Vanishing Cohomology Groups

Question

I have following two questions about two arguments concerning vanishing cohomology groups in a excerpt of Okonek, Schneider and Spindler's "Vector Bundles on Complex Projective Spaces"( page 20):

Firstly, the property "beeing Stein" I can be replaced by "affine". Now the questions:

1. Why the vanishing of $H^q(\mathbb{P}_1,\mathcal{O}_{\mathbb{P}_1})$ and $H^q(U,\mathcal{O}_{U})$ for $q>0$ already imply that

$H^q(U \times \mathbb{P}_1,\mathcal{O}_{U \times \mathbb{P}_1})=0$?

Intuitivelly I would suppose a relative version of Serre's criterion for affiness...

2. Why $H^{q+1}(U \times \mathbb{P}_1,J)$ also vanishes if we know that $U \times \mathbb{P}_1$ can be covered by affine sets?

Clear is that the $q+1$-th cohomologies of the affine covering sets vanishes, but does it already imply that $H^{q+1}(U \times \mathbb{P}_1,J)$ vanishes?

Or does there exist a generalized version of "sheaf axiom" for higher cohomology groups.

Especially that

$$H^{q+1}(U \times \mathbb{P}_1,J) \to H^{q+1}(U \times U_0,J) \oplus H^{q+1}(U \times U_1,J)$$

is injective?

Answer 1

Answer to question 2:

It is true in general that:

If $\{ U_1, \cdots, U_k\}$ is an acyclic cover with respect to the sheaf $\mathcal F$, then $H^m(X, \mathcal F) = 0$ for $m\ge k$.

The reason is that $H^i(X, \mathcal F)$ can be computed using Cech-cohomology using the cover $\{U_1, \cdots, U_k\}$ and all $k+1$ intersection of the open cover is empty.

In your special case you have a Stein covering $\{ U \times U_0, U\times U_1\}$. Thus $H^{q+1}(U\times P^1 , J) = 0$ for $q+1 \ge 2$.