Schemes

Excercises 1

Sheaves

1. Carefully prove that a map of sheaves of sets is an epimorphisms (in the category of sheaves of sets) if and only if the induced maps on all the stalks are surjective.

2. Let f : X → Y be a map of topological spaces. Prove pushforward f_{∗} and pullback f^{−1} for sheaves of sets form an adjoint pair of functors.

3. Let j : U → X be an open immersion. Show that j^{−1} has a left adjoint j_{!} on the category of sheaves of sets. Characterize the stalks of j_{!}(G). (Hint: j_{!} is called extension by zero when you do this for abelian sheaves... )

4. Let F be an abelian sheaf on X. Show that F is the quotient of a (possibly very large) direct sum of sheaves all of whose terms are of the form

j_{!}(\underline{ℤ}_{U})

where U ⊂ X is open and \underline{ℤ}_{U} denotes the constant sheaf with value ℤ on U.

Remark. In the category of abelian sheaves the direct sum of a family of sheaves {F_{i}}_{i∈I} is the sheaf associated to the presheaf U ↦ ⊕F_{i}(U). Consequently the stalk of the direct sum at a point x is the direct sum of the stalks of the F_{i} at x.