Math 256B. Homework 13 (Corrected)

Due Wednesday 1 May

1. Let k be a field. Compute the dualizing sheaf of the (reduced) scheme

X = V (z) ∪ V (x, y) ⊆ ℙ_{k}^{3} .

(This is the union of a plane and a line in ℙ_{k}^{3} , which intersect at a point.)

2. Prove that the derived-functor cohomology class determined by the Čech cocycle α of (III, Remark 7.1.1) is invariant under automorphisms of ℙ_{k}^{n}. Do this by direct computation.

[Hints: You may find (II, Example 7.1.1) and Exercise 2 on Homework 12 useful. Remember elementary matrices. For information on differentials, see pages 172–173 and (II, 8.13) in Hartshorne.]

3(nc). Let k be a field, let n ∈ ℤ_{>0} , and let X = ℙ_{k}^{n}. Conclude from the previous exercise that there is a canonical trace map

t: H^{n}(X, ω_{X}) → k .