Math 256B. Homework 14

Due Thursday 9 May

1(nc). Let k be a field. Find the dualizing sheaf of X := V (xy) in ℙ_{k}^{2} (the union of two lines

in ℙ_{k}^{2} intersecting in a point, with reduced induced subscheme structure).

Either express it as the restriction to X of a line sheaf ℒ on ℙ_{k}^{2} , or show that no such ℒ exists.

[Hint: Don’t work too hard.]

2. Let A be a ring. Show that Tor_{i}^{A}(M, N) ≅ Tor_{i}^{A}(N, M) for all A-modules M and

N , and for all i ∈ ℕ (without looking it up anywhere). Use a spectral sequence. [Hint: Use the opposite(s) of one or more categories.]

3(nc). Hartshorne III Ex. 8.1.

4. Hartshorne III Ex. 8.4.

For part (b), you may assume that Remark 7.1.1 is true for arbitrary commutative rings. For part (d), assume that Y is a nonsingular variety.

5(nc). Give an explicit example showing that (III, Thm. 8.8) is false if X and Y are allowed to be locally noetherian schemes instead of noetherian schemes.