theorem Lec1ZariskiNullstellensatz.thm1_1_essential_nullstellensatz (k : Type u_1) (K : Type u_2) [Field k] [Field K] [Algebra k K] [Algebra.FiniteType k K] : Algebra.IsAlgebraic k K

Theorem 1.1 (Lecture 1, essential Nullstellensatz). Any field extension K/k that is finitely generated as a k-algebra is algebraic over k.

theorem Lec1ZariskiNullstellensatz.thm1_1_essential_nullstellensatz_algClosed (k : Type u_1) (K : Type u_2) [Field k] [IsAlgClosed k] [Field K] [Algebra k K] [Algebra.FiniteType k K] : Function.Bijective ⇑(algebraMap k K)

Algebraically closed form of Theorem 1.1: if k = k̄ is algebraically closed and K/k is a finitely generated k-algebra that is a field, then the structure map k → K is bijective, i.e. K = k.

def Lec1ZariskiNullstellensatz.IsZariskiClosed (k : Type u_1) [Field k] (σ : Type u_2) (V : Set (σ → k)) : Prop

Definition 1 (Lecture 1). A subset V ⊆ k^σ is Zariski closed if it is the zero set of some collection of polynomials in k[x_σ].

theorem Lec1ZariskiNullstellensatz.isZariskiClosed_iff_exists_ideal (k : Type u_1) [Field k] (σ : Type u_2) (V : Set (σ → k)) : IsZariskiClosed k σ V ↔ ∃ (I : Ideal (MvPolynomial σ k)), V = MvPolynomial.zeroLocus k I

A subset is Zariski closed iff it is the zero locus of some ideal of polynomials; equivalence between cutting out by a set of polynomials and by the ideal it generates.

theorem Lec1ZariskiNullstellensatz.vanishingIdeal_isRadical (k : Type u_1) [Field k] (σ : Type u_2) (V : Set (σ → k)) : (MvPolynomial.vanishingIdeal k V).IsRadical

The vanishing ideal I(V) of any subset of affine space is a radical ideal.

theorem Lec1ZariskiNullstellensatz.zeroLocus_radical_eq (k : Type u_1) [Field k] (σ : Type u_2) (I : Ideal (MvPolynomial σ k)) : MvPolynomial.zeroLocus k I.radical = MvPolynomial.zeroLocus k I

An ideal and its radical have the same zero locus: V(I) = V(√I).

theorem Lec1ZariskiNullstellensatz.nullstellensatz_correspondence_eq_radical (k : Type u_1) [Field k] [IsAlgClosed k] (σ : Type u_2) [Finite σ] (I : Ideal (MvPolynomial σ k)) : MvPolynomial.vanishingIdeal k (MvPolynomial.zeroLocus k I) = I.radical

Hilbert's Nullstellensatz (over an algebraically closed field): I(V(I)) = √I.

theorem Lec1ZariskiNullstellensatz.nullstellensatz_radical_inverse (k : Type u_1) [Field k] [IsAlgClosed k] (σ : Type u_2) [Finite σ] (I : Ideal (MvPolynomial σ k)) (hI : I.IsRadical) : MvPolynomial.vanishingIdeal k (MvPolynomial.zeroLocus k I) = I

For a radical ideal I over k̄, we have I(V(I)) = I: passing to the zero locus and back recovers a radical ideal.

theorem Lec1ZariskiNullstellensatz.nullstellensatz_closed_inverse (k : Type u_1) [Field k] [IsAlgClosed k] (σ : Type u_2) [Finite σ] (I : Ideal (MvPolynomial σ k)) : MvPolynomial.zeroLocus k (MvPolynomial.vanishingIdeal k (MvPolynomial.zeroLocus k I)) = MvPolynomial.zeroLocus k I

For any ideal I over k̄, the iterated operation V(I(V(I))) recovers V(I).

noncomputable def Lec1ZariskiNullstellensatz.nullstellensatz_bijection (k : Type u_1) [Field k] [IsAlgClosed k] (σ : Type u_2) [Finite σ] : { I : Ideal (MvPolynomial σ k) // I.IsRadical } ≃ { V : Set (σ → k) // ∃ (I : Ideal (MvPolynomial σ k)), V = MvPolynomial.zeroLocus k I }

Theorem 1.2 (Lecture 1). Over an algebraically closed field with finitely many variables, there is a bijection between radical ideals of k[x_σ] and Zariski-closed subsets of k^σ, given by I ↦ V(I) with inverse V ↦ I(V).

@[implicit_reducible]

noncomputable instance Lec1ZariskiNullstellensatz.zariskiTopology_classical (k : Type u_1) [Field k] (σ : Type u_2) [Fintype σ] : TopologicalSpace (σ → k)

The classical Zariski topology on affine space k^σ: closed sets are the zero loci of ideals in k[x_σ].

theorem Lec1ZariskiNullstellensatz.zariskiTopology_isClosed_iff (k : Type u_1) [Field k] (σ : Type u_2) [Fintype σ] (Z : Set (σ → k)) : IsClosed Z ↔ IsZariskiClosed k σ Z

Closed sets in the classical Zariski topology on k^σ are exactly the Zariski-closed subsets in the polynomial-zero-set sense.

noncomputable def Lec1ZariskiNullstellensatz.yonedaPullback (A : CommRingCat) (X : AlgebraicGeometry.Scheme) : (X ⟶ AlgebraicGeometry.Spec A) → (A ⟶ X.presheaf.obj (Opposite.op ⊤))

The Yoneda-style pullback sending a scheme morphism X → Spec A to the induced ring map A → Γ(X, 𝒪_X).

structure Lec1ZariskiNullstellensatz.IsAffineVariety' (k : Type u_1) [Field k] (A : Type u_2) [CommRing A] [Algebra k A] : Prop

A bundling of the affine variety condition: A is a finite-type k-algebra and Spec A represents the functor X ↦ Hom_{Ring}(A, Γ(X, 𝒪_X)).

• finiteType : Algebra.FiniteType k A
• yoneda_bijective (X : AlgebraicGeometry.Scheme) : Function.Bijective (yonedaPullback (CommRingCat.of A) X)

noncomputable def Lec1ZariskiNullstellensatz.affine_variety_yoneda (A : CommRingCat) (X : AlgebraicGeometry.Scheme) : (X ⟶ AlgebraicGeometry.Spec A) ≃ (A ⟶ X.presheaf.obj (Opposite.op ⊤))

The fundamental Spec-Γ adjunction packaged as a bijection: scheme morphisms X → Spec A correspond to ring maps A → Γ(X, 𝒪_X).

theorem Lec1ZariskiNullstellensatz.isAffineVariety_of_finiteType (k : Type u_1) [Field k] (A : Type u_2) [CommRing A] [Algebra k A] (hfin : Algebra.FiniteType k A) : IsAffineVariety' k A

Any finitely generated k-algebra A defines an affine variety in the sense of IsAffineVariety': Spec A represents Hom from A.

instance Lec1ZariskiNullstellensatz.spec_isAffine (A : CommRingCat) : AlgebraicGeometry.IsAffine (AlgebraicGeometry.Spec A)

Spec A is an affine scheme.