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 ⌀ |
|---|---|---|---|---|---|---|
preimage_specComap_zeroLocus_aux (f : R →+* S) (s : Set R) :
f.specComap ⁻¹' zeroLocus s = zeroLocus (f '' s) := by
ext x
simp only [mem_zeroLocus, Set.image_subset_iff, Set.mem_preimage, mem_zeroLocus, Ideal.coe_comap]
variable (f : R →+* S)
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | preimage_specComap_zeroLocus_aux | null |
specComap_asIdeal (y : PrimeSpectrum S) :
(f.specComap y).asIdeal = Ideal.comap f y.asIdeal :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | specComap_asIdeal | null |
specComap_id : (RingHom.id R).specComap = fun x => x :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | specComap_id | null |
specComap_comp (f : R →+* S) (g : S →+* S') :
(g.comp f).specComap = f.specComap.comp g.specComap :=
rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | specComap_comp | null |
specComap_comp_apply (f : R →+* S) (g : S →+* S') (x : PrimeSpectrum S') :
(g.comp f).specComap x = f.specComap (g.specComap x) :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | specComap_comp_apply | null |
preimage_specComap_zeroLocus (s : Set R) :
f.specComap ⁻¹' zeroLocus s = zeroLocus (f '' s) :=
preimage_specComap_zeroLocus_aux f s | theorem | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | preimage_specComap_zeroLocus | null |
specComap_injective_of_surjective (f : R →+* S) (hf : Function.Surjective f) :
Function.Injective f.specComap := fun x y h =>
PrimeSpectrum.ext
(Ideal.comap_injective_of_surjective f hf
(congr_arg PrimeSpectrum.asIdeal h : (f.specComap x).asIdeal = (f.specComap y).asIdeal)) | theorem | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | specComap_injective_of_surjective | null |
@[simps apply symm_apply]
comapEquiv (e : R ≃+* S) : PrimeSpectrum R ≃o PrimeSpectrum S where
toFun := e.symm.toRingHom.specComap
invFun := e.toRingHom.specComap
left_inv x := by
rw [← specComap_comp_apply, RingEquiv.toRingHom_eq_coe,
RingEquiv.toRingHom_eq_coe, RingEquiv.symm_comp]
rfl
right_inv x := by
rw [← specComap_comp_apply, RingEquiv.toRingHom_eq_coe,
RingEquiv.toRingHom_eq_coe, RingEquiv.comp_symm]
rfl
map_rel_iff' {I J} := Ideal.comap_le_comap_iff_of_surjective _ e.symm.surjective .. | def | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | comapEquiv | `RingHom.specComap` of an isomorphism of rings as an equivalence of their prime spectra. |
@[simps] sigmaToPi : (Σ i, PrimeSpectrum (R i)) → PrimeSpectrum (Π i, R i)
| ⟨i, p⟩ => (Pi.evalRingHom R i).specComap p | def | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | sigmaToPi | The canonical map from a disjoint union of prime spectra of commutative semirings to
the prime spectrum of the product semiring. -/
/- TODO: show this is always a topological embedding (even when ι is infinite)
and is a homeomorphism when ι is finite. |
sigmaToPi_injective : (sigmaToPi R).Injective := fun ⟨i, p⟩ ⟨j, q⟩ eq ↦ by
classical
obtain rfl | ne := eq_or_ne i j
· congr; ext x
simpa using congr_arg (Function.update (0 : ∀ i, R i) i x ∈ ·.asIdeal) eq
· refine (p.1.ne_top_iff_one.mp p.2.ne_top ?_).elim
have : Function.update (1 : ∀ i, R i) j 0 ∈ (sigmaToPi R ⟨j, q⟩).asIdeal := by simp
simpa [← eq, Function.update_of_ne ne]
variable [Infinite ι] [∀ i, Nontrivial (R i)] | theorem | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | sigmaToPi_injective | null |
exists_maximal_notMem_range_sigmaToPi_of_infinite :
∃ (I : Ideal (Π i, R i)) (_ : I.IsMaximal), ⟨I, inferInstance⟩ ∉ Set.range (sigmaToPi R) := by
classical
let J : Ideal (Π i, R i) := -- `J := Π₀ i, R i` is an ideal in `Π i, R i`
{ __ := AddMonoidHom.mrange DFinsupp.coeFnAddMonoidHom
smul_mem' := by
rintro r _ ⟨x, rfl⟩
refine ⟨.mk x.support fun i ↦ r i * x i, funext fun i ↦ show dite _ _ _ = _ from ?_⟩
simp_rw [DFinsupp.coeFnAddMonoidHom]
refine dite_eq_left_iff.mpr fun h ↦ ?_
rw [DFinsupp.notMem_support_iff.mp h, mul_zero] }
have ⟨I, max, le⟩ := J.exists_le_maximal <| (Ideal.ne_top_iff_one _).mpr <| by
rintro ⟨x, hx⟩
have ⟨i, hi⟩ := x.support.exists_notMem
simpa [DFinsupp.coeFnAddMonoidHom, DFinsupp.notMem_support_iff.mp hi] using congr_fun hx i
refine ⟨I, max, fun ⟨⟨i, p⟩, eq⟩ ↦ ?_⟩
have : ⇑(DFinsupp.single i 1) ∉ (sigmaToPi R ⟨i, p⟩).asIdeal := by
simpa using p.1.ne_top_iff_one.mp p.2.ne_top
rw [eq] at this
exact this (le ⟨.single i 1, rfl⟩)
@[deprecated (since := "2025-05-24")]
alias exists_maximal_nmem_range_sigmaToPi_of_infinite :=
exists_maximal_notMem_range_sigmaToPi_of_infinite | theorem | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | exists_maximal_notMem_range_sigmaToPi_of_infinite | An infinite product of nontrivial commutative semirings has a maximal ideal outside of the
range of `sigmaToPi`, i.e. is not of the form `πᵢ⁻¹(𝔭)` for some prime `𝔭 ⊂ R i`, where
`πᵢ : (Π i, R i) →+* R i` is the projection. For a complete description of all prime ideals,
see https://math.stackexchange.com/a/1563190. |
sigmaToPi_not_surjective_of_infinite : ¬ (sigmaToPi R).Surjective := fun surj ↦
have ⟨_, _, notMem⟩ := exists_maximal_notMem_range_sigmaToPi_of_infinite R
(Set.range_eq_univ.mpr surj ▸ notMem) ⟨⟩ | theorem | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | sigmaToPi_not_surjective_of_infinite | null |
exists_comap_evalRingHom_eq
{ι : Type*} {R : ι → Type*} [∀ i, CommRing (R i)] [Finite ι]
(p : PrimeSpectrum (Π i, R i)) :
∃ (i : ι) (q : PrimeSpectrum (R i)), (Pi.evalRingHom R i).specComap q = p := by
classical
cases nonempty_fintype ι
let e (i) : Π i, R i := Function.update 1 i 0
have H : ∏ i, e i = 0 := by
ext j
rw [Finset.prod_apply, Pi.zero_apply, Finset.prod_eq_zero (Finset.mem_univ j)]
simp [e]
obtain ⟨i, hi⟩ : ∃ i, e i ∈ p.asIdeal := by
simpa [← H, Ideal.IsPrime.prod_mem_iff] using p.asIdeal.zero_mem
let h₁ : Function.Surjective (Pi.evalRingHom R i) := RingHomSurjective.is_surjective
have h₂ : RingHom.ker (Pi.evalRingHom R i) ≤ p.asIdeal := by
intro x hx
convert p.asIdeal.mul_mem_left x hi
ext j
by_cases hj : i = j
· subst hj; simpa [e]
· simp [e, Function.update_of_ne (.symm hj)]
have : (p.asIdeal.map (Pi.evalRingHom R i)).comap (Pi.evalRingHom R i) = p.asIdeal := by
rwa [Ideal.comap_map_of_surjective _ h₁, sup_eq_left]
exact ⟨i, ⟨_, Ideal.map_isPrime_of_surjective h₁ h₂⟩, PrimeSpectrum.ext this⟩ | lemma | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | exists_comap_evalRingHom_eq | null |
sigmaToPi_bijective {ι : Type*} (R : ι → Type*) [∀ i, CommRing (R i)] [Finite ι] :
Function.Bijective (sigmaToPi R) := by
refine ⟨sigmaToPi_injective R, ?_⟩
intro q
obtain ⟨i, q, rfl⟩ := exists_comap_evalRingHom_eq q
exact ⟨⟨i, q⟩, rfl⟩ | lemma | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | sigmaToPi_bijective | null |
iUnion_range_specComap_comp_evalRingHom
{ι : Type*} {R : ι → Type*} [∀ i, CommRing (R i)] [Finite ι]
{S : Type*} [CommRing S] (f : S →+* Π i, R i) :
⋃ i, Set.range ((Pi.evalRingHom R i).comp f).specComap = Set.range f.specComap := by
simp_rw [specComap_comp]
apply subset_antisymm
· exact Set.iUnion_subset fun _ ↦ Set.range_comp_subset_range _ _
· rintro _ ⟨p, rfl⟩
obtain ⟨i, p, rfl⟩ := exists_comap_evalRingHom_eq p
exact Set.mem_iUnion_of_mem i ⟨p, rfl⟩ | lemma | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | iUnion_range_specComap_comp_evalRingHom | null |
image_specComap_zeroLocus_eq_zeroLocus_comap (hf : Surjective f) (I : Ideal S) :
f.specComap '' zeroLocus I = zeroLocus (I.comap f) := by
simp only [Set.ext_iff, Set.mem_image, mem_zeroLocus, SetLike.coe_subset_coe]
refine fun p => ⟨?_, fun h_I_p => ?_⟩
· rintro ⟨p, hp, rfl⟩ a ha
exact hp ha
· have hp : ker f ≤ p.asIdeal := (Ideal.comap_mono bot_le).trans h_I_p
refine ⟨⟨p.asIdeal.map f, Ideal.map_isPrime_of_surjective hf hp⟩, fun x hx => ?_, ?_⟩
· obtain ⟨x', rfl⟩ := hf x
exact Ideal.mem_map_of_mem f (h_I_p hx)
· ext x
rw [specComap_asIdeal, Ideal.mem_comap, Ideal.mem_map_iff_of_surjective f hf]
refine ⟨?_, fun hx => ⟨x, hx, rfl⟩⟩
rintro ⟨x', hx', heq⟩
rw [← sub_sub_cancel x' x]
refine p.asIdeal.sub_mem hx' (hp ?_)
rwa [mem_ker, map_sub, sub_eq_zero] | theorem | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | image_specComap_zeroLocus_eq_zeroLocus_comap | null |
range_specComap_of_surjective (hf : Surjective f) :
Set.range f.specComap = zeroLocus (ker f) := by
rw [← Set.image_univ]
convert image_specComap_zeroLocus_eq_zeroLocus_comap _ _ hf _
rw [zeroLocus_bot]
variable {S} | theorem | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | range_specComap_of_surjective | null |
noncomputable Ideal.primeSpectrumOrderIsoZeroLocusOfSurj (hf : Surjective f) {I : Ideal R}
(hI : RingHom.ker f = I) : PrimeSpectrum S ≃o (PrimeSpectrum.zeroLocus (R := R) I) where
toFun p := ⟨f.specComap p, hI.symm.trans_le (Ideal.ker_le_comap f)⟩
invFun := fun ⟨⟨p, _⟩, hp⟩ ↦ ⟨p.map f, p.map_isPrime_of_surjective hf (hI.trans_le hp)⟩
left_inv := by
intro ⟨p, _⟩
simp only [PrimeSpectrum.mk.injEq]
exact p.map_comap_of_surjective f hf
right_inv := by
intro ⟨⟨p, _⟩, hp⟩
simp only [Subtype.mk.injEq, PrimeSpectrum.mk.injEq]
exact (p.comap_map_of_surjective f hf).trans <| sup_eq_left.mpr (hI.trans_le hp)
map_rel_iff' {a b} := by
change a.asIdeal.comap _ ≤ b.asIdeal.comap _ ↔ a ≤ b
rw [← Ideal.map_le_iff_le_comap, Ideal.map_comap_of_surjective f hf,
PrimeSpectrum.asIdeal_le_asIdeal] | def | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | Ideal.primeSpectrumOrderIsoZeroLocusOfSurj | Let `f : R →+* S` be a surjective ring homomorphism, then `Spec S` is order-isomorphic to `Z(I)`
where `I = ker f`. |
noncomputable Ideal.primeSpectrumQuotientOrderIsoZeroLocus (I : Ideal R) :
PrimeSpectrum (R ⧸ I) ≃o (PrimeSpectrum.zeroLocus (R := R) I) :=
primeSpectrumOrderIsoZeroLocusOfSurj (Quotient.mk I) Quotient.mk_surjective I.mk_ker | def | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | Ideal.primeSpectrumQuotientOrderIsoZeroLocus | `Spec (R / I)` is order-isomorphic to `Z(I)`. |
PrimeSpectrum.mem_range_comap_iff {p : PrimeSpectrum R} :
p ∈ Set.range f.specComap ↔ (p.asIdeal.map f).comap f = p.asIdeal := by
refine ⟨fun ⟨q, hq⟩ ↦ by simp [← hq], ?_⟩
rw [Ideal.comap_map_eq_self_iff_of_isPrime]
rintro ⟨q, _, hq⟩
exact ⟨⟨q, inferInstance⟩, PrimeSpectrum.ext hq⟩
open TensorProduct | lemma | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | PrimeSpectrum.mem_range_comap_iff | `p` is in the image of `Spec S → Spec R` if and only if `p` extended to `S` and
restricted back to `R` is `p`. |
PrimeSpectrum.nontrivial_iff_mem_rangeComap {S : Type*} [CommRing S]
[Algebra R S] (p : PrimeSpectrum R) :
Nontrivial (p.asIdeal.ResidueField ⊗[R] S) ↔ p ∈ Set.range (algebraMap R S).specComap := by
let k := p.asIdeal.ResidueField
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· obtain ⟨m, hm⟩ := Ideal.exists_maximal (k ⊗[R] S)
use (Algebra.TensorProduct.includeRight).specComap ⟨m, hm.isPrime⟩
ext : 1
rw [← PrimeSpectrum.specComap_comp_apply,
← Algebra.TensorProduct.includeLeftRingHom_comp_algebraMap, specComap_comp_apply]
simp [Ideal.eq_bot_of_prime, k, ← RingHom.ker_eq_comap_bot]
· obtain ⟨q, rfl⟩ := h
let f : k ⊗[R] S →ₐ[R] q.asIdeal.ResidueField :=
Algebra.TensorProduct.lift (Ideal.ResidueField.mapₐ _ _ rfl)
(IsScalarTower.toAlgHom _ _ _) (fun _ _ ↦ Commute.all ..)
exact RingHom.domain_nontrivial f.toRingHom | lemma | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | PrimeSpectrum.nontrivial_iff_mem_rangeComap | A prime `p` is in the range of `Spec S → Spec R` if the fiber over `p` is nontrivial. |
PrimeSpectrum.residueField_specComap (I : PrimeSpectrum R) :
Set.range (algebraMap R I.asIdeal.ResidueField).specComap = {I} := by
rw [Set.range_unique, Set.singleton_eq_singleton_iff]
exact PrimeSpectrum.ext (Ideal.ext fun x ↦ Ideal.algebraMap_residueField_eq_zero) | lemma | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | PrimeSpectrum.residueField_specComap | null |
IsLocalHom.of_specComap_surjective [CommSemiring R] [CommSemiring S] (f : R →+* S)
(hf : Function.Surjective f.specComap) : IsLocalHom f where
map_nonunit x hfx := by
by_contra hx
obtain ⟨p, hp, _⟩ := exists_max_ideal_of_mem_nonunits hx
obtain ⟨⟨q, hqp⟩, hq⟩ := hf ⟨p, hp.isPrime⟩
simp only [PrimeSpectrum.mk.injEq] at hq
exact hqp.ne_top (q.eq_top_of_isUnit_mem (q.mem_comap.mp (by rwa [hq])) hfx) | theorem | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Basic",
"Mathlib.RingTheory.LocalRing.ResidueField.Ideal",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Spectrum/Prime/RingHom.lean | IsLocalHom.of_specComap_surjective | null |
noncomputable
PrimeSpectrum.tensorProductTo (x : PrimeSpectrum (S ⊗[R] T)) :
PrimeSpectrum S × PrimeSpectrum T :=
⟨comap (algebraMap _ _) x, comap Algebra.TensorProduct.includeRight.toRingHom x⟩
@[fun_prop] | def | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.RingTheory.SurjectiveOnStalks"
] | Mathlib/RingTheory/Spectrum/Prime/TensorProduct.lean | PrimeSpectrum.tensorProductTo | The canonical map from `Spec(S ⊗[R] T)` to the Cartesian product `Spec S × Spec T`. |
PrimeSpectrum.continuous_tensorProductTo : Continuous (tensorProductTo R S T) :=
(comap _).2.prodMk (comap _).2
variable (hRT : (algebraMap R T).SurjectiveOnStalks)
include hRT | lemma | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.RingTheory.SurjectiveOnStalks"
] | Mathlib/RingTheory/Spectrum/Prime/TensorProduct.lean | PrimeSpectrum.continuous_tensorProductTo | null |
PrimeSpectrum.isEmbedding_tensorProductTo_of_surjectiveOnStalks_aux
(p₁ p₂ : PrimeSpectrum (S ⊗[R] T))
(h : tensorProductTo R S T p₁ = tensorProductTo R S T p₂) :
p₁ ≤ p₂ := by
let g : T →+* S ⊗[R] T := Algebra.TensorProduct.includeRight.toRingHom
intro x hxp₁
by_contra hxp₂
obtain ⟨t, r, a, ht, e⟩ := hRT.exists_mul_eq_tmul x
(p₂.asIdeal.comap g) inferInstance
have h₁ : a ⊗ₜ[R] t ∈ p₁.asIdeal := e ▸ p₁.asIdeal.mul_mem_left (1 ⊗ₜ[R] (r • t)) hxp₁
have h₂ : a ⊗ₜ[R] t ∉ p₂.asIdeal := e ▸ p₂.asIdeal.primeCompl.mul_mem ht hxp₂
rw [← mul_one a, ← one_mul t, ← Algebra.TensorProduct.tmul_mul_tmul] at h₁ h₂
have h₃ : t ∉ p₂.asIdeal.comap g := fun h ↦ h₂ (Ideal.mul_mem_left _ _ h)
have h₄ : a ∉ p₂.asIdeal.comap (algebraMap S (S ⊗[R] T)) :=
fun h ↦ h₂ (Ideal.mul_mem_right _ _ h)
replace h₃ : t ∉ p₁.asIdeal.comap g := by
rwa [show p₁.asIdeal.comap g = p₂.asIdeal.comap g from congr($h.2.1)]
replace h₄ : a ∉ p₁.asIdeal.comap (algebraMap S (S ⊗[R] T)) := by
rwa [show p₁.asIdeal.comap (algebraMap S (S ⊗[R] T)) = p₂.asIdeal.comap _ from congr($h.1.1)]
exact p₁.asIdeal.primeCompl.mul_mem h₄ h₃ h₁ | lemma | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.RingTheory.SurjectiveOnStalks"
] | Mathlib/RingTheory/Spectrum/Prime/TensorProduct.lean | PrimeSpectrum.isEmbedding_tensorProductTo_of_surjectiveOnStalks_aux | null |
PrimeSpectrum.isEmbedding_tensorProductTo_of_surjectiveOnStalks :
IsEmbedding (tensorProductTo R S T) := by
refine ⟨?_, fun p₁ p₂ e ↦
(isEmbedding_tensorProductTo_of_surjectiveOnStalks_aux R S T hRT p₁ p₂ e).antisymm
(isEmbedding_tensorProductTo_of_surjectiveOnStalks_aux R S T hRT p₂ p₁ e.symm)⟩
let g : T →+* S ⊗[R] T := Algebra.TensorProduct.includeRight.toRingHom
refine ⟨(continuous_tensorProductTo ..).le_induced.antisymm (isBasis_basic_opens.le_iff.mpr ?_)⟩
rintro _ ⟨f, rfl⟩
rw [@isOpen_iff_forall_mem_open]
rintro J (hJ : f ∉ J.asIdeal)
obtain ⟨t, r, a, ht, e⟩ := hRT.exists_mul_eq_tmul f
(J.asIdeal.comap g) inferInstance
refine ⟨_, ?_, ⟨_, (basicOpen a).2.prod (basicOpen t).2, rfl⟩, ?_⟩
· rintro x ⟨hx₁ : a ⊗ₜ[R] (1 : T) ∉ x.asIdeal, hx₂ : (1 : S) ⊗ₜ[R] t ∉ x.asIdeal⟩
(hx₃ : f ∈ x.asIdeal)
apply x.asIdeal.primeCompl.mul_mem hx₁ hx₂
rw [Algebra.TensorProduct.tmul_mul_tmul, mul_one, one_mul, ← e]
exact x.asIdeal.mul_mem_left _ hx₃
· have : a ⊗ₜ[R] (1 : T) * (1 : S) ⊗ₜ[R] t ∉ J.asIdeal := by
rw [Algebra.TensorProduct.tmul_mul_tmul, mul_one, one_mul, ← e]
exact J.asIdeal.primeCompl.mul_mem ht hJ
rwa [J.isPrime.mul_mem_iff_mem_or_mem.not, not_or] at this | lemma | RingTheory | [
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.RingTheory.SurjectiveOnStalks"
] | Mathlib/RingTheory/Spectrum/Prime/TensorProduct.lean | PrimeSpectrum.isEmbedding_tensorProductTo_of_surjectiveOnStalks | null |
zariskiTopology : TopologicalSpace (PrimeSpectrum R) :=
TopologicalSpace.ofClosed (Set.range PrimeSpectrum.zeroLocus) ⟨Set.univ, by simp⟩
(by
intro Zs h
rw [Set.sInter_eq_iInter]
choose f hf using fun i : Zs => h i.prop
simp only [← hf]
exact ⟨_, zeroLocus_iUnion _⟩)
(by
rintro _ ⟨s, rfl⟩ _ ⟨t, rfl⟩
exact ⟨_, (union_zeroLocus s t).symm⟩) | instance | 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 | zariskiTopology | The Zariski topology on the prime spectrum of a commutative (semi)ring is defined
via the closed sets of the topology: they are exactly those sets that are the zero locus
of a subset of the ring. |
isOpen_iff (U : Set (PrimeSpectrum R)) : IsOpen U ↔ ∃ s, Uᶜ = zeroLocus s := by
simp only [@eq_comm _ Uᶜ]; rfl | 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 | isOpen_iff | null |
isClosed_iff_zeroLocus (Z : Set (PrimeSpectrum R)) : IsClosed Z ↔ ∃ s, Z = zeroLocus s := by
rw [← isOpen_compl_iff, isOpen_iff, compl_compl] | 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_iff_zeroLocus | null |
isClosed_iff_zeroLocus_ideal (Z : Set (PrimeSpectrum R)) :
IsClosed Z ↔ ∃ I : Ideal R, Z = zeroLocus I :=
(isClosed_iff_zeroLocus _).trans
⟨fun ⟨s, hs⟩ => ⟨_, (zeroLocus_span s).substr hs⟩, fun ⟨I, hI⟩ => ⟨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 | isClosed_iff_zeroLocus_ideal | null |
isClosed_iff_zeroLocus_radical_ideal (Z : Set (PrimeSpectrum R)) :
IsClosed Z ↔ ∃ I : Ideal R, I.IsRadical ∧ Z = zeroLocus I :=
(isClosed_iff_zeroLocus_ideal _).trans
⟨fun ⟨I, hI⟩ => ⟨_, I.radical_isRadical, (zeroLocus_radical I).substr hI⟩, fun ⟨I, _, hI⟩ =>
⟨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 | isClosed_iff_zeroLocus_radical_ideal | null |
isClosed_zeroLocus (s : Set R) : IsClosed (zeroLocus s) := by
rw [isClosed_iff_zeroLocus]
exact ⟨s, rfl⟩ | 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_zeroLocus | null |
zeroLocus_vanishingIdeal_eq_closure (t : Set (PrimeSpectrum R)) :
zeroLocus (vanishingIdeal t : Set R) = closure t := by
rcases isClosed_iff_zeroLocus (closure t) |>.mp isClosed_closure with ⟨I, hI⟩
rw [subset_antisymm_iff, (isClosed_zeroLocus _).closure_subset_iff, hI,
subset_zeroLocus_iff_subset_vanishingIdeal, (gc R).u_l_u_eq_u,
← subset_zeroLocus_iff_subset_vanishingIdeal, ← hI]
exact ⟨subset_closure, subset_zeroLocus_vanishingIdeal t⟩ | 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 | zeroLocus_vanishingIdeal_eq_closure | null |
vanishingIdeal_closure (t : Set (PrimeSpectrum R)) :
vanishingIdeal (closure t) = vanishingIdeal t :=
zeroLocus_vanishingIdeal_eq_closure t ▸ (gc R).u_l_u_eq_u t | 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 | vanishingIdeal_closure | null |
closure_singleton (x) : closure ({x} : Set (PrimeSpectrum R)) = zeroLocus x.asIdeal := by
rw [← zeroLocus_vanishingIdeal_eq_closure, vanishingIdeal_singleton] | 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 | closure_singleton | null |
isClosed_singleton_iff_isMaximal (x : PrimeSpectrum R) :
IsClosed ({x} : Set (PrimeSpectrum R)) ↔ x.asIdeal.IsMaximal := by
rw [← closure_subset_iff_isClosed, ← zeroLocus_vanishingIdeal_eq_closure,
vanishingIdeal_singleton]
constructor <;> intro H
· rcases x.asIdeal.exists_le_maximal x.2.1 with ⟨m, hm, hxm⟩
exact (congr_arg asIdeal (@H ⟨m, hm.isPrime⟩ hxm)) ▸ hm
· exact fun p hp ↦ PrimeSpectrum.ext (H.eq_of_le p.2.1 hp).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 | isClosed_singleton_iff_isMaximal | null |
isRadical_vanishingIdeal (s : Set (PrimeSpectrum R)) : (vanishingIdeal s).IsRadical := by
rw [← vanishingIdeal_closure, ← zeroLocus_vanishingIdeal_eq_closure,
vanishingIdeal_zeroLocus_eq_radical]
apply Ideal.radical_isRadical | 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 | isRadical_vanishingIdeal | null |
zeroLocus_eq_iff {I J : Ideal R} :
zeroLocus (I : Set R) = zeroLocus J ↔ I.radical = J.radical := by
constructor
· intro h; simp_rw [← vanishingIdeal_zeroLocus_eq_radical, h]
· intro h; rw [← zeroLocus_radical, h, zeroLocus_radical] | 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 | zeroLocus_eq_iff | null |
vanishingIdeal_anti_mono_iff {s t : Set (PrimeSpectrum R)} (ht : IsClosed t) :
s ⊆ t ↔ vanishingIdeal t ≤ vanishingIdeal s :=
⟨vanishingIdeal_anti_mono, fun h => by
rw [← ht.closure_subset_iff, ← ht.closure_eq]
convert ← zeroLocus_anti_mono_ideal h <;> apply zeroLocus_vanishingIdeal_eq_closure⟩ | 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 | vanishingIdeal_anti_mono_iff | null |
vanishingIdeal_strict_anti_mono_iff {s t : Set (PrimeSpectrum R)} (hs : IsClosed s)
(ht : IsClosed t) : s ⊂ t ↔ vanishingIdeal t < vanishingIdeal s := by
rw [Set.ssubset_def, vanishingIdeal_anti_mono_iff hs, vanishingIdeal_anti_mono_iff ht,
lt_iff_le_not_ge] | 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 | vanishingIdeal_strict_anti_mono_iff | null |
closedsEmbedding (R : Type*) [CommSemiring R] :
(TopologicalSpace.Closeds <| PrimeSpectrum R)ᵒᵈ ↪o Ideal R :=
OrderEmbedding.ofMapLEIff (fun s => vanishingIdeal ↑(OrderDual.ofDual s)) fun s _ =>
(vanishingIdeal_anti_mono_iff s.2).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 | closedsEmbedding | The antitone order embedding of closed subsets of `Spec R` into ideals of `R`. |
t1Space_iff_isField [IsDomain R] : T1Space (PrimeSpectrum R) ↔ IsField R := by
refine ⟨?_, fun h => ?_⟩
· intro h
have hbot : Ideal.IsPrime (⊥ : Ideal R) := Ideal.bot_prime
exact
Classical.not_not.1
(mt
(Ring.ne_bot_of_isMaximal_of_not_isField <|
(isClosed_singleton_iff_isMaximal _).1 (T1Space.t1 ⟨⊥, hbot⟩))
(by simp))
· refine ⟨fun x => (isClosed_singleton_iff_isMaximal x).2 ?_⟩
by_cases hx : x.asIdeal = ⊥
· letI := h.toSemifield
exact hx.symm ▸ Ideal.bot_isMaximal
· exact absurd h (Ring.not_isField_iff_exists_prime.2 ⟨x.asIdeal, ⟨hx, x.2⟩⟩)
local notation "Z(" a ")" => zeroLocus (a : Set 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 | t1Space_iff_isField | null |
isIrreducible_zeroLocus_iff_of_radical (I : Ideal R) (hI : I.IsRadical) :
IsIrreducible (zeroLocus (I : Set R)) ↔ I.IsPrime := by
rw [Ideal.isPrime_iff, IsIrreducible]
apply and_congr
· rw [Set.nonempty_iff_ne_empty, Ne, zeroLocus_empty_iff_eq_top]
· trans ∀ x y : Ideal R, Z(I) ⊆ Z(x) ∪ Z(y) → Z(I) ⊆ Z(x) ∨ Z(I) ⊆ Z(y)
· simp_rw [isPreirreducible_iff_isClosed_union_isClosed, isClosed_iff_zeroLocus_ideal]
constructor
· rintro h x y
exact h _ _ ⟨x, rfl⟩ ⟨y, rfl⟩
· rintro h _ _ ⟨x, rfl⟩ ⟨y, rfl⟩
exact h x y
· simp_rw [← zeroLocus_inf, subset_zeroLocus_iff_le_vanishingIdeal,
vanishingIdeal_zeroLocus_eq_radical, hI.radical]
constructor
· simp_rw [← SetLike.mem_coe, ← Set.singleton_subset_iff, ← Ideal.span_le, ←
Ideal.span_singleton_mul_span_singleton]
refine fun h x y h' => h _ _ ?_
rw [← hI.radical_le_iff] at h' ⊢
simpa only [Ideal.radical_inf, Ideal.radical_mul] using h'
· simp_rw [or_iff_not_imp_left, SetLike.not_le_iff_exists]
rintro h s t h' ⟨x, hx, hx'⟩ y hy
exact h (h' ⟨Ideal.mul_mem_right _ _ hx, Ideal.mul_mem_left _ _ hy⟩) 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 | isIrreducible_zeroLocus_iff_of_radical | null |
isIrreducible_zeroLocus_iff (I : Ideal R) :
IsIrreducible (zeroLocus (I : Set R)) ↔ I.radical.IsPrime :=
zeroLocus_radical I ▸ isIrreducible_zeroLocus_iff_of_radical _ I.radical_isRadical | 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 | isIrreducible_zeroLocus_iff | null |
isIrreducible_iff_vanishingIdeal_isPrime {s : Set (PrimeSpectrum R)} :
IsIrreducible s ↔ (vanishingIdeal s).IsPrime := by
rw [← isIrreducible_iff_closure, ← zeroLocus_vanishingIdeal_eq_closure,
isIrreducible_zeroLocus_iff_of_radical _ (isRadical_vanishingIdeal s)] | 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 | isIrreducible_iff_vanishingIdeal_isPrime | null |
vanishingIdeal_isIrreducible :
vanishingIdeal (R := R) '' {s | IsIrreducible s} = {P | P.IsPrime} :=
Set.ext fun I ↦ ⟨fun ⟨_, hs, e⟩ ↦ e ▸ isIrreducible_iff_vanishingIdeal_isPrime.mp hs,
fun h ↦ ⟨zeroLocus I, (isIrreducible_zeroLocus_iff_of_radical _ h.isRadical).mpr h,
(vanishingIdeal_zeroLocus_eq_radical I).trans h.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 | vanishingIdeal_isIrreducible | null |
vanishingIdeal_isClosed_isIrreducible :
vanishingIdeal (R := R) '' {s | IsClosed s ∧ IsIrreducible s} = {P | P.IsPrime} := by
refine (subset_antisymm ?_ ?_).trans vanishingIdeal_isIrreducible
· exact Set.image_mono fun _ ↦ And.right
rintro _ ⟨s, hs, rfl⟩
exact ⟨closure s, ⟨isClosed_closure, hs.closure⟩, vanishingIdeal_closure 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 | vanishingIdeal_isClosed_isIrreducible | null |
irreducibleSpace [IsDomain R] : IrreducibleSpace (PrimeSpectrum R) := by
rw [irreducibleSpace_def, Set.top_eq_univ, ← zeroLocus_bot, isIrreducible_zeroLocus_iff]
simpa using Ideal.bot_prime | instance | 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 | irreducibleSpace | null |
quasiSober : QuasiSober (PrimeSpectrum R) :=
⟨fun {S} h₁ h₂ =>
⟨⟨_, isIrreducible_iff_vanishingIdeal_isPrime.1 h₁⟩, by
rw [IsGenericPoint, closure_singleton, zeroLocus_vanishingIdeal_eq_closure, h₂.closure_eq]⟩⟩ | instance | 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 | quasiSober | null |
compactSpace : CompactSpace (PrimeSpectrum R) := by
refine compactSpace_of_finite_subfamily_closed fun S S_closed S_empty ↦ ?_
choose I hI using fun i ↦ (isClosed_iff_zeroLocus_ideal (S i)).mp (S_closed i)
simp_rw [hI, ← zeroLocus_iSup, zeroLocus_empty_iff_eq_top, ← top_le_iff] at S_empty ⊢
exact Ideal.isCompactElement_top.exists_finset_of_le_iSup _ _ S_empty | instance | 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 | compactSpace | The prime spectrum of a commutative (semi)ring is a compact topological space. |
discreteTopology_iff_finite_and_krullDimLE_zero : DiscreteTopology (PrimeSpectrum R) ↔
Finite (PrimeSpectrum R) ∧ Ring.KrullDimLE 0 R :=
⟨fun _ ↦ ⟨finite_of_compact_of_discrete, .mk₀ fun I h ↦ isClosed_singleton_iff_isMaximal ⟨I, h⟩
|>.mp <| discreteTopology_iff_forall_isClosed.mp ‹_› _⟩, fun ⟨_, _⟩ ↦
.of_finite_of_isClosed_singleton fun p ↦ (isClosed_singleton_iff_isMaximal p).mpr 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 | discreteTopology_iff_finite_and_krullDimLE_zero | The prime spectrum of a commutative semiring has discrete Zariski topology iff it is finite and
the semiring has Krull dimension zero or is trivial. |
discreteTopology_iff_finite_isMaximal_and_sInf_le_nilradical :
letI s := {I : Ideal R | I.IsMaximal}
DiscreteTopology (PrimeSpectrum R) ↔ Finite s ∧ sInf s ≤ nilradical R := by
rw [discreteTopology_iff_finite_and_krullDimLE_zero, Ring.krullDimLE_zero_iff,
(equivSubtype R).finite_iff, ← Set.coe_setOf, Set.finite_coe_iff, Set.finite_coe_iff]
refine ⟨fun h ↦ ⟨h.1.subset fun _ h ↦ h.isPrime, nilradical_eq_sInf R ▸ sInf_le_sInf h.2⟩,
fun ⟨fin, le⟩ ↦ ?_⟩
have hpm (I : Ideal R) (hI : I.IsPrime): I.IsMaximal := by
replace le := le.trans (nilradical_le_prime I)
rw [← fin.coe_toFinset, ← Finset.inf_id_eq_sInf, hI.inf_le'] at le
have ⟨M, hM, hMI⟩ := le
rw [fin.mem_toFinset] at hM
rwa [← hM.eq_of_le hI.1 hMI]
exact ⟨fin.subset hpm, hpm⟩ | 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_finite_isMaximal_and_sInf_le_nilradical | The prime spectrum of a semiring has discrete Zariski topology iff there are only
finitely many maximal ideals and their intersection is contained in the nilradical. |
discreteTopology_of_toLocalization_surjective
(surj : Function.Surjective (toPiLocalization R)) :
DiscreteTopology (PrimeSpectrum R) :=
discreteTopology_iff_finite_and_krullDimLE_zero.mpr ⟨finite_of_toPiLocalization_surjective
surj, .mk₀ fun I prime ↦ isMaximal_of_toPiLocalization_surjective surj ⟨I, prime⟩⟩ | 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_of_toLocalization_surjective | null |
comap (f : R →+* S) : C(PrimeSpectrum S, PrimeSpectrum R) where
toFun := f.specComap
continuous_toFun := by
simp only [continuous_iff_isClosed, isClosed_iff_zeroLocus]
rintro _ ⟨s, rfl⟩
exact ⟨_, preimage_specComap_zeroLocus_aux f s⟩ | 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 | comap | The continuous function between prime spectra of commutative (semi)rings induced by a ring
homomorphism. |
coe_comap (f : R →+* S) : comap f = f.specComap := 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_comap | null |
comap_apply (f : R →+* S) (x) : comap f x = f.specComap x := rfl
variable (f : R →+* S)
@[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 | comap_apply | null |
comap_asIdeal (y : PrimeSpectrum S) : (comap f y).asIdeal = Ideal.comap f y.asIdeal :=
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 | comap_asIdeal | null |
comap_id : comap (RingHom.id R) = ContinuousMap.id _ := by
ext
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 | comap_id | null |
comap_comp (f : R →+* S) (g : S →+* S') : comap (g.comp f) = (comap f).comp (comap g) :=
rfl | 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_comp | null |
comap_comp_apply (f : R →+* S) (g : S →+* S') (x : PrimeSpectrum S') :
PrimeSpectrum.comap (g.comp f) x = (PrimeSpectrum.comap f) (PrimeSpectrum.comap g x) :=
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 | comap_comp_apply | null |
preimage_comap_zeroLocus (s : Set R) : comap f ⁻¹' zeroLocus s = zeroLocus (f '' s) :=
preimage_specComap_zeroLocus_aux f s | 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 | preimage_comap_zeroLocus | null |
comap_injective_of_surjective (f : R →+* S) (hf : Function.Surjective f) :
Function.Injective (comap f) := fun _ _ h => specComap_injective_of_surjective _ hf h
variable (S) | 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_injective_of_surjective | null |
localization_specComap_injective [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
Function.Injective (algebraMap R S).specComap := by
intro p q h
replace h := _root_.congr_arg (fun x : PrimeSpectrum R => Ideal.map (algebraMap R S) x.asIdeal) h
dsimp only [RingHom.specComap] at h
rw [IsLocalization.map_comap M S, IsLocalization.map_comap M S] at h
ext1
exact h | 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_specComap_injective | null |
localization_specComap_range [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
Set.range (algebraMap R S).specComap = { p | Disjoint (M : Set R) p.asIdeal } := by
refine Set.ext fun x ↦ ⟨?_, fun h ↦ ?_⟩
· rintro ⟨p, rfl⟩
exact ((IsLocalization.isPrime_iff_isPrime_disjoint ..).mp p.2).2
· use ⟨x.asIdeal.map (algebraMap R S), IsLocalization.isPrime_of_isPrime_disjoint M S _ x.2 h⟩
ext1
exact IsLocalization.comap_map_of_isPrime_disjoint M S _ x.2 h | 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_specComap_range | null |
localization_comap_isInducing [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
IsInducing (comap (algebraMap R S)) := by
refine ⟨TopologicalSpace.ext_isClosed fun Z ↦ ?_⟩
simp_rw [isClosed_induced_iff, isClosed_iff_zeroLocus, @eq_comm _ _ (zeroLocus _),
exists_exists_eq_and, preimage_comap_zeroLocus]
constructor
· rintro ⟨s, rfl⟩
refine ⟨(Ideal.span s).comap (algebraMap R S), ?_⟩
rw [← zeroLocus_span, ← zeroLocus_span s, ← Ideal.map, IsLocalization.map_comap M S]
· rintro ⟨s, rfl⟩
exact ⟨_, rfl⟩ | 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_comap_isInducing | null |
localization_comap_injective [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
Function.Injective (comap (algebraMap R S)) :=
fun _ _ h => localization_specComap_injective S M h | 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_comap_injective | null |
localization_comap_isEmbedding [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
IsEmbedding (comap (algebraMap R S)) :=
⟨localization_comap_isInducing S M, localization_comap_injective S M⟩ | 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_comap_isEmbedding | null |
localization_comap_range [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
Set.range (comap (algebraMap R S)) = { p | Disjoint (M : Set R) p.asIdeal } :=
localization_specComap_range ..
open Function RingHom | 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_comap_range | null |
comap_isInducing_of_surjective (hf : Surjective f) : IsInducing (comap f) where
eq_induced := by
simp only [TopologicalSpace.ext_iff, ← isClosed_compl_iff, isClosed_iff_zeroLocus,
isClosed_induced_iff]
refine fun s =>
⟨fun ⟨F, hF⟩ =>
⟨zeroLocus (f ⁻¹' F), ⟨f ⁻¹' F, rfl⟩, by
rw [preimage_comap_zeroLocus, Function.Surjective.image_preimage hf, hF]⟩,
?_⟩
rintro ⟨-, ⟨F, rfl⟩, hF⟩
exact ⟨f '' F, hF.symm.trans (preimage_comap_zeroLocus f F)⟩ | 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_isInducing_of_surjective | null |
isEmbedding_comap_of_surjective (hf : Surjective f) : IsEmbedding (comap f) :=
(isEmbedding_iff _).2 ⟨comap_isInducing_of_surjective _ _ hf, comap_injective_of_surjective f hf⟩ | 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 | isEmbedding_comap_of_surjective | The embedding has closed range if the domain (and therefore the codomain) is a ring,
see `PrimeSpectrum.isClosedEmbedding_comap_of_surjective`.
On the other hand, `comap (Nat.castRingHom (ZMod 2))` does not have closed range. |
homeomorphOfRingEquiv (e : R ≃+* S) : PrimeSpectrum R ≃ₜ PrimeSpectrum S where
toFun := comap (e.symm : S →+* R)
invFun := comap (e : R →+* S)
left_inv _ := (comap_comp_apply ..).symm.trans (by simp)
right_inv _ := (comap_comp_apply ..).symm.trans (by simp) | 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 | homeomorphOfRingEquiv | Homeomorphism between prime spectra induced by an isomorphism of semirings. |
isHomeomorph_comap_of_bijective {f : R →+* S} (hf : Function.Bijective f) :
IsHomeomorph (comap f) := (homeomorphOfRingEquiv (.ofBijective f hf)).symm.isHomeomorph | 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 | isHomeomorph_comap_of_bijective | null |
comap_singleton_isClosed_of_surjective (f : R →+* S) (hf : Function.Surjective f)
(x : PrimeSpectrum S) (hx : IsClosed ({x} : Set (PrimeSpectrum S))) :
IsClosed ({comap f x} : Set (PrimeSpectrum R)) :=
haveI : x.asIdeal.IsMaximal := (isClosed_singleton_iff_isMaximal x).1 hx
(isClosed_singleton_iff_isMaximal _).2 (Ideal.comap_isMaximal_of_surjective f hf) | 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_singleton_isClosed_of_surjective | null |
image_comap_zeroLocus_eq_zeroLocus_comap (hf : Surjective f) (I : Ideal S) :
comap f '' zeroLocus I = zeroLocus (I.comap f) :=
image_specComap_zeroLocus_eq_zeroLocus_comap _ f hf I | 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 | image_comap_zeroLocus_eq_zeroLocus_comap | null |
range_comap_of_surjective (hf : Surjective f) :
Set.range (comap f) = zeroLocus (ker f) :=
range_specComap_of_surjective _ f hf | 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 | range_comap_of_surjective | null |
comap_quotientMk_bijective_of_le_nilradical {I : Ideal R} (hle : I ≤ nilradical R) :
Function.Bijective (comap <| Ideal.Quotient.mk I) := by
refine ⟨comap_injective_of_surjective _ Ideal.Quotient.mk_surjective, ?_⟩
simpa [← Set.range_eq_univ, range_comap_of_surjective _ _ Ideal.Quotient.mk_surjective,
zeroLocus_eq_univ_iff] | 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_quotientMk_bijective_of_le_nilradical | null |
isClosed_range_comap_of_surjective (hf : Surjective f) :
IsClosed (Set.range (comap f)) := by
rw [range_comap_of_surjective _ f hf]
exact isClosed_zeroLocus _ | 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_range_comap_of_surjective | null |
isClosedEmbedding_comap_of_surjective (hf : Surjective f) : IsClosedEmbedding (comap f) where
toIsInducing := comap_isInducing_of_surjective S f hf
injective := comap_injective_of_surjective f hf
isClosed_range := isClosed_range_comap_of_surjective S f hf | 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 | isClosedEmbedding_comap_of_surjective | null |
primeSpectrumProd_symm_inl (x) :
(primeSpectrumProd R S).symm (.inl x) = comap (RingHom.fst R S) x := by
ext; simp [Ideal.prod] | 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 | primeSpectrumProd_symm_inl | null |
primeSpectrumProd_symm_inr (x) :
(primeSpectrumProd R S).symm (.inr x) = comap (RingHom.snd R S) x := by
ext; simp [Ideal.prod] | 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 | primeSpectrumProd_symm_inr | null |
range_comap_fst :
Set.range (comap (RingHom.fst R S)) = zeroLocus (RingHom.ker (RingHom.fst R S)) := by
refine Set.ext fun p ↦ ⟨?_, fun h ↦ ?_⟩
· rintro ⟨I, hI, rfl⟩; exact Ideal.comap_mono bot_le
obtain ⟨p, hp, eq⟩ | ⟨p, hp, eq⟩ := p.1.ideal_prod_prime.mp p.2
· exact ⟨⟨p, hp⟩, PrimeSpectrum.ext <| by simpa [Ideal.prod] using eq.symm⟩
· refine (hp.ne_top <| (Ideal.eq_top_iff_one _).mpr ?_).elim
simpa [eq] using h (show (0, 1) ∈ RingHom.ker (RingHom.fst R S) by 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 | range_comap_fst | null |
range_comap_snd :
Set.range (comap (RingHom.snd R S)) = zeroLocus (RingHom.ker (RingHom.snd R S)) := by
refine Set.ext fun p ↦ ⟨?_, fun h ↦ ?_⟩
· rintro ⟨I, hI, rfl⟩; exact Ideal.comap_mono bot_le
obtain ⟨p, hp, eq⟩ | ⟨p, hp, eq⟩ := p.1.ideal_prod_prime.mp p.2
· refine (hp.ne_top <| (Ideal.eq_top_iff_one _).mpr ?_).elim
simpa [eq] using h (show (1, 0) ∈ RingHom.ker (RingHom.snd R S) by simp)
· exact ⟨⟨p, hp⟩, PrimeSpectrum.ext <| by simpa [Ideal.prod] using eq.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 | range_comap_snd | null |
isClosedEmbedding_comap_fst : IsClosedEmbedding (comap (RingHom.fst R S)) :=
(isClosedEmbedding_iff _).mpr ⟨isEmbedding_comap_of_surjective _ _ Prod.fst_surjective, by
simp_rw [range_comap_fst, 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 | isClosedEmbedding_comap_fst | null |
isClosedEmbedding_comap_snd : IsClosedEmbedding (comap (RingHom.snd R S)) :=
(isClosedEmbedding_iff _).mpr ⟨isEmbedding_comap_of_surjective _ _ Prod.snd_surjective, by
simp_rw [range_comap_snd, 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 | isClosedEmbedding_comap_snd | null |
noncomputable
primeSpectrumProdHomeo :
PrimeSpectrum (R × S) ≃ₜ PrimeSpectrum R ⊕ PrimeSpectrum S := by
refine ((primeSpectrumProd R S).symm.toHomeomorphOfIsInducing ?_).symm
refine (IsClosedEmbedding.of_continuous_injective_isClosedMap ?_
(Equiv.injective _) ?_).isInducing
· rw [continuous_sum_dom]
simp only [Function.comp_def, primeSpectrumProd_symm_inl, primeSpectrumProd_symm_inr]
exact ⟨(comap _).2, (comap _).2⟩
· simp_rw [isClosedMap_sum, primeSpectrumProd_symm_inl, primeSpectrumProd_symm_inr]
exact ⟨isClosedEmbedding_comap_fst.isClosedMap, isClosedEmbedding_comap_snd.isClosedMap⟩ | 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 | primeSpectrumProdHomeo | The prime spectrum of `R × S` is homeomorphic
to the disjoint union of `PrimeSpectrum R` and `PrimeSpectrum S`. |
basicOpen (r : R) : TopologicalSpace.Opens (PrimeSpectrum R) where
carrier := { x | r ∉ x.asIdeal }
is_open' := ⟨{r}, Set.ext fun _ => Set.singleton_subset_iff.trans <| Classical.not_not.symm⟩
@[simp] | 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 | basicOpen | `basicOpen r` is the open subset containing all prime ideals not containing `r`. |
mem_basicOpen (f : R) (x : PrimeSpectrum R) : x ∈ basicOpen f ↔ f ∉ x.asIdeal :=
Iff.rfl | 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 | mem_basicOpen | null |
isOpen_basicOpen {a : R} : IsOpen (basicOpen a : Set (PrimeSpectrum R)) :=
(basicOpen a).isOpen
@[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 | isOpen_basicOpen | null |
basicOpen_eq_zeroLocus_compl (r : R) :
(basicOpen r : Set (PrimeSpectrum R)) = (zeroLocus {r})ᶜ :=
Set.ext fun x => by simp only [SetLike.mem_coe, mem_basicOpen, Set.mem_compl_iff, mem_zeroLocus,
Set.singleton_subset_iff]
@[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 | basicOpen_eq_zeroLocus_compl | null |
basicOpen_one : basicOpen (1 : R) = ⊤ :=
TopologicalSpace.Opens.ext <| by simp
@[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 | basicOpen_one | null |
basicOpen_zero : basicOpen (0 : R) = ⊥ :=
TopologicalSpace.Opens.ext <| by 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 | basicOpen_zero | null |
basicOpen_le_basicOpen_iff (f g : R) :
basicOpen f ≤ basicOpen g ↔ f ∈ (Ideal.span ({g} : Set R)).radical := by
rw [← SetLike.coe_subset_coe, basicOpen_eq_zeroLocus_compl, basicOpen_eq_zeroLocus_compl,
Set.compl_subset_compl, zeroLocus_subset_zeroLocus_singleton_iff] | 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_le_basicOpen_iff | null |
basicOpen_le_basicOpen_iff_algebraMap_isUnit {f g : R} [Algebra R S]
[IsLocalization.Away f S] : basicOpen f ≤ basicOpen g ↔ IsUnit (algebraMap R S g) := by
simp_rw [basicOpen_le_basicOpen_iff, Ideal.mem_radical_iff, Ideal.mem_span_singleton,
IsLocalization.Away.algebraMap_isUnit_iff f] | 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_le_basicOpen_iff_algebraMap_isUnit | null |
basicOpen_mul (f g : R) : basicOpen (f * g) = basicOpen f ⊓ basicOpen g :=
TopologicalSpace.Opens.ext <| by simp [zeroLocus_singleton_mul] | 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_mul | null |
basicOpen_mul_le_left (f g : R) : basicOpen (f * g) ≤ basicOpen f := by
rw [basicOpen_mul f g]
exact inf_le_left | 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_mul_le_left | null |
basicOpen_mul_le_right (f g : R) : basicOpen (f * g) ≤ basicOpen g := by
rw [basicOpen_mul f g]
exact inf_le_right
@[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 | basicOpen_mul_le_right | null |
basicOpen_pow (f : R) (n : ℕ) (hn : 0 < n) : basicOpen (f ^ n) = basicOpen f :=
TopologicalSpace.Opens.ext <| by simpa using zeroLocus_singleton_pow f n hn | 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_pow | null |
isTopologicalBasis_basic_opens :
TopologicalSpace.IsTopologicalBasis
(Set.range fun r : R => (basicOpen r : Set (PrimeSpectrum R))) := by
apply TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds
· rintro _ ⟨r, rfl⟩
exact isOpen_basicOpen
· rintro p U hp ⟨s, hs⟩
rw [← compl_compl U, Set.mem_compl_iff, ← hs, mem_zeroLocus, Set.not_subset] at hp
obtain ⟨f, hfs, hfp⟩ := hp
refine ⟨basicOpen f, ⟨f, rfl⟩, hfp, ?_⟩
rw [← Set.compl_subset_compl, ← hs, basicOpen_eq_zeroLocus_compl, compl_compl]
exact zeroLocus_anti_mono (Set.singleton_subset_iff.mpr hfs) | 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 | isTopologicalBasis_basic_opens | null |
eq_biUnion_of_isOpen {s : Set (PrimeSpectrum R)} (hs : IsOpen s) :
s = ⋃ (r : R) (_ : ↑(basicOpen r) ⊆ s), basicOpen r :=
(isTopologicalBasis_basic_opens.open_eq_sUnion' hs).trans <| by aesop | 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 | eq_biUnion_of_isOpen | null |
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