class Definition3_AlgebraicVariety (k : Type u) [Field k] (X : AlgebraicGeometry.Scheme) (f : X ⟶ AlgebraicGeometry.Spec (CommRingCat.of k)) : Prop

Definition 3: An algebraic variety over k is a scheme X with a morphism f : X → Spec k that is locally of finite type, quasicompact, and whose total space is reduced.

• locallyOfFiniteType : AlgebraicGeometry.LocallyOfFiniteType f
• quasiCompact : AlgebraicGeometry.QuasiCompact f
• reduced : AlgebraicGeometry.IsReduced X

theorem Theorem2_2_forward (k : Type u) [Field k] (X : AlgebraicGeometry.Scheme) (f : X ⟶ AlgebraicGeometry.Spec (CommRingCat.of k)) [hvar : 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

Theorem 2.2 (forward direction): An affine algebraic variety X over k is canonically isomorphic to Spec Γ(X, ⊤), with reduced and finite-type global sections.

theorem Theorem2_2_backward (k : Type u) [Field k] (A : Type u) [CommRing A] [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)))

Theorem 2.2 (backward direction): For any finitely generated reduced k-algebra A, the affine scheme Spec A is an algebraic variety over k.

theorem thm22_affine_variety_characterization (k : Type u) [Field 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 u) [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)))

Theorem 2.2 (full characterisation): A scheme over k is an affine algebraic variety iff it is Spec A for a finitely generated reduced k-algebra A.

theorem thm31_affine_variety_characterization (k : Type u) [Field 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 u) [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)))

Restatement of the affine-variety characterisation, packaged under the index used in Lecture 3.

theorem Lemma2_ClosedSubspaceAffine {X Z : AlgebraicGeometry.Scheme} (i : Z ⟶ X) [AlgebraicGeometry.IsClosedImmersion i] [AlgebraicGeometry.IsAffine X] : AlgebraicGeometry.IsAffine Z

Lemma 2 (part 1): A closed subscheme of an affine scheme is affine.

theorem Lemma2_GlobalSectionsSurjective {X Z : AlgebraicGeometry.Scheme} (i : Z ⟶ X) [AlgebraicGeometry.IsClosedImmersion i] [AlgebraicGeometry.IsAffine X] : Function.Surjective ⇑(CategoryTheory.ConcreteCategory.hom (AlgebraicGeometry.Scheme.Hom.appTop i))

Lemma 2 (part 2): A closed immersion into an affine scheme induces a surjection on global sections.

theorem Lemma2_combined {X Z : AlgebraicGeometry.Scheme} (i : Z ⟶ X) [AlgebraicGeometry.IsClosedImmersion i] [AlgebraicGeometry.IsAffine X] : AlgebraicGeometry.IsAffine Z ∧ Function.Surjective ⇑(CategoryTheory.ConcreteCategory.hom (AlgebraicGeometry.Scheme.Hom.appTop i))

Lemma 2 (combined): A closed subscheme of an affine scheme is affine and the closed immersion is surjective on global sections.

theorem Corollary2_ClosedSubspaceOfVariety (k : Type u) [Field k] (X Z : AlgebraicGeometry.Scheme) (f : X ⟶ AlgebraicGeometry.Spec (CommRingCat.of k)) [Definition3_AlgebraicVariety k X f] (i : Z ⟶ X) [AlgebraicGeometry.IsClosedImmersion i] [AlgebraicGeometry.IsReduced Z] : Definition3_AlgebraicVariety k Z (CategoryTheory.CategoryStruct.comp i f)

Corollary 2: A reduced closed subscheme of an algebraic variety is again an algebraic variety (with composed structure map to Spec k).

theorem Theorem2_3_HilbertBasis_PolynomialRing (k : Type u) [Field k] (n : ℕ) : IsNoetherianRing (MvPolynomial (Fin n) k)

Theorem 2.3 (Hilbert basis): The polynomial ring k[x₁, …, xₙ] over a field is Noetherian.

def polynomialGrading (k : Type u) [Field k] (n n✝ : ℕ) : Submodule k (MvPolynomial (Fin (n + 1)) k)

The natural ℕ-grading of k[x₀, …, xₙ] by total degree, used to define ℙⁿ_k = Proj.

@[implicit_reducible]

noncomputable instance polynomialGradedRing (k : Type u) [Field k] (n : ℕ) : GradedRing (polynomialGrading k n)

The degree-grading of k[x₀, …, xₙ] gives it the structure of a graded ring.

noncomputable def Definition4_ProjectiveSpace (k : Type u) [Field k] (n : ℕ) : AlgebraicGeometry.Scheme

Definition 4: The projective n-space ℙⁿ_k, realised as Proj k[x₀, …, xₙ].

@[implicit_reducible]

noncomputable instance Definition4_ProjectiveSpace_topologicalSpace (k : Type u) [Field k] (n : ℕ) : TopologicalSpace ↥(Definition4_ProjectiveSpace k n)

The underlying topological space of ℙⁿ_k.

noncomputable def Definition4_structureSheaf (k : Type u) [Field k] (n : ℕ) : TopCat.Presheaf CommRingCat ↑(Definition4_ProjectiveSpace k n).toPresheafedSpace

The structure sheaf O_{ℙⁿ_k} of projective space.