Q: Computation of relative Picard Group Let $X=\mathbb{P}_{\mathbb{C}}^1$ and let $A$ be a scheme of finite type over $\mathbb{C}$. 
Suppose I want to compute the relative Picard group $\operatorname{Pic}(X \times A)/p_1^*(\operatorname{Pic}(A))$. 
I know that $X \times A \to A$ is covered by open subset $U \cong \operatorname{Spec}{A}[z]$ and $V \cong \operatorname{Spec}{A}[w]$. 
Can I still use Cech cohomology to compute $\operatorname{Pic}(X \times A)/p_1^*(\operatorname{Pic}(A))$? 
It seems like the only difference would be that now my invertible functions include stuff coming from $A$.
 However, this doesn't really affect my computation, I still get $\mathbb{Z}$ every time. 
