Homework for Week 7

Part 1

Let X be the topological space with points {0,1} and open sets {}, {0}, and {0,1}. What can you say about the sheaves on X? Determine the stalks.

Note: You can think about the problems in Part 2 now.

Part 2

1. If f: F → G is a morphism of sheaves on X, then f can be surjective even if there is an open set U for which the morphism f_U: F(U) → G(U) is NOT surjective. Find an example of this when X consists of three points.
2. Prove that f: F → G is injective if and only if f_U: F(U) → G(U) is injective for each open set U in X.

• Part 1: Due Tuesday
• Part 2: Due Thursday