theorem
Lec7ClosedProductSubvariety.lemma16_tensorProduct_map_surjective
{R : Type u_1}
{M : Type u_2}
{N : Type u_3}
{M' : Type u_4}
{N' : Type u_5}
[CommSemiring R]
[AddCommMonoid M]
[Module R M]
[AddCommMonoid N]
[Module R N]
[AddCommMonoid M']
[Module R M']
[AddCommMonoid N']
[Module R N']
{f : M →ₗ[R] M'}
{g : N →ₗ[R] N'}
(hf : Function.Surjective ⇑f)
(hg : Function.Surjective ⇑g)
:
Helper for Lec 7, Lem 16: the tensor product of two surjective linear maps is surjective.
theorem
Lec7ClosedProductSubvariety.lemma16_algebra_tensorProduct_map_surjective
{R : Type u_1}
{A₁ : Type u_2}
{B₁ : Type u_3}
{A₂ : Type u_4}
{B₂ : Type u_5}
[CommSemiring R]
[CommSemiring A₁]
[CommSemiring B₁]
[CommSemiring A₂]
[CommSemiring B₂]
[Algebra R A₁]
[Algebra R B₁]
[Algebra R A₂]
[Algebra R B₂]
(φ : A₁ →ₐ[R] B₁)
(ψ : A₂ →ₐ[R] B₂)
(hφ : Function.Surjective ⇑φ)
(hψ : Function.Surjective ⇑ψ)
:
Helper for Lec 7, Lem 16: the tensor product of two surjective algebra homomorphisms is surjective.
instance
Lec7ClosedProductSubvariety.isClosedImmersion_prod_map_id_right
{X₁ Y₁ Z : AlgebraicGeometry.Scheme}
(f : X₁ ⟶ Y₁)
[AlgebraicGeometry.IsClosedImmersion f]
:
Product of a closed immersion with the identity on the right is a closed immersion (used in Lec 7, Lem 16).
instance
Lec7ClosedProductSubvariety.isClosedImmersion_prod_map_id_left
{X₂ Y₂ Z : AlgebraicGeometry.Scheme}
(g : X₂ ⟶ Y₂)
[AlgebraicGeometry.IsClosedImmersion g]
:
Product of the identity on the left with a closed immersion is a closed immersion (used in Lec 7, Lem 16).
theorem
Lec7ClosedProductSubvariety.lemma16_closed_product_subvariety
{X₁ X₂ Y₁ Y₂ : AlgebraicGeometry.Scheme}
(i₁ : X₁ ⟶ Y₁)
(i₂ : X₂ ⟶ Y₂)
[AlgebraicGeometry.IsClosedImmersion i₁]
[AlgebraicGeometry.IsClosedImmersion i₂]
:
Lec 7, Lem 16: the product of two closed immersions
i₁ : X₁ ↪ Y₁ and i₂ : X₂ ↪ Y₂ is a closed immersion
X₁ × X₂ ↪ Y₁ × Y₂.