Math 256B. Homework 7

Due Wednesday 13 March

1. Show directly from the definition (without using cohomology) that if i: X → Y is a closed embedding of noetherian schemes and if ℒ is an ample line sheaf on Y , then the line sheaf i^{∗}ℒ is ample on X.

2. Hartshorne II Ex. 7.5. Do not use cohomology for this exercise.

In part (d), for partial credit you may assume that X is proper over Spec A, or for slightly more partial credit you may instead assume that X is reduced. For parts (d) and (e), “very ample” means very ample over Spec A.

3. Hartshorne III Ex. 5.7(a)–(c). For part (a), use cohomology.