isIdempotentElemEquivClopens_mul | null |
isIdempotentElemEquivClopens_one_sub (e : {e : R | IsIdempotentElem e}) :
isIdempotentElemEquivClopens ⟨_, e.2.one_sub⟩ = (isIdempotentElemEquivClopens e)ᶜ :=
map_compl .. | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isIdempotentElemEquivClopens_one_sub | null |
isIdempotentElemEquivClopens_symm_inf (s₁ s₂) :
letI e := isIdempotentElemEquivClopens (R := R).symm
e (s₁ ⊓ s₂) = ⟨_, (e s₁).2.mul (e s₂).2⟩ :=
map_inf .. | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isIdempotentElemEquivClopens_symm_inf | null |
isIdempotentElemEquivClopens_symm_compl (s : Clopens (PrimeSpectrum R)) :
isIdempotentElemEquivClopens.symm sᶜ = ⟨_, (isIdempotentElemEquivClopens.symm s).2.one_sub⟩ :=
map_compl .. | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isIdempotentElemEquivClopens_symm_compl | null |
isIdempotentElemEquivClopens_symm_top :
isIdempotentElemEquivClopens.symm ⊤ = ⟨(1 : R), .one⟩ :=
map_top _ | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isIdempotentElemEquivClopens_symm_top | null |
isIdempotentElemEquivClopens_symm_bot :
isIdempotentElemEquivClopens.symm ⊥ = ⟨(0 : R), .zero⟩ :=
map_bot _ | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isIdempotentElemEquivClopens_symm_bot | null |
isIdempotentElemEquivClopens_symm_sup (s₁ s₂ : Clopens (PrimeSpectrum R)) :
letI e := isIdempotentElemEquivClopens (R := R).symm
e (s₁ ⊔ s₂) = ⟨_, (e s₁).2.add_sub_mul (e s₂).2⟩ :=
map_sup .. | lemma | RingTheory | [
"Mathlib.Algebra.Order.Ring.Idempotent",
"Mathlib.Order.Heyting.Hom",
"Mathlib.RingTheory.Finiteness.Ideal",
"Mathlib.RingTheory.Ideal.GoingUp",
"Mathlib.RingTheory.Ideal.MinimalPrime.Localization",
"Mathlib.RingTheory.KrullDimension.Basic",
"Mathlib.RingTheory.Localization.Algebra",
"Mathlib.RingTheo... | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | isIdempotentElemEquivClopens_symm_sup | null |
IsRankOneDiscrete : Prop where
exists_generator_lt_one' : ∃ (γ : Γˣ), zpowers γ = (valueGroup v) ∧ γ < 1 | class | RingTheory | [
"Mathlib.Algebra.GroupWithZero.Range",
"Mathlib.Algebra.Order.Group.Cyclic",
"Mathlib.RingTheory.DedekindDomain.AdicValuation",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.PrincipalIdealDomainOfPrime"
] | Mathlib/RingTheory/Valuation/Discrete/Basic.lean | IsRankOneDiscrete | Given a linearly ordered commutative group with zero `Γ` such that `Γˣ` is
nontrivial cyclic, a valuation `v : A → Γ` on a ring `A` is *discrete*, if
`genLTOne Γˣ` belongs to the image. Note that the latter is equivalent to
asking that `1 : ℤ` belongs to the image of the corresponding additive valuation. |
exists_generator_lt_one : ∃ (γ : Γˣ), zpowers γ = valueGroup v ∧ γ < 1 :=
exists_generator_lt_one' | lemma | RingTheory | [
"Mathlib.Algebra.GroupWithZero.Range",
"Mathlib.Algebra.Order.Group.Cyclic",
"Mathlib.RingTheory.DedekindDomain.AdicValuation",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.PrincipalIdealDomainOfPrime"
] | Mathlib/RingTheory/Valuation/Discrete/Basic.lean | exists_generator_lt_one | null |
noncomputable generator : Γˣ := (exists_generator_lt_one v).choose | def | RingTheory | [
"Mathlib.Algebra.GroupWithZero.Range",
"Mathlib.Algebra.Order.Group.Cyclic",
"Mathlib.RingTheory.DedekindDomain.AdicValuation",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.PrincipalIdealDomainOfPrime"
] | Mathlib/RingTheory/Valuation/Discrete/Basic.lean | generator | Given a discrete valuation `v`, `Valuation.IsRankOneDiscrete.generator` is a generator of
the value group that is `< 1`. |
generator_zpowers_eq_valueGroup :
zpowers (generator v) = valueGroup v :=
(exists_generator_lt_one v).choose_spec.1 | lemma | RingTheory | [
"Mathlib.Algebra.GroupWithZero.Range",
"Mathlib.Algebra.Order.Group.Cyclic",
"Mathlib.RingTheory.DedekindDomain.AdicValuation",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.PrincipalIdealDomainOfPrime"
] | Mathlib/RingTheory/Valuation/Discrete/Basic.lean | generator_zpowers_eq_valueGroup | null |
generator_mem_valueGroup :
(IsRankOneDiscrete.generator v) ∈ valueGroup v := by
rw [← IsRankOneDiscrete.generator_zpowers_eq_valueGroup]
exact mem_zpowers (IsRankOneDiscrete.generator v) | lemma | RingTheory | [
"Mathlib.Algebra.GroupWithZero.Range",
"Mathlib.Algebra.Order.Group.Cyclic",
"Mathlib.RingTheory.DedekindDomain.AdicValuation",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.PrincipalIdealDomainOfPrime"
] | Mathlib/RingTheory/Valuation/Discrete/Basic.lean | generator_mem_valueGroup | null |
generator_lt_one : generator v < 1 :=
(exists_generator_lt_one v).choose_spec.2 | lemma | RingTheory | [
"Mathlib.Algebra.GroupWithZero.Range",
"Mathlib.Algebra.Order.Group.Cyclic",
"Mathlib.RingTheory.DedekindDomain.AdicValuation",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.PrincipalIdealDomainOfPrime"
] | Mathlib/RingTheory/Valuation/Discrete/Basic.lean | generator_lt_one | null |
generator_ne_one : generator v ≠ 1 :=
ne_of_lt <| generator_lt_one v | lemma | RingTheory | [
"Mathlib.Algebra.GroupWithZero.Range",
"Mathlib.Algebra.Order.Group.Cyclic",
"Mathlib.RingTheory.DedekindDomain.AdicValuation",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.PrincipalIdealDomainOfPrime"
] | Mathlib/RingTheory/Valuation/Discrete/Basic.lean | generator_ne_one | null |
generator_zpowers_eq_range (K : Type*) [Field K] (w : Valuation K Γ) [IsRankOneDiscrete w] :
Units.val '' (zpowers (generator w)) = range w \ {0} := by
rw [generator_zpowers_eq_valueGroup, valueGroup_eq_range] | lemma | RingTheory | [
"Mathlib.Algebra.GroupWithZero.Range",
"Mathlib.Algebra.Order.Group.Cyclic",
"Mathlib.RingTheory.DedekindDomain.AdicValuation",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.PrincipalIdealDomainOfPrime"
] | Mathlib/RingTheory/Valuation/Discrete/Basic.lean | generator_zpowers_eq_range | null |
generator_mem_range (K : Type*) [Field K] (w : Valuation K Γ) [IsRankOneDiscrete w] :
↑(generator w) ∈ range w := by
apply diff_subset
rw [← generator_zpowers_eq_range]
exact ⟨generator w, by simp⟩ | lemma | RingTheory | [
"Mathlib.Algebra.GroupWithZero.Range",
"Mathlib.Algebra.Order.Group.Cyclic",
"Mathlib.RingTheory.DedekindDomain.AdicValuation",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.PrincipalIdealDomainOfPrime"
] | Mathlib/RingTheory/Valuation/Discrete/Basic.lean | generator_mem_range | null |
generator_ne_zero : (generator v : Γ) ≠ 0 := by simp | lemma | RingTheory | [
"Mathlib.Algebra.GroupWithZero.Range",
"Mathlib.Algebra.Order.Group.Cyclic",
"Mathlib.RingTheory.DedekindDomain.AdicValuation",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.PrincipalIdealDomainOfPrime"
] | Mathlib/RingTheory/Valuation/Discrete/Basic.lean | generator_ne_zero | null |
valueGroup_genLTOne_eq_generator : (valueGroup v).genLTOne = generator v :=
((valueGroup v).genLTOne_unique (generator_lt_one v) (generator_zpowers_eq_valueGroup v)).symm | lemma | RingTheory | [
"Mathlib.Algebra.GroupWithZero.Range",
"Mathlib.Algebra.Order.Group.Cyclic",
"Mathlib.RingTheory.DedekindDomain.AdicValuation",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.PrincipalIdealDomainOfPrime"
] | Mathlib/RingTheory/Valuation/Discrete/Basic.lean | valueGroup_genLTOne_eq_generator | null |
IsUniformizer (π : A) : Prop := v π = hv.generator
variable {v} {π : A} | def | RingTheory | [
"Mathlib.Algebra.GroupWithZero.Range",
"Mathlib.Algebra.Order.Group.Cyclic",
"Mathlib.RingTheory.DedekindDomain.AdicValuation",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.PrincipalIdealDomainOfPrime"
] | Mathlib/RingTheory/Valuation/Discrete/Basic.lean | IsUniformizer | An element `π : A` is a uniformizer if `v π` is a generator of the value group that is `< 1`. |
iff : v.IsUniformizer π ↔ v π = hv.generator := refl _ | theorem | RingTheory | [
"Mathlib.Algebra.GroupWithZero.Range",
"Mathlib.Algebra.Order.Group.Cyclic",
"Mathlib.RingTheory.DedekindDomain.AdicValuation",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.PrincipalIdealDomainOfPrime"
] | Mathlib/RingTheory/Valuation/Discrete/Basic.lean | iff | null |
ne_zero (hπ : IsUniformizer v π) : π ≠ 0 := by
intro h0
rw [h0, IsUniformizer, map_zero] at hπ
exact (Units.ne_zero _).symm hπ
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.GroupWithZero.Range",
"Mathlib.Algebra.Order.Group.Cyclic",
"Mathlib.RingTheory.DedekindDomain.AdicValuation",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.PrincipalIdealDomainOfPrime"
] | Mathlib/RingTheory/Valuation/Discrete/Basic.lean | ne_zero | null |
val (hπ : v.IsUniformizer π) : v π = hv.generator := hπ | lemma | RingTheory | [
"Mathlib.Algebra.GroupWithZero.Range",
"Mathlib.Algebra.Order.Group.Cyclic",
"Mathlib.RingTheory.DedekindDomain.AdicValuation",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.PrincipalIdealDomainOfPrime"
] | Mathlib/RingTheory/Valuation/Discrete/Basic.lean | val | null |
val_lt_one (hπ : v.IsUniformizer π) : v π < 1 := hπ ▸ hv.generator_lt_one | lemma | RingTheory | [
"Mathlib.Algebra.GroupWithZero.Range",
"Mathlib.Algebra.Order.Group.Cyclic",
"Mathlib.RingTheory.DedekindDomain.AdicValuation",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.PrincipalIdealDomainOfPrime"
] | Mathlib/RingTheory/Valuation/Discrete/Basic.lean | val_lt_one | null |
val_ne_zero (hπ : v.IsUniformizer π) : v π ≠ 0 := by
by_contra h0
simp only [IsUniformizer, h0] at hπ
exact (Units.ne_zero _).symm hπ | lemma | RingTheory | [
"Mathlib.Algebra.GroupWithZero.Range",
"Mathlib.Algebra.Order.Group.Cyclic",
"Mathlib.RingTheory.DedekindDomain.AdicValuation",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.PrincipalIdealDomainOfPrime"
] | Mathlib/RingTheory/Valuation/Discrete/Basic.lean | val_ne_zero | null |
val_pos (hπ : IsUniformizer v π) : 0 < v π := by
rw [IsUniformizer.iff] at hπ; simp [zero_lt_iff, ne_eq, hπ] | theorem | RingTheory | [
"Mathlib.Algebra.GroupWithZero.Range",
"Mathlib.Algebra.Order.Group.Cyclic",
"Mathlib.RingTheory.DedekindDomain.AdicValuation",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.PrincipalIdealDomainOfPrime"
] | Mathlib/RingTheory/Valuation/Discrete/Basic.lean | val_pos | null |
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