theorem Lec7Products.tensor_product_map_surjective {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {A : Type u_3} {B : Type u_4} {C : Type u_5} {D : Type u_6} [Ring A] [Ring B] [Ring C] [Ring D] [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D] [Algebra S A] [Algebra S B] [IsScalarTower R S A] [IsScalarTower R S B] (f : A →ₐ[S] B) (g : C →ₐ[R] D) (hf : Function.Surjective ⇑f) (hg : Function.Surjective ⇑g) : Function.Surjective ⇑(Algebra.TensorProduct.map f g)

Helper: the tensor product of two surjective algebra homomorphisms is surjective (used for products of closed immersions).

instance Lec7Products.isClosedImmersion_prod_map_id_right {X₁ Y₁ Z : AlgebraicGeometry.Scheme} (f : X₁ ⟶ Y₁) [AlgebraicGeometry.IsClosedImmersion f] : AlgebraicGeometry.IsClosedImmersion (CategoryTheory.Limits.prod.map f (CategoryTheory.CategoryStruct.id Z))

Product of a closed immersion with the identity on the right is a closed immersion (Lec 7, used for Prop 8).

instance Lec7Products.isClosedImmersion_prod_map_id_left {X₂ Y₂ Z : AlgebraicGeometry.Scheme} (g : X₂ ⟶ Y₂) [AlgebraicGeometry.IsClosedImmersion g] : AlgebraicGeometry.IsClosedImmersion (CategoryTheory.Limits.prod.map (CategoryTheory.CategoryStruct.id Z) g)

Product of the identity on the left with a closed immersion is a closed immersion (Lec 7, used for Prop 8).

theorem Lec7Products.isClosedImmersion_prod_map {X₁ X₂ Y₁ Y₂ : AlgebraicGeometry.Scheme} (i₁ : X₁ ⟶ Y₁) (i₂ : X₂ ⟶ Y₂) [AlgebraicGeometry.IsClosedImmersion i₁] [AlgebraicGeometry.IsClosedImmersion i₂] : AlgebraicGeometry.IsClosedImmersion (CategoryTheory.Limits.prod.map i₁ i₂)

The product of two closed immersions is a closed immersion (Lec 7, Lem 16 / used in Prop 8).

class Lec7Products.Scheme.IsProjective (X : AlgebraicGeometry.Scheme) : Prop

A scheme X is projective if it admits a closed immersion into some Proj 𝒜 of a graded ring (Lec 7, Prop 8 setting).

• exists_closedImmersion_into_proj : ∃ (A : Type) (σ : Type) (x : CommRing A) (x_1 : SetLike σ A) (x_2 : AddSubgroupClass σ A) (𝒜 : ℕ → σ) (x_3 : GradedRing 𝒜) (f : X ⟶ AlgebraicGeometry.Proj 𝒜), AlgebraicGeometry.IsClosedImmersion f

instance Lec7Products.proj_isProjective {A σ : Type} [CommRing A] [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] : Scheme.IsProjective (AlgebraicGeometry.Proj 𝒜)

Proj 𝒜 is projective via the identity closed immersion.

theorem Lec7Products.isProjective_of_closedImmersion {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsClosedImmersion f] [Scheme.IsProjective Y] : Scheme.IsProjective X

A closed subscheme of a projective scheme is projective.

noncomputable def Lec7Products.segreAlgHom (R : Type u_1) [CommRing R] (n m : ℕ) : MvPolynomial (Fin n × Fin m) R →ₐ[R] TensorProduct R (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R)

The Segre algebra homomorphism sending X_{(i,j)} ↦ X_i ⊗ X_j, underlying the Segre embedding for Lec 7, Prop 8.

theorem Lec7Products.segreAlgHom_X (R : Type u_1) [CommRing R] (n m : ℕ) (i : Fin n) (j : Fin m) : (segreAlgHom R n m) (MvPolynomial.X (i, j)) = MvPolynomial.X i ⊗ₜ[R] MvPolynomial.X j

The Segre map sends the generator X_{(i,j)} to X_i ⊗ X_j.

theorem Lec7Products.segreAlgHom_relation (R : Type u_1) [CommRing R] (n m : ℕ) (i k : Fin n) (j l : Fin m) : (segreAlgHom R n m) (MvPolynomial.X (i, j) * MvPolynomial.X (k, l) - MvPolynomial.X (k, j) * MvPolynomial.X (i, l)) = 0

The defining Segre relations X_{(i,j)} X_{(k,l)} - X_{(k,j)} X_{(i,l)} lie in the kernel of the Segre map.

theorem Lec7Products.segreRelation_isHomogeneous (R : Type u_1) [CommRing R] (n m : ℕ) (i k : Fin n) (j l : Fin m) : (MvPolynomial.X (i, j) * MvPolynomial.X (k, l) - MvPolynomial.X (k, j) * MvPolynomial.X (i, l)).IsHomogeneous 2

The Segre relations are homogeneous of degree 2 in the standard grading on R[X_{(i,j)}].

theorem Lec7Products.product_projective_is_projective (X Y : AlgebraicGeometry.Scheme) [hX : Scheme.IsProjective X] [hY : Scheme.IsProjective Y] : Scheme.IsProjective (X ⨯ Y)

Lec 7, Prop 8: the product of two projective schemes is projective (via the Segre embedding).

noncomputable def Lec7Products.graphMorphism {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) : X ⟶ X ⨯ Y

The graph morphism Γ_f : X ⟶ X × Y of f : X ⟶ Y (Lec 7, Def 18).

theorem Lec7Products.graphMorphism_fst {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) : CategoryTheory.CategoryStruct.comp (graphMorphism f) CategoryTheory.Limits.prod.fst = CategoryTheory.CategoryStruct.id X

The first projection of the graph morphism is the identity on X.

theorem Lec7Products.graphMorphism_snd {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) : CategoryTheory.CategoryStruct.comp (graphMorphism f) CategoryTheory.Limits.prod.snd = f

The second projection of the graph morphism recovers f.

theorem Lec7Products.graphMorphism_id (X : AlgebraicGeometry.Scheme) : graphMorphism (CategoryTheory.CategoryStruct.id X) = CategoryTheory.Limits.prod.lift (CategoryTheory.CategoryStruct.id X) (CategoryTheory.CategoryStruct.id X)

The graph morphism of 𝟙 X is the diagonal X ⟶ X × X.