Math 256B. Homework 6 Due Wednesday 6 March

1. [This exercise generalizes II Ex. 5.9b.] Let A be a noetherian ring, let S be a graded ring, finitely generated by S_{1} over S_{0} , and assume that S_{0} = A. Let M be a finitelygenerated graded S-module. By (II Ex. 5.9a), there is a natural map α: M → Γ_{∗}(M̃) . Let X = Proj S.

Show that the map α is an isomorphism in all large enough degrees; i.e., α_{d}: M_{d} → Γ(X, M̃(d))

is an isomorphism for all d ≫ 0 . Use cohomology. [Hint: Use methods from the proof of (III, 5.2).]

2. Hartshorne III Ex. 5.3.

3. [WITHDRAWN] 4(nc). Hartshorne III Ex. 5.10.