Schemes

Exercises 5

Schemes – invertible sheaves are important

Feel free to quote results from Hartshorne or EGA.

An invertible O_{X}-module on a locally ringed space (X, O_{X}) is a sheaf of O_{X}-modules L such that every point has an open neighbourhood U ⊂ X such that L|_{U} is isomorphic to O_{U} as O_{U}-module. We say that L is trivial if it is isomorphic to O_{X} as a O_{X}-module.

1. General facts.

(a) Show that an invertible O_{X}-module on a scheme X is quasi-coherent.

(b) Suppose X → Y is a morphism of ringed spaces, and L an invertible O_{Y} -module. Show that f^{∗}L is an invertible O_{X} module.

2. Algebra.

(a) Show that an invertible O_{X}-module on an affine scheme Spec A corresponds to an A-module M which is (i) finite, (ii) projective, (iii) locally free of rank 1, and hence (iv) flat, and (v) finitely presented. (Feel free to quote things from last semesters course; or from algebra books.)

(b) Suppose that A is a domain and that M is a module as in (a). Show that M is isomorphic as an A-module to an ideal I ⊂ A such that IA_{𝔭} is principal for every prime 𝔭.

3. Simple examples. (a) Let k be a field. Let A = k[x]. Show that X = Spec A has only trivial invertible O_{X}-modules.

(b) Let A be the ring

A = {f ∈ k[x] | f(0) = f(1)}.

Show that X = Spec A has a nontrivial invertible O_{X}-module, unless k = 𝔽_{2}. (Hint: Think about Spec A as identifying 0 and 1 in 𝔸_{k}^{1} = Spec k[x].)

(c) Same question for the ring A = k[x^{2}, x^{3}] ⊂ k[x] (except now k = 𝔽_{2} works as well).

4. Higher dimensions.

(a) Prove that every invertible sheaf on two dimensional affine space is trivial. More precisely, let 𝔸_{k}^{2} = Spec k[x, y] where k is a field. Show that every invertible sheaf on 𝔸_{k}^{2} is trivial. (Hint: One way to do this is to consider the corresponding module M, to look at M ⊗_{k[x,y]} k(x)[y], and then use 3(a) to find a generator for this; then you still have to think. Another way to is to use 2(b) and use what we know about ideals of the polynomial ring: primes of height one are generated by an irreducible polynomial; then you still have to think.)

(b) Prove that every invertible sheaf on any open subscheme of two dimensional affine space is trivial. More precisely, let U ⊂ 𝔸_{k}^{2} be an open subscheme where k is a field. Show that every invertible sheaf on U is trivial. Hint: Show that every invertible sheaf on U extends to one on 𝔸_{k}^{2}. Not easy; but you can find it in Hartshorne.

(c) Find an example of a nontrivial invertible sheaf on a punctured cone over a field. More precisely, let k be a field and let C = Spec k[x, y, z]/(xy−z^{2}). Let U = C \{(x, y, z)}. Find a nontrivial invertible sheaf on U. Hint: It may be easier to compute the group of isomorphism classes of invertible sheaves on U than to just find one. Note that U is covered by the opens Spec k[x, y, z, 1/x]/(xy − z^{2}) and Spec k[x, y, z, 1/y]/(xy − z^{2}) which are “easy” to deal with.