Schemes

Exercises 2

Schemes – examples are important

Let C be the category of locally ringed spaces. An affine scheme is an object in C isomorphic in C to a pair of the form (Spec A, O_{A}). A scheme is an object (X, O_{X}) of C such that every point x ∈ X has an open neighbourhood U ⊂ X such that the pair (U, O_{X}|_{U}) is an affine scheme.

1. Suppose that X is a scheme whose underlying topological space has 2 points. Show that X is an affine scheme.

2. Give an example of an affine scheme (X, O_{X}) and an open U ⊂ X such that (U, O_{X}|U) is not an affine scheme.

3. Given an example of a pair of affine schemes (X, O_{X}), (Y, O_{Y}), an open subscheme (U, O_{X}|_{U}) of X and a morphism of schemes (U, O_{X}|_{U}) → (Y, O_{Y}) that does not extend to a morphism of schemes (X, O_{X}) → (Y, O_{Y}).

4. Give an example of a scheme X, a field K, and a morphism of ringed spaces Spec K → X which is NOT a morphism of schemes.

5. Do all the exercises in Hartshorne, Chapter II, Sections 1 and 2... Just kidding!

Remark. When (X, O_{X}) is a ringed space and U ⊂ X is an open subset then (U, O_{X}|_{U}) is a ringed space. Notation: O_{U} = O_{X}|_{U}. There is a canonical morphisms of ringed spaces

j : (U, O_{U}) ⟶ (X, O_{X}).

If (X, O_{X}) is a locally ringed space, so is (U, O_{U}) and j is a morphism of locally ringed spaces. If (X, O_{X}) is a scheme so is (U, O_{U}) and j is a morphism of schemes. We say that (U, O_{U}) is an open subscheme of (X, O_{X}) and that j is an open immersion. More generally, any morphism j′ : (V, O_{V}) → (X, O_{X}) that is isomorphic to a morphism j : (U, O_{U}) → (X, O_{X}) as above is called an open immersion.