Prime spectrum of a commutative (semi)ring

The prime spectrum of a commutative (semi)ring is the type of all prime ideals. It is naturally endowed with a topology: the Zariski topology.

(It is also naturally endowed with a sheaf of rings, which is constructed in AlgebraicGeometry.StructureSheaf.)

Main definitions

• PrimeSpectrum R: The prime spectrum of a commutative (semi)ring R, i.e., the set of all prime ideals of R.
• zeroLocus s: The zero locus of a subset s of R is the subset of PrimeSpectrum R consisting of all prime ideals that contain s.
• vanishingIdeal t: The vanishing ideal of a subset t of PrimeSpectrum R is the intersection of points in t (viewed as prime ideals).

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).

theorem PrimeSpectrum.ext {R : Type u} : ∀ {inst : CommSemiring R} (x y : PrimeSpectrum R), x.asIdeal = y.asIdeal → x = y

theorem PrimeSpectrum.ext_iff {R : Type u} : ∀ {inst : CommSemiring R} (x y : PrimeSpectrum R), x = y ↔ x.asIdeal = y.asIdeal

structure PrimeSpectrum (R : Type u) [CommSemiring R] : Type u

The prime spectrum of a commutative (semi)ring R is the type of all prime ideals of R.

It is naturally endowed with a topology (the Zariski topology), and a sheaf of commutative rings (see AlgebraicGeometry.StructureSheaf). It is a fundamental building block in algebraic geometry.

• asIdeal : Ideal R
• isPrime : self.asIdeal.IsPrime

theorem PrimeSpectrum.isPrime {R : Type u} [CommSemiring R] (self : PrimeSpectrum R) : self.asIdeal.IsPrime

@[deprecated PrimeSpectrum.isPrime]

theorem PrimeSpectrum.IsPrime {R : Type u} [CommSemiring R] (self : PrimeSpectrum R) : self.asIdeal.IsPrime

Alias of PrimeSpectrum.isPrime.

instance PrimeSpectrum.instNonemptyOfNontrivial {R : Type u} [CommSemiring R] [Nontrivial R] : Nonempty (PrimeSpectrum R)

Equations

• ⋯ = ⋯

instance PrimeSpectrum.instIsEmptyOfSubsingleton {R : Type u} [CommSemiring R] [Subsingleton R] : IsEmpty (PrimeSpectrum R)

The prime spectrum of the zero ring is empty.

Equations

• ⋯ = ⋯

def PrimeSpectrum.primeSpectrumProdOfSum (R : Type u) (S : Type v) [CommSemiring R] [CommSemiring S] : PrimeSpectrum R ⊕ PrimeSpectrum S → PrimeSpectrum (R × S)

The map from the direct sum of prime spectra to the prime spectrum of a direct product.

Equations

• One or more equations did not get rendered due to their size.

noncomputable def PrimeSpectrum.primeSpectrumProd (R : Type u) (S : Type v) [CommSemiring R] [CommSemiring S] : PrimeSpectrum (R × S) ≃ PrimeSpectrum R ⊕ PrimeSpectrum S

The prime spectrum of R × S is in bijection with the disjoint unions of the prime spectrum of R and the prime spectrum of S.

Equations

• PrimeSpectrum.primeSpectrumProd R S = (Equiv.ofBijective (PrimeSpectrum.primeSpectrumProdOfSum R S) ⋯).symm

@[simp]

theorem PrimeSpectrum.primeSpectrumProd_symm_inl_asIdeal {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] (x : PrimeSpectrum R) : ((PrimeSpectrum.primeSpectrumProd R S).symm (Sum.inl x)).asIdeal = x.asIdeal.prod ⊤

@[simp]

theorem PrimeSpectrum.primeSpectrumProd_symm_inr_asIdeal {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] (x : PrimeSpectrum S) : ((PrimeSpectrum.primeSpectrumProd R S).symm (Sum.inr x)).asIdeal = ⊤.prod x.asIdeal

def PrimeSpectrum.zeroLocus {R : Type u} [CommSemiring R] (s : Set R) : Set (PrimeSpectrum R)

The zero locus of a set s of elements of a commutative (semi)ring R is the set of all prime ideals of the ring that contain the set s.

An element f of R can be thought of as a dependent function on the prime spectrum of R. At a point x (a prime ideal) the function (i.e., element) f takes values in the quotient ring R modulo the prime ideal x. In this manner, zeroLocus s is exactly the subset of PrimeSpectrum R where all "functions" in s vanish simultaneously.

Equations

• PrimeSpectrum.zeroLocus s = {x : PrimeSpectrum R | s ⊆ ↑x.asIdeal}

@[simp]

theorem PrimeSpectrum.mem_zeroLocus {R : Type u} [CommSemiring R] (x : PrimeSpectrum R) (s : Set R) : x ∈ PrimeSpectrum.zeroLocus s ↔ s ⊆ ↑x.asIdeal

@[simp]

theorem PrimeSpectrum.zeroLocus_span {R : Type u} [CommSemiring R] (s : Set R) : PrimeSpectrum.zeroLocus ↑(Ideal.span s) = PrimeSpectrum.zeroLocus s

def PrimeSpectrum.vanishingIdeal {R : Type u} [CommSemiring R] (t : Set (PrimeSpectrum R)) : Ideal R

The vanishing ideal of a set t of points of the prime spectrum of a commutative ring R is the intersection of all the prime ideals in the set t.

An element f of R can be thought of as a dependent function on the prime spectrum of R. At a point x (a prime ideal) the function (i.e., element) f takes values in the quotient ring R modulo the prime ideal x. In this manner, vanishingIdeal t is exactly the ideal of R consisting of all "functions" that vanish on all of t.

Equations

• PrimeSpectrum.vanishingIdeal t = ⨅ x ∈ t, x.asIdeal

theorem PrimeSpectrum.coe_vanishingIdeal {R : Type u} [CommSemiring R] (t : Set (PrimeSpectrum R)) : ↑(PrimeSpectrum.vanishingIdeal t) = {f : R | ∀ x ∈ t, f ∈ x.asIdeal}

theorem PrimeSpectrum.mem_vanishingIdeal {R : Type u} [CommSemiring R] (t : Set (PrimeSpectrum R)) (f : R) : f ∈ PrimeSpectrum.vanishingIdeal t ↔ ∀ x ∈ t, f ∈ x.asIdeal

@[simp]

theorem PrimeSpectrum.vanishingIdeal_singleton {R : Type u} [CommSemiring R] (x : PrimeSpectrum R) : PrimeSpectrum.vanishingIdeal {x} = x.asIdeal

theorem PrimeSpectrum.subset_zeroLocus_iff_le_vanishingIdeal {R : Type u} [CommSemiring R] (t : Set (PrimeSpectrum R)) (I : Ideal R) : t ⊆ PrimeSpectrum.zeroLocus ↑I ↔ I ≤ PrimeSpectrum.vanishingIdeal t

theorem PrimeSpectrum.gc (R : Type u) [CommSemiring R] : GaloisConnection (fun (I : Ideal R) => PrimeSpectrum.zeroLocus ↑I) fun (t : (Set (PrimeSpectrum R))ᵒᵈ) => PrimeSpectrum.vanishingIdeal t

zeroLocus and vanishingIdeal form a galois connection.

theorem PrimeSpectrum.gc_set (R : Type u) [CommSemiring R] : GaloisConnection (fun (s : Set R) => PrimeSpectrum.zeroLocus s) fun (t : (Set (PrimeSpectrum R))ᵒᵈ) => ↑(PrimeSpectrum.vanishingIdeal t)

zeroLocus and vanishingIdeal form a galois connection.

theorem PrimeSpectrum.subset_zeroLocus_iff_subset_vanishingIdeal (R : Type u) [CommSemiring R] (t : Set (PrimeSpectrum R)) (s : Set R) : t ⊆ PrimeSpectrum.zeroLocus s ↔ s ⊆ ↑(PrimeSpectrum.vanishingIdeal t)

theorem PrimeSpectrum.subset_vanishingIdeal_zeroLocus {R : Type u} [CommSemiring R] (s : Set R) : s ⊆ ↑(PrimeSpectrum.vanishingIdeal (PrimeSpectrum.zeroLocus s))

theorem PrimeSpectrum.le_vanishingIdeal_zeroLocus {R : Type u} [CommSemiring R] (I : Ideal R) : I ≤ PrimeSpectrum.vanishingIdeal (PrimeSpectrum.zeroLocus ↑I)

@[simp]

theorem PrimeSpectrum.vanishingIdeal_zeroLocus_eq_radical {R : Type u} [CommSemiring R] (I : Ideal R) : PrimeSpectrum.vanishingIdeal (PrimeSpectrum.zeroLocus ↑I) = I.radical

@[simp]

theorem PrimeSpectrum.zeroLocus_radical {R : Type u} [CommSemiring R] (I : Ideal R) : PrimeSpectrum.zeroLocus ↑I.radical = PrimeSpectrum.zeroLocus ↑I

theorem PrimeSpectrum.subset_zeroLocus_vanishingIdeal {R : Type u} [CommSemiring R] (t : Set (PrimeSpectrum R)) : t ⊆ PrimeSpectrum.zeroLocus ↑(PrimeSpectrum.vanishingIdeal t)

theorem PrimeSpectrum.zeroLocus_anti_mono {R : Type u} [CommSemiring R] {s : Set R} {t : Set R} (h : s ⊆ t) : PrimeSpectrum.zeroLocus t ⊆ PrimeSpectrum.zeroLocus s

theorem PrimeSpectrum.zeroLocus_anti_mono_ideal {R : Type u} [CommSemiring R] {s : Ideal R} {t : Ideal R} (h : s ≤ t) : PrimeSpectrum.zeroLocus ↑t ⊆ PrimeSpectrum.zeroLocus ↑s

theorem PrimeSpectrum.vanishingIdeal_anti_mono {R : Type u} [CommSemiring R] {s : Set (PrimeSpectrum R)} {t : Set (PrimeSpectrum R)} (h : s ⊆ t) : PrimeSpectrum.vanishingIdeal t ≤ PrimeSpectrum.vanishingIdeal s

theorem PrimeSpectrum.zeroLocus_subset_zeroLocus_iff {R : Type u} [CommSemiring R] (I : Ideal R) (J : Ideal R) : PrimeSpectrum.zeroLocus ↑I ⊆ PrimeSpectrum.zeroLocus ↑J ↔ J ≤ I.radical

theorem PrimeSpectrum.zeroLocus_subset_zeroLocus_singleton_iff {R : Type u} [CommSemiring R] (f : R) (g : R) : PrimeSpectrum.zeroLocus {f} ⊆ PrimeSpectrum.zeroLocus {g} ↔ g ∈ (Ideal.span {f}).radical

theorem PrimeSpectrum.zeroLocus_bot {R : Type u} [CommSemiring R] : PrimeSpectrum.zeroLocus ↑⊥ = Set.univ

@[simp]

theorem PrimeSpectrum.zeroLocus_singleton_zero {R : Type u} [CommSemiring R] : PrimeSpectrum.zeroLocus {0} = Set.univ

@[simp]

theorem PrimeSpectrum.zeroLocus_empty {R : Type u} [CommSemiring R] : PrimeSpectrum.zeroLocus ∅ = Set.univ

@[simp]

theorem PrimeSpectrum.vanishingIdeal_empty {R : Type u} [CommSemiring R] : PrimeSpectrum.vanishingIdeal ∅ = ⊤

theorem PrimeSpectrum.zeroLocus_empty_of_one_mem {R : Type u} [CommSemiring R] {s : Set R} (h : 1 ∈ s) : PrimeSpectrum.zeroLocus s = ∅

@[simp]

theorem PrimeSpectrum.zeroLocus_singleton_one {R : Type u} [CommSemiring R] : PrimeSpectrum.zeroLocus {1} = ∅

theorem PrimeSpectrum.zeroLocus_empty_iff_eq_top {R : Type u} [CommSemiring R] {I : Ideal R} : PrimeSpectrum.zeroLocus ↑I = ∅ ↔ I = ⊤

@[simp]

theorem PrimeSpectrum.zeroLocus_univ {R : Type u} [CommSemiring R] : PrimeSpectrum.zeroLocus Set.univ = ∅

theorem PrimeSpectrum.vanishingIdeal_eq_top_iff {R : Type u} [CommSemiring R] {s : Set (PrimeSpectrum R)} : PrimeSpectrum.vanishingIdeal s = ⊤ ↔ s = ∅

theorem PrimeSpectrum.zeroLocus_sup {R : Type u} [CommSemiring R] (I : Ideal R) (J : Ideal R) : PrimeSpectrum.zeroLocus ↑(I ⊔ J) = PrimeSpectrum.zeroLocus ↑I ∩ PrimeSpectrum.zeroLocus ↑J

theorem PrimeSpectrum.zeroLocus_union {R : Type u} [CommSemiring R] (s : Set R) (s' : Set R) : PrimeSpectrum.zeroLocus (s ∪ s') = PrimeSpectrum.zeroLocus s ∩ PrimeSpectrum.zeroLocus s'

theorem PrimeSpectrum.vanishingIdeal_union {R : Type u} [CommSemiring R] (t : Set (PrimeSpectrum R)) (t' : Set (PrimeSpectrum R)) : PrimeSpectrum.vanishingIdeal (t ∪ t') = PrimeSpectrum.vanishingIdeal t ⊓ PrimeSpectrum.vanishingIdeal t'

theorem PrimeSpectrum.zeroLocus_iSup {R : Type u} [CommSemiring R] {ι : Sort u_1} (I : ι → Ideal R) : PrimeSpectrum.zeroLocus ↑(⨆ (i : ι), I i) = ⋂ (i : ι), PrimeSpectrum.zeroLocus ↑(I i)

theorem PrimeSpectrum.zeroLocus_iUnion {R : Type u} [CommSemiring R] {ι : Sort u_1} (s : ι → Set R) : PrimeSpectrum.zeroLocus (⋃ (i : ι), s i) = ⋂ (i : ι), PrimeSpectrum.zeroLocus (s i)

theorem PrimeSpectrum.zeroLocus_bUnion {R : Type u} [CommSemiring R] (s : Set (Set R)) : PrimeSpectrum.zeroLocus (⋃ s' ∈ s, s') = ⋂ s' ∈ s, PrimeSpectrum.zeroLocus s'

theorem PrimeSpectrum.vanishingIdeal_iUnion {R : Type u} [CommSemiring R] {ι : Sort u_1} (t : ι → Set (PrimeSpectrum R)) : PrimeSpectrum.vanishingIdeal (⋃ (i : ι), t i) = ⨅ (i : ι), PrimeSpectrum.vanishingIdeal (t i)

theorem PrimeSpectrum.zeroLocus_inf {R : Type u} [CommSemiring R] (I : Ideal R) (J : Ideal R) : PrimeSpectrum.zeroLocus ↑(I ⊓ J) = PrimeSpectrum.zeroLocus ↑I ∪ PrimeSpectrum.zeroLocus ↑J

theorem PrimeSpectrum.union_zeroLocus {R : Type u} [CommSemiring R] (s : Set R) (s' : Set R) : PrimeSpectrum.zeroLocus s ∪ PrimeSpectrum.zeroLocus s' = PrimeSpectrum.zeroLocus ↑(Ideal.span s ⊓ Ideal.span s')

theorem PrimeSpectrum.zeroLocus_mul {R : Type u} [CommSemiring R] (I : Ideal R) (J : Ideal R) : PrimeSpectrum.zeroLocus ↑(I * J) = PrimeSpectrum.zeroLocus ↑I ∪ PrimeSpectrum.zeroLocus ↑J

theorem PrimeSpectrum.zeroLocus_singleton_mul {R : Type u} [CommSemiring R] (f : R) (g : R) : PrimeSpectrum.zeroLocus {f * g} = PrimeSpectrum.zeroLocus {f} ∪ PrimeSpectrum.zeroLocus {g}

@[simp]

theorem PrimeSpectrum.zeroLocus_pow {R : Type u} [CommSemiring R] (I : Ideal R) {n : ℕ} (hn : n ≠ 0) : PrimeSpectrum.zeroLocus ↑(I ^ n) = PrimeSpectrum.zeroLocus ↑I

@[simp]

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

theorem PrimeSpectrum.sup_vanishingIdeal_le {R : Type u} [CommSemiring R] (t : Set (PrimeSpectrum R)) (t' : Set (PrimeSpectrum R)) : PrimeSpectrum.vanishingIdeal t ⊔ PrimeSpectrum.vanishingIdeal t' ≤ PrimeSpectrum.vanishingIdeal (t ∩ t')

theorem PrimeSpectrum.mem_compl_zeroLocus_iff_not_mem {R : Type u} [CommSemiring R] {f : R} {I : PrimeSpectrum R} : I ∈ (PrimeSpectrum.zeroLocus {f})ᶜ ↔ f ∉ I.asIdeal

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.

Equations

• PrimeSpectrum.zariskiTopology = TopologicalSpace.ofClosed (Set.range PrimeSpectrum.zeroLocus) ⋯ ⋯ ⋯

theorem PrimeSpectrum.isOpen_iff {R : Type u} [CommSemiring R] (U : Set (PrimeSpectrum R)) : IsOpen U ↔ ∃ (s : Set R), Uᶜ = PrimeSpectrum.zeroLocus s

theorem PrimeSpectrum.isClosed_iff_zeroLocus {R : Type u} [CommSemiring R] (Z : Set (PrimeSpectrum R)) : IsClosed Z ↔ ∃ (s : Set R), Z = PrimeSpectrum.zeroLocus s

theorem PrimeSpectrum.isClosed_iff_zeroLocus_ideal {R : Type u} [CommSemiring R] (Z : Set (PrimeSpectrum R)) : IsClosed Z ↔ ∃ (I : Ideal R), Z = PrimeSpectrum.zeroLocus ↑I

theorem PrimeSpectrum.isClosed_iff_zeroLocus_radical_ideal {R : Type u} [CommSemiring R] (Z : Set (PrimeSpectrum R)) : IsClosed Z ↔ ∃ (I : Ideal R), I.IsRadical ∧ Z = PrimeSpectrum.zeroLocus ↑I

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

theorem PrimeSpectrum.zeroLocus_vanishingIdeal_eq_closure {R : Type u} [CommSemiring R] (t : Set (PrimeSpectrum R)) : PrimeSpectrum.zeroLocus ↑(PrimeSpectrum.vanishingIdeal t) = closure t

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

theorem PrimeSpectrum.closure_singleton {R : Type u} [CommSemiring R] (x : PrimeSpectrum R) : closure {x} = PrimeSpectrum.zeroLocus ↑x.asIdeal

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

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

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

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

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.

Equations

• PrimeSpectrum.closedsEmbedding R = OrderEmbedding.ofMapLEIff (fun (s : (TopologicalSpace.Closeds (PrimeSpectrum R))ᵒᵈ) => PrimeSpectrum.vanishingIdeal ↑(OrderDual.ofDual s)) ⋯

theorem PrimeSpectrum.t1Space_iff_isField {R : Type u} [CommSemiring R] [IsDomain R] : T1Space (PrimeSpectrum R) ↔ IsField R

theorem PrimeSpectrum.isIrreducible_zeroLocus_iff_of_radical {R : Type u} [CommSemiring R] (I : Ideal R) (hI : I.IsRadical) : IsIrreducible (PrimeSpectrum.zeroLocus ↑I) ↔ I.IsPrime

theorem PrimeSpectrum.isIrreducible_zeroLocus_iff {R : Type u} [CommSemiring R] (I : Ideal R) : IsIrreducible (PrimeSpectrum.zeroLocus ↑I) ↔ I.radical.IsPrime

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

theorem PrimeSpectrum.vanishingIdeal_isIrreducible {R : Type u} [CommSemiring R] : PrimeSpectrum.vanishingIdeal '' {s : Set (PrimeSpectrum R) | IsIrreducible s} = {P : Ideal R | P.IsPrime}

theorem PrimeSpectrum.vanishingIdeal_isClosed_isIrreducible {R : Type u} [CommSemiring R] : PrimeSpectrum.vanishingIdeal '' {s : Set (PrimeSpectrum R) | IsClosed s ∧ IsIrreducible s} = {P : Ideal R | P.IsPrime}

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

Equations

• ⋯ = ⋯

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

Equations

• ⋯ = ⋯

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

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

Equations

• ⋯ = ⋯

theorem PrimeSpectrum.preimage_comap_zeroLocus_aux {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] (f : R →+* S) (s : Set R) : (fun (y : PrimeSpectrum S) => { asIdeal := Ideal.comap f y.asIdeal, isPrime := ⋯ }) ⁻¹' PrimeSpectrum.zeroLocus s = PrimeSpectrum.zeroLocus (⇑f '' s)

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

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

Equations

• PrimeSpectrum.comap f = { toFun := fun (y : PrimeSpectrum S) => { asIdeal := Ideal.comap f y.asIdeal, isPrime := ⋯ }, continuous_toFun := ⋯ }

@[simp]

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

@[simp]

theorem PrimeSpectrum.comap_id {R : Type u} [CommSemiring R] : PrimeSpectrum.comap (RingHom.id R) = ContinuousMap.id (PrimeSpectrum R)

@[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') : PrimeSpectrum.comap (g.comp f) = (PrimeSpectrum.comap f).comp (PrimeSpectrum.comap g)

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') : (PrimeSpectrum.comap (g.comp f)) x = (PrimeSpectrum.comap f) ((PrimeSpectrum.comap g) x)

@[simp]

theorem PrimeSpectrum.preimage_comap_zeroLocus {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] (f : R →+* S) (s : Set R) : ⇑(PrimeSpectrum.comap f) ⁻¹' PrimeSpectrum.zeroLocus s = PrimeSpectrum.zeroLocus (⇑f '' s)

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 ⇑(PrimeSpectrum.comap f)

theorem PrimeSpectrum.localization_comap_inducing {R : Type u} (S : Type v) [CommSemiring R] [CommSemiring S] [Algebra R S] (M : Submonoid R) [IsLocalization M S] : Inducing ⇑(PrimeSpectrum.comap (algebraMap R S))

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 ⇑(PrimeSpectrum.comap (algebraMap R S))

theorem PrimeSpectrum.localization_comap_embedding {R : Type u} (S : Type v) [CommSemiring R] [CommSemiring S] [Algebra R S] (M : Submonoid R) [IsLocalization M S] : Embedding ⇑(PrimeSpectrum.comap (algebraMap R S))

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

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

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

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 {(PrimeSpectrum.comap f) x}

theorem PrimeSpectrum.comap_singleton_isClosed_of_isIntegral {R : Type u} (S : Type v) [CommRing R] [CommRing S] (f : R →+* S) (hf : f.IsIntegral) (x : PrimeSpectrum S) (hx : IsClosed {x}) : IsClosed {(PrimeSpectrum.comap f) x}

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

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 ⇑(PrimeSpectrum.comap f) = PrimeSpectrum.zeroLocus ↑(RingHom.ker f)

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 ⇑(PrimeSpectrum.comap f))

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

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.

Equations

• PrimeSpectrum.basicOpen r = { carrier := {x : PrimeSpectrum R | r ∉ x.asIdeal}, is_open' := ⋯ }

@[simp]

theorem PrimeSpectrum.mem_basicOpen {R : Type u} [CommSemiring R] (f : R) (x : PrimeSpectrum R) : x ∈ PrimeSpectrum.basicOpen f ↔ f ∉ x.asIdeal

theorem PrimeSpectrum.isOpen_basicOpen {R : Type u} [CommSemiring R] {a : R} : IsOpen ↑(PrimeSpectrum.basicOpen a)

@[simp]

theorem PrimeSpectrum.basicOpen_eq_zeroLocus_compl {R : Type u} [CommSemiring R] (r : R) : ↑(PrimeSpectrum.basicOpen r) = (PrimeSpectrum.zeroLocus {r})ᶜ

@[simp]

theorem PrimeSpectrum.basicOpen_one {R : Type u} [CommSemiring R] : PrimeSpectrum.basicOpen 1 = ⊤

@[simp]

theorem PrimeSpectrum.basicOpen_zero {R : Type u} [CommSemiring R] : PrimeSpectrum.basicOpen 0 = ⊥

theorem PrimeSpectrum.basicOpen_le_basicOpen_iff {R : Type u} [CommSemiring R] (f : R) (g : R) : PrimeSpectrum.basicOpen f ≤ PrimeSpectrum.basicOpen g ↔ f ∈ (Ideal.span {g}).radical

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

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

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

@[simp]

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

theorem PrimeSpectrum.isTopologicalBasis_basic_opens {R : Type u} [CommSemiring R] : TopologicalSpace.IsTopologicalBasis (Set.range fun (r : R) => ↑(PrimeSpectrum.basicOpen r))

theorem PrimeSpectrum.isBasis_basic_opens {R : Type u} [CommSemiring R] : TopologicalSpace.Opens.IsBasis (Set.range PrimeSpectrum.basicOpen)

@[simp]

theorem PrimeSpectrum.basicOpen_eq_bot_iff {R : Type u} [CommSemiring R] (f : R) : PrimeSpectrum.basicOpen f = ⊥ ↔ IsNilpotent f

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

theorem PrimeSpectrum.localization_away_openEmbedding {R : Type u} [CommSemiring R] (S : Type v) [CommSemiring S] [Algebra R S] (r : R) [IsLocalization.Away r S] : OpenEmbedding ⇑(PrimeSpectrum.comap (algebraMap R S))

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

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

The specialization order

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

instance PrimeSpectrum.instPartialOrder {R : Type u} [CommSemiring R] : PartialOrder (PrimeSpectrum R)

Equations

• PrimeSpectrum.instPartialOrder = PartialOrder.lift PrimeSpectrum.asIdeal ⋯

@[simp]

theorem PrimeSpectrum.asIdeal_le_asIdeal {R : Type u} [CommSemiring R] (x : PrimeSpectrum R) (y : PrimeSpectrum R) : x.asIdeal ≤ y.asIdeal ↔ x ≤ y

@[simp]

theorem PrimeSpectrum.asIdeal_lt_asIdeal {R : Type u} [CommSemiring R] (x : PrimeSpectrum R) (y : PrimeSpectrum R) : x.asIdeal < y.asIdeal ↔ x < y

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

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

@[simp]

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

def PrimeSpectrum.nhdsOrderEmbedding {R : Type u} [CommSemiring R] : PrimeSpectrum R ↪o Filter (PrimeSpectrum R)

nhds as an order embedding.

Equations

• PrimeSpectrum.nhdsOrderEmbedding = OrderEmbedding.ofMapLEIff nhds ⋯

instance PrimeSpectrum.instT0Space {R : Type u} [CommSemiring R] : T0Space (PrimeSpectrum R)

Equations

• ⋯ = ⋯

instance PrimeSpectrum.instOrderBotOfIsDomain {R : Type u} [CommSemiring R] [IsDomain R] : OrderBot (PrimeSpectrum R)

Equations

• PrimeSpectrum.instOrderBotOfIsDomain = OrderBot.mk ⋯

instance PrimeSpectrum.instUnique {R : Type u_1} [Field R] : Unique (PrimeSpectrum R)

Equations

• PrimeSpectrum.instUnique = { default := ⊥, uniq := ⋯ }

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

Equations

• PrimeSpectrum.localizationMapOfSpecializes h = IsLocalization.lift ⋯

def PrimeSpectrum.pointsEquivIrreducibleCloseds (R : Type u) [CommSemiring R] : PrimeSpectrum R ≃o (↑{s : Set (PrimeSpectrum R) | IsIrreducible s ∧ IsClosed s})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.

Equations

• PrimeSpectrum.pointsEquivIrreducibleCloseds R = let __spread.0 := irreducibleSetEquivPoints.symm.trans OrderDual.toDual; { toEquiv := __spread.0, map_rel_iff' := ⋯ }

def PrimeSpectrum.primeSpectrum_unique_of_localization_at_minimal {R : Type u} [CommSemiring R] {I : Ideal R} [hI : I.IsPrime] (h : I ∈ minimalPrimes R) : Unique (PrimeSpectrum (Localization.AtPrime I))

Localizations at minimal primes have single-point prime spectra.

Equations

• PrimeSpectrum.primeSpectrum_unique_of_localization_at_minimal h = { default := { asIdeal := LocalRing.maximalIdeal (Localization I.primeCompl), isPrime := ⋯ }, uniq := ⋯ }

def minimalPrimes.equivIrreducibleComponents (R : Type u) [CommSemiring R] : ↑(minimalPrimes R) ≃o (↑(irreducibleComponents (PrimeSpectrum R)))ᵒᵈ

Stacks: Lemma 00ES (3) 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.

Equations

• One or more equations did not get rendered due to their size.

theorem PrimeSpectrum.vanishingIdeal_irreducibleComponents (R : Type u) [CommSemiring R] : PrimeSpectrum.vanishingIdeal '' irreducibleComponents (PrimeSpectrum R) = minimalPrimes R

theorem PrimeSpectrum.zeroLocus_minimalPrimes (R : Type u) [CommSemiring R] : PrimeSpectrum.zeroLocus ∘ SetLike.coe '' minimalPrimes R = irreducibleComponents (PrimeSpectrum R)

theorem PrimeSpectrum.vanishingIdeal_mem_minimalPrimes {R : Type u} [CommSemiring R] {s : Set (PrimeSpectrum R)} : PrimeSpectrum.vanishingIdeal s ∈ minimalPrimes R ↔ closure s ∈ irreducibleComponents (PrimeSpectrum R)

theorem PrimeSpectrum.zeroLocus_ideal_mem_irreducibleComponents {R : Type u} [CommSemiring R] {I : Ideal R} : PrimeSpectrum.zeroLocus ↑I ∈ irreducibleComponents (PrimeSpectrum R) ↔ I.radical ∈ minimalPrimes R

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

The closed point in the prime spectrum of a local ring.

Equations

• LocalRing.closedPoint R = { asIdeal := LocalRing.maximalIdeal R, isPrime := ⋯ }

theorem LocalRing.isLocalRingHom_iff_comap_closedPoint {R : Type u} [CommSemiring R] [LocalRing R] {S : Type v} [CommSemiring S] [LocalRing S] (f : R →+* S) : IsLocalRingHom f ↔ (PrimeSpectrum.comap f) (LocalRing.closedPoint S) = LocalRing.closedPoint R

@[simp]

theorem LocalRing.comap_closedPoint {R : Type u} [CommSemiring R] [LocalRing R] {S : Type v} [CommSemiring S] [LocalRing S] (f : R →+* S) [IsLocalRingHom f] : (PrimeSpectrum.comap f) (LocalRing.closedPoint S) = LocalRing.closedPoint R

theorem LocalRing.specializes_closedPoint {R : Type u} [CommSemiring R] [LocalRing R] (x : PrimeSpectrum R) : x ⤳ LocalRing.closedPoint R

theorem LocalRing.closedPoint_mem_iff {R : Type u} [CommSemiring R] [LocalRing R] (U : TopologicalSpace.Opens (PrimeSpectrum R)) : LocalRing.closedPoint R ∈ U ↔ U = ⊤

@[simp]

theorem LocalRing.PrimeSpectrum.comap_residue (T : Type u) [CommRing T] [LocalRing T] (x : PrimeSpectrum (LocalRing.ResidueField T)) : (PrimeSpectrum.comap (LocalRing.residue T)) x = LocalRing.closedPoint T