theorem thm3_1_affine_iff_spec (k : Type v) [Field k] [IsAlgClosed k] : (∀ (X : AlgebraicGeometry.Scheme) (f : X ⟶ AlgebraicGeometry.Spec (CommRingCat.of k)) [Definition3_AlgebraicVariety k X f] [AlgebraicGeometry.IsAffine X], CategoryTheory.IsIso X.toSpecΓ ∧ IsReduced ↑(X.presheaf.obj (Opposite.op ⊤)) ∧ (CommRingCat.Hom.hom (AlgebraicGeometry.Scheme.Hom.appTop f)).FiniteType) ∧ ∀ (A : Type v) [inst : CommRing A] [inst_1 : Algebra k A] [Algebra.FiniteType k A] [IsReduced A], Definition3_AlgebraicVariety k (AlgebraicGeometry.Spec (CommRingCat.of A)) (AlgebraicGeometry.Spec.map (CommRingCat.ofHom (algebraMap k A)))

Lecture 3, Theorem 3.1: an algebraic variety X over an algebraically closed field k is affine iff X ≅ Spec A for A = Γ(X, O_X), and conversely Spec A is a variety whenever A is a reduced finitely generated k-algebra.

theorem thm3_2_noether_normalization (k : Type u_1) [Field k] (A : Type u_2) [CommRing A] [Nontrivial A] [Algebra k A] [Algebra.FiniteType k A] : ∃ (d : ℕ) (g : MvPolynomial (Fin d) k →ₐ[k] A), Function.Injective ⇑g ∧ g.Finite

Lecture 3, Theorem 3.2 (Noether normalization): any finitely generated k-algebra A admits an injective finite k-algebra map from a polynomial ring k[x_1, …, x_d].

theorem lemma3_linear_change_monic {k : Type u_1} [Field k] [Infinite k] {n : ℕ} (P : MvPolynomial (Fin (n + 1)) k) (hP : 0 < P.totalDegree) : ∃ (φ : MvPolynomial (Fin (n + 1)) k ≃ₐ[k] MvPolynomial (Fin (n + 1)) k), IsUnit ((MvPolynomial.finSuccEquiv k n) (φ P)).leadingCoeff

Lecture 3, Lemma 3 (linear change of variables): over an infinite field, any nonconstant polynomial in n + 1 variables can be transformed by an algebra automorphism so that, viewed as a polynomial in x_0 over k[x_1, …, x_n], its leading coefficient is a unit.

theorem lemma3_nonzero_polynomial_takes_nonzero_values {k : Type u_1} [Field k] [Infinite k] {σ : Type u_2} [Fintype σ] {f : MvPolynomial σ k} (hf : f ≠ 0) : ∃ (v : σ → k), (MvPolynomial.eval v) f ≠ 0

Companion to Lemma 3: over an infinite field, any nonzero polynomial in finitely many variables takes a nonzero value at some point.

theorem prop2_hilbert_basis_theorem (k : Type u_1) [Field k] (n : ℕ) : IsNoetherianRing (MvPolynomial (Fin n) k)

Lecture 3, Proposition 2 (Hilbert basis theorem): polynomial rings in finitely many variables over a field are Noetherian.

theorem lemma4_nakayama {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [Module.Finite R M] (I : Ideal R) (hIM : I • ⊤ = ⊤) : ∃ (a : R), a - 1 ∈ I ∧ ∀ (m : M), a • m = 0

Lecture 3, Lemma 4 (determinant form of Nakayama's lemma): if I is an ideal of R and M is a finitely generated R-module with IM = M, then some a ∈ R with a ≡ 1 mod I annihilates all of M.

theorem thm3_3_essential_nullstellensatz (k : Type u_1) [Field k] (A : Type u_2) [Field A] [Algebra k A] [Algebra.FiniteType k A] : Algebra.IsAlgebraic k A

Lecture 3, Theorem 3.3 (essential Nullstellensatz): any field that is a finitely generated algebra over a field k is algebraic over k.

theorem def7_irreducible_space_iff (X : Type u_1) [TopologicalSpace X] : IrreducibleSpace X ↔ IsIrreducible Set.univ

Lecture 3, Definition 7: a topological space is irreducible iff the whole space, viewed as a subset, is irreducible.

theorem def7_irreducible_not_union_proper_closed (X : Type u_1) [TopologicalSpace X] [IrreducibleSpace X] (s t : Set X) (hs : IsClosed s) (ht : IsClosed t) (hst : s ∪ t = Set.univ) : s = Set.univ ∨ t = Set.univ

Equivalent form of Definition 7: an irreducible space cannot be written as a union of two proper closed subsets.

theorem prop3_spec_irreducible_iff_domain (A : Type u_1) [CommRing A] [IsReduced A] : IrreducibleSpace (PrimeSpectrum A) ↔ IsDomain A

Lecture 3, Proposition 3: for a reduced ring A, the prime spectrum Spec A is irreducible iff A is a domain.

theorem def8_component_characterization (X : Type u_1) [TopologicalSpace X] (s : Set X) : s ∈ irreducibleComponents X ↔ IsIrreducible s ∧ ∀ (t : Set X), IsIrreducible t → s ⊆ t → t ⊆ s

Lecture 3, Definition 8: a subset is an irreducible component iff it is irreducible and maximal among irreducible subsets.

theorem def8_component_is_closed (X : Type u_1) [TopologicalSpace X] (s : Set X) (hs : s ∈ irreducibleComponents X) : IsClosed s

Companion to Definition 8: every irreducible component is closed.

theorem prop4_noetherian_finite_union_components (X : Type u_1) [TopologicalSpace X] [TopologicalSpace.NoetherianSpace X] : (irreducibleComponents X).Finite ∧ ⋃₀ irreducibleComponents X = Set.univ

Lecture 3, Proposition 4: a Noetherian topological space has finitely many irreducible components, whose union is the whole space.

theorem cor5_irreducible_closed_iff_prime (A : Type u_1) [CommRing A] (I : Ideal A) (hI : I.IsRadical) : IsIrreducible (PrimeSpectrum.zeroLocus ↑I) ↔ I.IsPrime

Lecture 3, Corollary 5 (form 1): the zero locus of a radical ideal I ⊆ A is irreducible iff I is prime.

theorem cor5_closed_irreducible_eq_primes (A : Type u_1) [CommRing A] : PrimeSpectrum.vanishingIdeal '' {s : Set (PrimeSpectrum A) | IsClosed s ∧ IsIrreducible s} = {P : Ideal A | P.IsPrime}

Corollary 5 (form 2): the vanishing-ideal map gives a bijection between closed irreducible subsets of Spec A and prime ideals of A.

theorem cor5_components_eq_minimal_primes (A : Type u_1) [CommRing A] [IsNoetherianRing A] : PrimeSpectrum.vanishingIdeal '' irreducibleComponents (PrimeSpectrum A) = minimalPrimes A

Corollary 5 (form 3): under this bijection, irreducible components of Spec A correspond to minimal primes of A when A is Noetherian.

theorem cor6_zero_eq_inf_minimal_primes (A : Type u_1) [CommRing A] [IsReduced A] : sInf ⊥.minimalPrimes = ⊥

Lecture 3, Corollary 6: in a reduced ring, the intersection of the minimal primes is zero.