Prime spectrum of a commutative (semi)ring as a type

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

For the Zariski topology, see AlgebraicGeometry.PrimeSpectrum.Basic.

(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).

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.ext {R : Type u} {inst✝ : CommSemiring R} {x y : PrimeSpectrum R} (asIdeal : x.asIdeal = y.asIdeal) : x = y

@[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

• PrimeSpectrum.primeSpectrumProdOfSum R S (Sum.inl { asIdeal := I, isPrime := isPrime }) = { asIdeal := I.prod ⊤, isPrime := ⋯ }
• PrimeSpectrum.primeSpectrumProdOfSum R S (Sum.inr { asIdeal := J, isPrime := isPrime }) = { asIdeal := ⊤.prod J, isPrime := ⋯ }

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 t : Set R} (h : s ⊆ t) : PrimeSpectrum.zeroLocus t ⊆ PrimeSpectrum.zeroLocus s

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

theorem PrimeSpectrum.vanishingIdeal_anti_mono {R : Type u} [CommSemiring R] {s 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 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 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_eq_top_iff {R : Type u} [CommSemiring R] (s : Set R) : PrimeSpectrum.zeroLocus s = ⊤ ↔ s ⊆ ↑(nilradical R)

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

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

theorem PrimeSpectrum.vanishingIdeal_union {R : Type u} [CommSemiring R] (t 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_iUnion₂ {R : Type u} [CommSemiring R] {ι : Sort u_1} {κ : ι → Sort u_2} (s : (i : ι) → κ i → Set R) : PrimeSpectrum.zeroLocus (⋃ (i : ι), ⋃ (j : κ i), s i j) = ⋂ (i : ι), ⋂ (j : κ i), PrimeSpectrum.zeroLocus (s i j)

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 J : Ideal R) : PrimeSpectrum.zeroLocus ↑(I ⊓ J) = PrimeSpectrum.zeroLocus ↑I ∪ PrimeSpectrum.zeroLocus ↑J

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

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

The specialization order

We endow PrimeSpectrum R with a partial order induced from the ideal lattice. This is exactly the specialization order. See the corresponding section at Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic.

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 y : PrimeSpectrum R) : x.asIdeal ≤ y.asIdeal ↔ x ≤ y

@[simp]

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

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 := ⋯ }

theorem PrimeSpectrum.exists_primeSpectrum_prod_le (R : Type u) [CommRing R] [IsNoetherianRing R] (I : Ideal R) : ∃ (Z : Multiset (PrimeSpectrum R)), (Multiset.map PrimeSpectrum.asIdeal Z).prod ≤ I

In a noetherian ring, every ideal contains a product of prime ideals ([samuel, § 3.3, Lemma 3])

theorem PrimeSpectrum.exists_primeSpectrum_prod_le_and_ne_bot_of_domain {A : Type u} [CommRing A] [IsDomain A] [IsNoetherianRing A] (h_fA : ¬IsField A) {I : Ideal A} (h_nzI : I ≠ ⊥) : ∃ (Z : Multiset (PrimeSpectrum A)), (Multiset.map PrimeSpectrum.asIdeal Z).prod ≤ I ∧ (Multiset.map PrimeSpectrum.asIdeal Z).prod ≠ ⊥

In a noetherian integral domain which is not a field, every non-zero ideal contains a non-zero product of prime ideals; in a field, the whole ring is a non-zero ideal containing only 0 as product or prime ideals ([samuel, § 3.3, Lemma 3])

@[reducible, inline]

abbrev RingHom.specComap {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (f : R →+* S) : PrimeSpectrum S → PrimeSpectrum R

The pullback of an element of PrimeSpectrum S along a ring homomorphism f : R →+* S. The bundled continuous version is PrimeSpectrum.comap.

Equations

• f.specComap y = { asIdeal := Ideal.comap f y.asIdeal, isPrime := ⋯ }

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

@[simp]

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

@[simp]

theorem PrimeSpectrum.specComap_id {R : Type u} [CommSemiring R] : (RingHom.id R).specComap = fun (x : PrimeSpectrum R) => x

@[simp]

theorem PrimeSpectrum.specComap_comp {R : Type u} {S : Type v} {S' : Type u_1} [CommSemiring R] [CommSemiring S] [CommSemiring S'] (f : R →+* S) (g : S →+* S') : (g.comp f).specComap = f.specComap ∘ g.specComap

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

@[simp]

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

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

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

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}

theorem image_specComap_zeroLocus_eq_zeroLocus_comap {R : Type u} (S : Type v) [CommRing R] [CommRing S] (f : R →+* S) (hf : Function.Surjective ⇑f) (I : Ideal S) : f.specComap '' PrimeSpectrum.zeroLocus ↑I = PrimeSpectrum.zeroLocus ↑(Ideal.comap f I)

theorem range_specComap_of_surjective {R : Type u} (S : Type v) [CommRing R] [CommRing S] (f : R →+* S) (hf : Function.Surjective ⇑f) : Set.range f.specComap = PrimeSpectrum.zeroLocus ↑(RingHom.ker f)