The Zariski topology on the prime spectrum of a commutative (semi)ring #

Conventions #

We denote subsets of (semi)rings with s, s', etc... whereas we denote subsets of prime spectra with t, t', etc...

Inspiration/contributors #

The contents of this file draw inspiration from https://github.com/ramonfmir/lean-scheme which has contributions from Ramon Fernandez Mir, Kevin Buzzard, Kenny Lau, and Chris Hughes (on an earlier repository).

Main definitions #

PrimeSpectrum.zariskiTopology: the Zariski topology on the prime spectrum, whose closed sets are zero loci (zeroLocus).
PrimeSpectrum.basicOpen: the complement of the zero locus of a single element. The basicOpens form a topological basis of the Zariski topology: PrimeSpectrum.isTopologicalBasis_basic_opens.
PrimeSpectrum.comap: the continuous map between prime spectra induced by a ring homomorphism.
IsLocalRing.closedPoint: the maximal ideal of a local ring is the unique closed point in its prime spectrum.

Main results #

PrimeSpectrum.instSpectralSpace: every prime spectrum is a spectral space, i.e. it is quasi-compact, sober (in particular T0), quasi-separated, and its compact open subsets form a topological basis.
PrimeSpectrum.discreteTopology_iff_finite_and_krullDimLE_zero: the prime spectrum of a commutative semiring is discrete iff it is finite and the semiring has zero Krull dimension or is trivial.
PrimeSpectrum.localization_comap_range, PrimeSpectrum.localization_comap_isEmbedding: localization at a submonoid of a commutative semiring induces an embedding between the prime spectra, with range consisting of prime ideals disjoint from the submonoid.
PrimeSpectrum.localization_away_comap_range: for localization away from an element, the range of the embedding is the basicOpen associated to the element.
PrimeSpectrum.comap_isEmbedding_of_surjective: a surjective ring homomorphism between commutative semirings induces an embedding between the prime spectra.
PrimeSpectrum.isClosedEmbedding_comap_of_surjective: a surjective ring homomorphism between commutative rings induces a closed embedding between the prime spectra.
PrimeSpectrum.primeSpectrumProdHomeo: the prime spectrum of a product semiring is homeomorphic to the disjoint union of the prime spectra.
PrimeSpectrum.stableUnderSpecialization_range_iff: the range of PrimeSpectrum.comap _ is closed iff it is stable under specialization.
PrimeSpectrum.denseRange_comap_iff_minimalPrimes, PrimeSpectrum.denseRange_comap_iff_ker_le_nilRadical: the range of comap f is dense iff it contains all minimal primes, iff the kernel of f is contained in the nilradical.
PrimeSpectrum.isClosedMap_comap_of_isIntegral: comap f is a closed map if f is integral.
PrimeSpectrum.isIntegral_of_isClosedMap_comap_mapRingHom: f : R →+* S is integral if comap (Polynomial.mapRingHom f : R[X] →+* S[X]) is a closed map.

In the prime spectrum of a commutative semiring:

PrimeSpectrum.isClosed_iff_zeroLocus_radical_ideal, PrimeSpectrum.isRadical_vanishingIdeal, PrimeSpectrum.zeroLocus_eq_iff, PrimeSpectrum.vanishingIdeal_anti_mono_iff: closed subsets correspond to radical ideals.
PrimeSpectrum.isClosed_singleton_iff_isMaximal: closed points correspond to maximal ideals.
PrimeSpectrum.isIrreducible_iff_vanishingIdeal_isPrime: irreducible closed subsets correspond to prime ideals.
minimalPrimes.equivIrreducibleComponents: irreducible components correspond to minimal primes.
PrimeSpectrum.mulZeroAddOneEquivClopens: clopen subsets correspond to pairs of elements that add up to 1 and multiply to 0 in the semiring.
PrimeSpectrum.isIdempotentElemEquivClopens: (if the semiring is a ring) clopen subsets correspond to idempotents in the ring.

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instance PrimeSpectrum.zariskiTopology {R : Type u} [CommSemiring R] :

TopologicalSpace (PrimeSpectrum R)

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.

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theorem PrimeSpectrum.isOpen_iff {R : Type u} [CommSemiring R] (U : Set (PrimeSpectrum R)) :

IsOpen U ↔ ∃ (s : Set R), Uᶜ = zeroLocus s

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theorem PrimeSpectrum.isClosed_iff_zeroLocus {R : Type u} [CommSemiring R] (Z : Set (PrimeSpectrum R)) :

IsClosed Z ↔ ∃ (s : Set R), Z = zeroLocus s

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theorem PrimeSpectrum.isClosed_iff_zeroLocus_ideal {R : Type u} [CommSemiring R] (Z : Set (PrimeSpectrum R)) :

IsClosed Z ↔ ∃ (I : Ideal R), Z = zeroLocus ↑I

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theorem PrimeSpectrum.isClosed_iff_zeroLocus_radical_ideal {R : Type u} [CommSemiring R] (Z : Set (PrimeSpectrum R)) :

IsClosed Z ↔ ∃ (I : Ideal R), I.IsRadical ∧ Z = zeroLocus ↑I

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theorem PrimeSpectrum.isClosed_zeroLocus {R : Type u} [CommSemiring R] (s : Set R) :

IsClosed (zeroLocus s)

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theorem PrimeSpectrum.zeroLocus_vanishingIdeal_eq_closure {R : Type u} [CommSemiring R] (t : Set (PrimeSpectrum R)) :

zeroLocus ↑(vanishingIdeal t) = closure t

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theorem PrimeSpectrum.vanishingIdeal_closure {R : Type u} [CommSemiring R] (t : Set (PrimeSpectrum R)) :

vanishingIdeal (closure t) = vanishingIdeal t

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theorem PrimeSpectrum.closure_singleton {R : Type u} [CommSemiring R] (x : PrimeSpectrum R) :

closure {x} = zeroLocus ↑x.asIdeal

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theorem PrimeSpectrum.isClosed_singleton_iff_isMaximal {R : Type u} [CommSemiring R] (x : PrimeSpectrum R) :

IsClosed {x} ↔ x.asIdeal.IsMaximal

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theorem PrimeSpectrum.isRadical_vanishingIdeal {R : Type u} [CommSemiring R] (s : Set (PrimeSpectrum R)) :

(vanishingIdeal s).IsRadical

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theorem PrimeSpectrum.zeroLocus_eq_iff {R : Type u} [CommSemiring R] {I J : Ideal R} :

zeroLocus ↑I = zeroLocus ↑J ↔ I.radical = J.radical

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theorem PrimeSpectrum.vanishingIdeal_anti_mono_iff {R : Type u} [CommSemiring R] {s t : Set (PrimeSpectrum R)} (ht : IsClosed t) :

s ⊆ t ↔ vanishingIdeal t ≤ vanishingIdeal s

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theorem PrimeSpectrum.vanishingIdeal_strict_anti_mono_iff {R : Type u} [CommSemiring R] {s t : Set (PrimeSpectrum R)} (hs : IsClosed s) (ht : IsClosed t) :

s ⊂ t ↔ vanishingIdeal t < vanishingIdeal s

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def PrimeSpectrum.closedsEmbedding (R : Type u_1) [CommSemiring R] :

(TopologicalSpace.Closeds (PrimeSpectrum R))ᵒᵈ ↪o Ideal R

The antitone order embedding of closed subsets of Spec R into ideals of R.

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theorem PrimeSpectrum.t1Space_iff_isField {R : Type u} [CommSemiring R] [IsDomain R] :

T1Space (PrimeSpectrum R) ↔ IsField R

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theorem PrimeSpectrum.isIrreducible_zeroLocus_iff_of_radical {R : Type u} [CommSemiring R] (I : Ideal R) (hI : I.IsRadical) :

IsIrreducible (zeroLocus ↑I) ↔ I.IsPrime

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theorem PrimeSpectrum.isIrreducible_zeroLocus_iff {R : Type u} [CommSemiring R] (I : Ideal R) :

IsIrreducible (zeroLocus ↑I) ↔ I.radical.IsPrime

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theorem PrimeSpectrum.isIrreducible_iff_vanishingIdeal_isPrime {R : Type u} [CommSemiring R] {s : Set (PrimeSpectrum R)} :

IsIrreducible s ↔ (vanishingIdeal s).IsPrime

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theorem PrimeSpectrum.vanishingIdeal_isIrreducible {R : Type u} [CommSemiring R] :

vanishingIdeal '' {s : Set (PrimeSpectrum R) | IsIrreducible s} = {P : Ideal R | P.IsPrime}

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theorem PrimeSpectrum.vanishingIdeal_isClosed_isIrreducible {R : Type u} [CommSemiring R] :

vanishingIdeal '' {s : Set (PrimeSpectrum R) | IsClosed s ∧ IsIrreducible s} = {P : Ideal R | P.IsPrime}

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instance PrimeSpectrum.irreducibleSpace {R : Type u} [CommSemiring R] [IsDomain R] :

IrreducibleSpace (PrimeSpectrum R)

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instance PrimeSpectrum.quasiSober {R : Type u} [CommSemiring R] :

QuasiSober (PrimeSpectrum R)

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instance PrimeSpectrum.compactSpace {R : Type u} [CommSemiring R] :

CompactSpace (PrimeSpectrum R)

The prime spectrum of a commutative (semi)ring is a compact topological space.

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theorem PrimeSpectrum.discreteTopology_iff_finite_and_krullDimLE_zero {R : Type u} [CommSemiring R] :

DiscreteTopology (PrimeSpectrum R) ↔ Finite (PrimeSpectrum R) ∧ Ring.KrullDimLE 0 R

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.

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@[deprecated PrimeSpectrum.discreteTopology_iff_finite_and_krullDimLE_zero (since := "2025-02-05")]

theorem PrimeSpectrum.discreteTopology_iff_finite_and_isPrime_imp_isMaximal {R : Type u} [CommSemiring R] :

DiscreteTopology (PrimeSpectrum R) ↔ Finite (PrimeSpectrum R) ∧ ∀ (I : Ideal R), I.IsPrime → I.IsMaximal

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theorem PrimeSpectrum.discreteTopology_iff_finite_isMaximal_and_sInf_le_nilradical {R : Type u} [CommSemiring R] :

DiscreteTopology (PrimeSpectrum R) ↔ Finite ↑{I : Ideal R | I.IsMaximal} ∧ sInf {I : Ideal R | I.IsMaximal} ≤ nilradical R

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.

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theorem PrimeSpectrum.discreteTopology_of_toLocalization_surjective {R : Type u} [CommSemiring R] (surj : Function.Surjective ⇑(toPiLocalization R)) :

DiscreteTopology (PrimeSpectrum R)

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def PrimeSpectrum.comap {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] (f : R →+* S) :

C(PrimeSpectrum S, PrimeSpectrum R)

The continuous function between prime spectra of commutative (semi)rings induced by a ring homomorphism.

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theorem PrimeSpectrum.coe_comap {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] (f : R →+* S) :

⇑(comap f) = f.specComap

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theorem PrimeSpectrum.comap_apply {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] (f : R →+* S) (x : PrimeSpectrum S) :

(comap f) x = f.specComap x

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@[simp]

theorem PrimeSpectrum.comap_asIdeal {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] (f : R →+* S) (y : PrimeSpectrum S) :

((comap f) y).asIdeal = Ideal.comap f y.asIdeal

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@[simp]

theorem PrimeSpectrum.comap_id {R : Type u} [CommSemiring R] :

comap (RingHom.id R) = ContinuousMap.id (PrimeSpectrum R)

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@[simp]

theorem PrimeSpectrum.comap_comp {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] {S' : Type u_1} [CommSemiring S'] (f : R →+* S) (g : S →+* S') :

comap (g.comp f) = (comap f).comp (comap g)

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theorem PrimeSpectrum.comap_comp_apply {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] {S' : Type u_1} [CommSemiring S'] (f : R →+* S) (g : S →+* S') (x : PrimeSpectrum S') :

(comap (g.comp f)) x = (comap f) ((comap g) x)

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@[simp]

theorem PrimeSpectrum.preimage_comap_zeroLocus {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] (f : R →+* S) (s : Set R) :

⇑(comap f) ⁻¹' zeroLocus s = zeroLocus (⇑f '' s)

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theorem PrimeSpectrum.comap_injective_of_surjective {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] (f : R →+* S) (hf : Function.Surjective ⇑f) :

Function.Injective ⇑(comap f)

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theorem PrimeSpectrum.localization_specComap_injective {R : Type u} (S : Type v) [CommSemiring R] [CommSemiring S] [Algebra R S] (M : Submonoid R) [IsLocalization M S] :

Function.Injective (algebraMap R S).specComap

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theorem PrimeSpectrum.localization_specComap_range {R : Type u} (S : Type v) [CommSemiring R] [CommSemiring S] [Algebra R S] (M : Submonoid R) [IsLocalization M S] :

Set.range (algebraMap R S).specComap = {p : PrimeSpectrum R | Disjoint ↑M ↑p.asIdeal}

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theorem PrimeSpectrum.localization_comap_isInducing {R : Type u} (S : Type v) [CommSemiring R] [CommSemiring S] [Algebra R S] (M : Submonoid R) [IsLocalization M S] :

Topology.IsInducing ⇑(comap (algebraMap R S))

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theorem PrimeSpectrum.localization_comap_injective {R : Type u} (S : Type v) [CommSemiring R] [CommSemiring S] [Algebra R S] (M : Submonoid R) [IsLocalization M S] :

Function.Injective ⇑(comap (algebraMap R S))

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theorem PrimeSpectrum.localization_comap_isEmbedding {R : Type u} (S : Type v) [CommSemiring R] [CommSemiring S] [Algebra R S] (M : Submonoid R) [IsLocalization M S] :

Topology.IsEmbedding ⇑(comap (algebraMap R S))

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theorem PrimeSpectrum.localization_comap_range {R : Type u} (S : Type v) [CommSemiring R] [CommSemiring S] [Algebra R S] (M : Submonoid R) [IsLocalization M S] :

Set.range ⇑(comap (algebraMap R S)) = {p : PrimeSpectrum R | Disjoint ↑M ↑p.asIdeal}

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theorem PrimeSpectrum.comap_isInducing_of_surjective {R : Type u} (S : Type v) [CommSemiring R] [CommSemiring S] (f : R →+* S) (hf : Function.Surjective ⇑f) :

Topology.IsInducing ⇑(comap f)

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theorem PrimeSpectrum.isEmbedding_comap_of_surjective {R : Type u} (S : Type v) [CommSemiring R] [CommSemiring S] (f : R →+* S) (hf : Function.Surjective ⇑f) :

Topology.IsEmbedding ⇑(comap f)

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.

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def PrimeSpectrum.homeomorphOfRingEquiv {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] (e : R ≃+* S) :

PrimeSpectrum R ≃ₜ PrimeSpectrum S

Homeomorphism between prime spectra induced by an isomorphism of semirings.

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theorem PrimeSpectrum.isHomeomorph_comap_of_bijective {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] {f : R →+* S} (hf : Function.Bijective ⇑f) :

IsHomeomorph ⇑(comap f)

The comap of a surjective ring homomorphism is a closed embedding between the prime spectra.

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theorem PrimeSpectrum.comap_singleton_isClosed_of_surjective {R : Type u} (S : Type v) [CommRing R] [CommRing S] (f : R →+* S) (hf : Function.Surjective ⇑f) (x : PrimeSpectrum S) (hx : IsClosed {x}) :

IsClosed {(comap f) x}

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theorem PrimeSpectrum.image_comap_zeroLocus_eq_zeroLocus_comap {R : Type u} (S : Type v) [CommRing R] [CommRing S] (f : R →+* S) (hf : Function.Surjective ⇑f) (I : Ideal S) :

⇑(comap f) '' zeroLocus ↑I = zeroLocus ↑(Ideal.comap f I)

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theorem PrimeSpectrum.range_comap_of_surjective {R : Type u} (S : Type v) [CommRing R] [CommRing S] (f : R →+* S) (hf : Function.Surjective ⇑f) :

Set.range ⇑(comap f) = zeroLocus ↑(RingHom.ker f)

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theorem PrimeSpectrum.comap_quotientMk_bijective_of_le_nilradical {R : Type u} [CommRing R] {I : Ideal R} (hle : I ≤ nilradical R) :

Function.Bijective ⇑(comap (Ideal.Quotient.mk I))

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theorem PrimeSpectrum.isClosed_range_comap_of_surjective {R : Type u} (S : Type v) [CommRing R] [CommRing S] (f : R →+* S) (hf : Function.Surjective ⇑f) :

IsClosed (Set.range ⇑(comap f))

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theorem PrimeSpectrum.isClosedEmbedding_comap_of_surjective {R : Type u} (S : Type v) [CommRing R] [CommRing S] (f : R →+* S) (hf : Function.Surjective ⇑f) :

Topology.IsClosedEmbedding ⇑(comap f)

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theorem PrimeSpectrum.primeSpectrumProd_symm_inl {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] (x : PrimeSpectrum R) :

(primeSpectrumProd R S).symm (Sum.inl x) = (comap (RingHom.fst R S)) x

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theorem PrimeSpectrum.primeSpectrumProd_symm_inr {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] (x : PrimeSpectrum S) :

(primeSpectrumProd R S).symm (Sum.inr x) = (comap (RingHom.snd R S)) x

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theorem PrimeSpectrum.range_comap_fst {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] :

Set.range ⇑(comap (RingHom.fst R S)) = zeroLocus ↑(RingHom.ker (RingHom.fst R S))

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theorem PrimeSpectrum.range_comap_snd {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] :

Set.range ⇑(comap (RingHom.snd R S)) = zeroLocus ↑(RingHom.ker (RingHom.snd R S))

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theorem PrimeSpectrum.isClosedEmbedding_comap_fst {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] :

Topology.IsClosedEmbedding ⇑(comap (RingHom.fst R S))

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theorem PrimeSpectrum.isClosedEmbedding_comap_snd {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] :

Topology.IsClosedEmbedding ⇑(comap (RingHom.snd R S))

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noncomputable def PrimeSpectrum.primeSpectrumProdHomeo {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] :

PrimeSpectrum (R × S) ≃ₜ PrimeSpectrum R ⊕ PrimeSpectrum S

The prime spectrum of R × S is homeomorphic to the disjoint union of PrimeSpectrum R and PrimeSpectrum S.

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def PrimeSpectrum.basicOpen {R : Type u} [CommSemiring R] (r : R) :

TopologicalSpace.Opens (PrimeSpectrum R)

basicOpen r is the open subset containing all prime ideals not containing r.

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@[simp]

theorem PrimeSpectrum.mem_basicOpen {R : Type u} [CommSemiring R] (f : R) (x : PrimeSpectrum R) :

x ∈ basicOpen f ↔ f ∉ x.asIdeal

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theorem PrimeSpectrum.isOpen_basicOpen {R : Type u} [CommSemiring R] {a : R} :

IsOpen ↑(basicOpen a)

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@[simp]

theorem PrimeSpectrum.basicOpen_eq_zeroLocus_compl {R : Type u} [CommSemiring R] (r : R) :

↑(basicOpen r) = (zeroLocus {r})ᶜ

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@[simp]

theorem PrimeSpectrum.basicOpen_one {R : Type u} [CommSemiring R] :

basicOpen 1 = ⊤

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@[simp]

theorem PrimeSpectrum.basicOpen_zero {R : Type u} [CommSemiring R] :

basicOpen 0 = ⊥

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theorem PrimeSpectrum.basicOpen_le_basicOpen_iff {R : Type u} [CommSemiring R] (f g : R) :

basicOpen f ≤ basicOpen g ↔ f ∈ (Ideal.span {g}).radical

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theorem PrimeSpectrum.basicOpen_le_basicOpen_iff_algebraMap_isUnit {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] {f g : R} [Algebra R S] [IsLocalization.Away f S] :

basicOpen f ≤ basicOpen g ↔ IsUnit ((algebraMap R S) g)

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theorem PrimeSpectrum.basicOpen_mul {R : Type u} [CommSemiring R] (f g : R) :

basicOpen (f * g) = basicOpen f ⊓ basicOpen g

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theorem PrimeSpectrum.basicOpen_mul_le_left {R : Type u} [CommSemiring R] (f g : R) :

basicOpen (f * g) ≤ basicOpen f

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theorem PrimeSpectrum.basicOpen_mul_le_right {R : Type u} [CommSemiring R] (f g : R) :

basicOpen (f * g) ≤ basicOpen g

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@[simp]

theorem PrimeSpectrum.basicOpen_pow {R : Type u} [CommSemiring R] (f : R) (n : ℕ) (hn : 0 < n) :

basicOpen (f ^ n) = basicOpen f

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theorem PrimeSpectrum.isTopologicalBasis_basic_opens {R : Type u} [CommSemiring R] :

TopologicalSpace.IsTopologicalBasis (Set.range fun (r : R) => ↑(basicOpen r))

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theorem PrimeSpectrum.eq_biUnion_of_isOpen {R : Type u} [CommSemiring R] {s : Set (PrimeSpectrum R)} (hs : IsOpen s) :

s = ⋃ (r : R), ⋃ (_ : ↑(basicOpen r) ⊆ s), ↑(basicOpen r)

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theorem PrimeSpectrum.isBasis_basic_opens {R : Type u} [CommSemiring R] :

TopologicalSpace.Opens.IsBasis (Set.range basicOpen)

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@[simp]

theorem PrimeSpectrum.basicOpen_eq_bot_iff {R : Type u} [CommSemiring R] (f : R) :

basicOpen f = ⊥ ↔ IsNilpotent f

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theorem PrimeSpectrum.localization_away_comap_range {R : Type u} [CommSemiring R] (S : Type v) [CommSemiring S] [Algebra R S] (r : R) [IsLocalization.Away r S] :

Set.range ⇑(comap (algebraMap R S)) = ↑(basicOpen r)

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theorem PrimeSpectrum.localization_away_isOpenEmbedding {R : Type u} [CommSemiring R] (S : Type v) [CommSemiring S] [Algebra R S] (r : R) [IsLocalization.Away r S] :

Topology.IsOpenEmbedding ⇑(comap (algebraMap R S))

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theorem PrimeSpectrum.isCompact_basicOpen {R : Type u} [CommSemiring R] (f : R) :

IsCompact ↑(basicOpen f)

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theorem PrimeSpectrum.comap_basicOpen {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] (f : R →+* S) (x : R) :

(TopologicalSpace.Opens.comap (comap f)) (basicOpen x) = basicOpen (f x)

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theorem PrimeSpectrum.iSup_basicOpen_eq_top_iff {R : Type u} [CommSemiring R] {ι : Type u_1} {f : ι → R} :

⨆ (i : ι), basicOpen (f i) = ⊤ ↔ Ideal.span (Set.range f) = ⊤

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theorem PrimeSpectrum.iSup_basicOpen_eq_top_iff' {R : Type u} [CommSemiring R] {s : Set R} :

⨆ i ∈ s, basicOpen i = ⊤ ↔ Ideal.span s = ⊤

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theorem PrimeSpectrum.isLocalization_away_iff_atPrime_of_basicOpen_eq_singleton {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] [Algebra R S] {f : R} {p : PrimeSpectrum R} (h : (basicOpen f).carrier = {p}) :

IsLocalization.Away f S ↔ IsLocalization.AtPrime S p.asIdeal

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theorem PrimeSpectrum.range_comap_algebraMap_localization_compl_eq_range_comap_quotientMk {R : Type u_1} [CommRing R] (c : R) :

(Set.range ⇑(comap (algebraMap (Polynomial R) (Polynomial (Localization.Away c)))))ᶜ = Set.range ⇑(comap (Polynomial.mapRingHom (Ideal.Quotient.mk (Ideal.span {c}))))

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instance PrimeSpectrum.instQuasiSeparatedSpace {R : Type u} [CommSemiring R] :

QuasiSeparatedSpace (PrimeSpectrum R)

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theorem PrimeSpectrum.toPiLocalization_surjective_of_discreteTopology (R : Type u) [CommSemiring R] [DiscreteTopology (PrimeSpectrum R)] :

Function.Surjective ⇑(toPiLocalization R)

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theorem PrimeSpectrum.maximalSpectrumToPiLocalization_surjective_of_discreteTopology (R : Type u) [CommSemiring R] [DiscreteTopology (PrimeSpectrum R)] :

Function.Surjective ⇑(MaximalSpectrum.toPiLocalization R)

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def MaximalSpectrum.toPiLocalizationEquiv (R : Type u) [CommSemiring R] [DiscreteTopology (PrimeSpectrum R)] :

R ≃+* PiLocalization R

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.

Stacks Tag 00JA (See also PrimeSpectrum.discreteTopology_iff_finite_isMaximal_and_sInf_le_nilradical.)

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@[deprecated MaximalSpectrum.toPiLocalizationEquiv (since := "2025-02-12")]

def PrimeSpectrum.MaximalSpectrum.toPiLocalizationEquivtoLocalizationEquiv (R : Type u) [CommSemiring R] [DiscreteTopology (PrimeSpectrum R)] :

R ≃+* MaximalSpectrum.PiLocalization R

Alias of 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.

Stacks Tag 00JA (See also PrimeSpectrum.discreteTopology_iff_finite_isMaximal_and_sInf_le_nilradical.)

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theorem PrimeSpectrum.discreteTopology_iff_toPiLocalization_surjective {R : Type u_1} [CommSemiring R] :

DiscreteTopology (PrimeSpectrum R) ↔ Function.Surjective ⇑(toPiLocalization R)

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theorem PrimeSpectrum.discreteTopology_iff_toPiLocalization_bijective {R : Type u_1} [CommSemiring R] :

DiscreteTopology (PrimeSpectrum R) ↔ Function.Bijective ⇑(toPiLocalization R)

The specialization order #

We endow PrimeSpectrum R with a partial order, where x ≤ y if and only if y ∈ closure {x}.

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theorem PrimeSpectrum.le_iff_mem_closure {R : Type u} [CommSemiring R] (x y : PrimeSpectrum R) :

x ≤ y ↔ y ∈ closure {x}

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theorem PrimeSpectrum.le_iff_specializes {R : Type u} [CommSemiring R] (x y : PrimeSpectrum R) :

x ≤ y ↔ x ⤳ y

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def PrimeSpectrum.nhdsOrderEmbedding {R : Type u} [CommSemiring R] :

PrimeSpectrum R ↪o Filter (PrimeSpectrum R)

nhds as an order embedding.

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@[simp]

theorem PrimeSpectrum.nhdsOrderEmbedding_apply {R : Type u} [CommSemiring R] (x : PrimeSpectrum R) :

nhdsOrderEmbedding x = nhds x

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instance PrimeSpectrum.instT0Space {R : Type u} [CommSemiring R] :

T0Space (PrimeSpectrum R)

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instance PrimeSpectrum.instPrespectralSpace {R : Type u} [CommSemiring R] :

PrespectralSpace (PrimeSpectrum R)

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instance PrimeSpectrum.instSpectralSpace {R : Type u} [CommSemiring R] :

SpectralSpace (PrimeSpectrum R)

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def PrimeSpectrum.localizationMapOfSpecializes {R : Type u} [CommSemiring R] {x y : PrimeSpectrum R} (h : x ⤳ y) :

Localization.AtPrime y.asIdeal →+* Localization.AtPrime x.asIdeal

If x specializes to y, then there is a natural map from the localization of y to the localization of x.

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theorem PrimeSpectrum.isClosed_image_of_stableUnderSpecialization {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (f : R →+* S) (Z : Set (PrimeSpectrum S)) (hZ : IsClosed Z) (hf : StableUnderSpecialization (⇑(comap f) '' Z)) :

IsClosed (⇑(comap f) '' Z)

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theorem PrimeSpectrum.isClosed_range_of_stableUnderSpecialization {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (f : R →+* S) (hf : StableUnderSpecialization (Set.range ⇑(comap f))) :

IsClosed (Set.range ⇑(comap f))

Stacks Tag 00HY

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theorem PrimeSpectrum.stableUnderSpecialization_range_iff {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] {f : R →+* S} :

StableUnderSpecialization (Set.range ⇑(comap f)) ↔ IsClosed (Set.range ⇑(comap f))

Stacks Tag 00HY

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theorem PrimeSpectrum.stableUnderSpecialization_image_iff {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (f : R →+* S) (Z : Set (PrimeSpectrum S)) (hZ : IsClosed Z) :

StableUnderSpecialization (⇑(comap f) '' Z) ↔ IsClosed (⇑(comap f) '' Z)

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theorem PrimeSpectrum.isQuotientMap_of_specializingMap {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] {f : R →+* S} (h₁ : Function.Surjective ⇑(comap f)) (h₂ : SpecializingMap ⇑(comap f)) :

Topology.IsQuotientMap ⇑(comap f)

If f : Spec S → Spec R is specializing and surjective, the topology on Spec R is the quotient topology induced by f.

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theorem PrimeSpectrum.isQuotientMap_of_generalizingMap {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] {f : R →+* S} (h₁ : Function.Surjective ⇑(comap f)) (h₂ : GeneralizingMap ⇑(comap f)) :

Topology.IsQuotientMap ⇑(comap f)

If f : Spec S → Spec R is generalizing and surjective, the topology on Spec R is the quotient topology induced by f.

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theorem PrimeSpectrum.vanishingIdeal_range_comap {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (f : R →+* S) :

vanishingIdeal (Set.range ⇑(comap f)) = (RingHom.ker f).radical

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theorem PrimeSpectrum.closure_range_comap {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (f : R →+* S) :

closure (Set.range ⇑(comap f)) = zeroLocus ↑(RingHom.ker f)

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theorem PrimeSpectrum.denseRange_comap_iff_ker_le_nilRadical {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (f : R →+* S) :

DenseRange ⇑(comap f) ↔ RingHom.ker f ≤ nilradical R

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theorem PrimeSpectrum.denseRange_comap_iff_minimalPrimes {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (f : R →+* S) :

DenseRange ⇑(comap f) ↔ ∀ (I : Ideal R) (h : I ∈ minimalPrimes R), { asIdeal := I, isPrime := ⋯ } ∈ Set.range ⇑(comap f)

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def PrimeSpectrum.pointsEquivIrreducibleCloseds (R : Type u) [CommSemiring R] :

PrimeSpectrum R ≃o (TopologicalSpace.IrreducibleCloseds (PrimeSpectrum R))ᵒᵈ

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.

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theorem PrimeSpectrum.stableUnderSpecialization_singleton {R : Type u} [CommSemiring R] {x : PrimeSpectrum R} :

StableUnderSpecialization {x} ↔ x.asIdeal.IsMaximal

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theorem PrimeSpectrum.stableUnderGeneralization_singleton {R : Type u} [CommSemiring R] {x : PrimeSpectrum R} :

StableUnderGeneralization {x} ↔ x.asIdeal ∈ minimalPrimes R

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theorem PrimeSpectrum.isCompact_isOpen_iff {R : Type u} [CommSemiring R] {s : Set (PrimeSpectrum R)} :

IsCompact s ∧ IsOpen s ↔ ∃ (t : Finset R), (zeroLocus ↑t)ᶜ = s

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theorem PrimeSpectrum.isCompact_isOpen_iff_ideal {R : Type u} [CommSemiring R] {s : Set (PrimeSpectrum R)} :

IsCompact s ∧ IsOpen s ↔ ∃ (I : Ideal R), I.FG ∧ (zeroLocus ↑I)ᶜ = s

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theorem PrimeSpectrum.basicOpen_eq_zeroLocus_of_mul_add {R : Type u} [CommSemiring R] (e f : R) (mul : e * f = 0) (add : e + f = 1) :

↑(basicOpen e) = zeroLocus {f}

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theorem PrimeSpectrum.zeroLocus_eq_basicOpen_of_mul_add {R : Type u} [CommSemiring R] (e f : R) (mul : e * f = 0) (add : e + f = 1) :

zeroLocus {e} = ↑(basicOpen f)

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theorem PrimeSpectrum.isClopen_basicOpen_of_mul_add {R : Type u} [CommSemiring R] (e f : R) (mul : e * f = 0) (add : e + f = 1) :

IsClopen ↑(basicOpen e)

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theorem PrimeSpectrum.basicOpen_injOn_isIdempotentElem {R : Type u} [CommSemiring R] :

Set.InjOn basicOpen {e : R | IsIdempotentElem e}

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theorem PrimeSpectrum.exists_mul_eq_zero_add_eq_one_basicOpen_eq_of_isClopen {R : Type u} [CommSemiring R] {s : Set (PrimeSpectrum R)} (hs : IsClopen s) :

∃ (e : R) (f : R), e * f = 0 ∧ e + f = 1 ∧ s = ↑(basicOpen e) ∧ sᶜ = ↑(basicOpen f)

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theorem PrimeSpectrum.exists_idempotent_basicOpen_eq_of_isClopen {R : Type u} [CommSemiring R] {s : Set (PrimeSpectrum R)} (hs : IsClopen s) :

∃ (e : R), IsIdempotentElem e ∧ s = ↑(basicOpen e)

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theorem PrimeSpectrum.existsUnique_idempotent_basicOpen_eq_of_isClopen {R : Type u} [CommSemiring R] {s : Set (PrimeSpectrum R)} (hs : IsClopen s) :

∃! e : R, IsIdempotentElem e ∧ s = ↑(basicOpen e)

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theorem PrimeSpectrum.isClopen_iff_mul_add {R : Type u} [CommSemiring R] {s : Set (PrimeSpectrum R)} :

IsClopen s ↔ ∃ (e : R) (f : R), e * f = 0 ∧ e + f = 1 ∧ s = ↑(basicOpen e)

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theorem PrimeSpectrum.isClopen_iff_mul_add_zeroLocus {R : Type u} [CommSemiring R] {s : Set (PrimeSpectrum R)} :

IsClopen s ↔ ∃ (e : R) (f : R), e * f = 0 ∧ e + f = 1 ∧ s = zeroLocus {e}

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def PrimeSpectrum.mulZeroAddOneEquivClopens {R : Type u} [CommSemiring R] :

{ e : R × R // e.1 * e.2 = 0 ∧ e.1 + e.2 = 1 } ≃o TopologicalSpace.Clopens (PrimeSpectrum R)

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.

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theorem PrimeSpectrum.isRetrocompact_zeroLocus_compl {R : Type u} [CommSemiring R] {s : Set R} (hs : s.Finite) :

IsRetrocompact (zeroLocus s)ᶜ

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theorem PrimeSpectrum.isRetrocompact_zeroLocus_compl_of_fg {R : Type u} [CommSemiring R] {I : Ideal R} (hI : I.FG) :

IsRetrocompact (zeroLocus ↑I)ᶜ

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theorem PrimeSpectrum.isRetrocompact_basicOpen {R : Type u} [CommSemiring R] {f : R} :

IsRetrocompact ↑(basicOpen f)

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theorem PrimeSpectrum.isConstructible_basicOpen {R : Type u} [CommSemiring R] {f : R} :

Topology.IsConstructible ↑(basicOpen f)

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theorem PrimeSpectrum.isClosedMap_comap_of_isIntegral {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (f : R →+* S) (hf : f.IsIntegral) :

IsClosedMap ⇑(comap f)

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theorem PrimeSpectrum.isClosed_comap_singleton_of_isIntegral {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (f : R →+* S) (hf : f.IsIntegral) (x : PrimeSpectrum S) (hx : IsClosed {x}) :

IsClosed {(comap f) x}

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theorem PrimeSpectrum.closure_image_comap_zeroLocus {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (f : R →+* S) (I : Ideal S) :

closure (⇑(comap f) '' zeroLocus ↑I) = zeroLocus ↑(Ideal.comap f I)

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theorem PrimeSpectrum.isIntegral_of_isClosedMap_comap_mapRingHom {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (f : R →+* S) (h : IsClosedMap ⇑(comap (Polynomial.mapRingHom f))) :

f.IsIntegral

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theorem RingHom.IsIntegral.specComap_surjective {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] {f : R →+* S} (hf : f.IsIntegral) (hinj : Function.Injective ⇑f) :

Function.Surjective f.specComap

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def minimalPrimes.equivIrreducibleComponents (R : Type u) [CommSemiring R] :

↑(minimalPrimes R) ≃o (↑(irreducibleComponents (PrimeSpectrum R)))ᵒᵈ

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.

Stacks Tag 00ES

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