def AlgebraicGeometry.IsIsomorphism {X Y : Scheme} (f : X ⟶ Y) : Prop

A morphism of schemes is an isomorphism (Lecture 3) if it has a two-sided inverse.

theorem AlgebraicGeometry.isIsomorphism_iff_isIso {X Y : Scheme} (f : X ⟶ Y) : IsIsomorphism f ↔ CategoryTheory.IsIso f

Our definition of IsIsomorphism agrees with the categorical predicate IsIso.

theorem AlgebraicGeometry.IsIsomorphism.inverse_unique {X Y : Scheme} {f : X ⟶ Y} (_hf : IsIsomorphism f) {g₁ g₂ : Y ⟶ X} (hg₁ : CategoryTheory.CategoryStruct.comp f g₁ = CategoryTheory.CategoryStruct.id X ∧ CategoryTheory.CategoryStruct.comp g₁ f = CategoryTheory.CategoryStruct.id Y) (hg₂ : CategoryTheory.CategoryStruct.comp f g₂ = CategoryTheory.CategoryStruct.id X ∧ CategoryTheory.CategoryStruct.comp g₂ f = CategoryTheory.CategoryStruct.id Y) : g₁ = g₂

The two-sided inverse of an isomorphism of schemes is unique.

theorem AlgebraicGeometry.isIsomorphism_id (X : Scheme) : IsIsomorphism (CategoryTheory.CategoryStruct.id X)

The identity morphism of a scheme is an isomorphism.

theorem AlgebraicGeometry.IsIsomorphism.comp {X Y Z : Scheme} {f : X ⟶ Y} {g : Y ⟶ Z} (hf : IsIsomorphism f) (hg : IsIsomorphism g) : IsIsomorphism (CategoryTheory.CategoryStruct.comp f g)

The composition of two isomorphisms of schemes is again an isomorphism.

theorem AlgebraicGeometry.IsIsomorphism.inverse {X Y : Scheme} {f : X ⟶ Y} (hf : IsIsomorphism f) : ∃ (g : Y ⟶ X), IsIsomorphism g

The inverse of an isomorphism is itself an isomorphism.