Let $\mathcal{E}$ be a locally free sheaf on $\mathbb{P}^n_A=\mathbb{P}^n\times_{Spec k} Spec A$, where $A$ is a finitely generated algebra over a field $k$. By a well known theorem (see e.g. Hartshorne's Algebraic Geometry, Thm 5.19) $H^0(\mathbb{P}^n_A, \mathcal{E})$ is a finitely generated $A$- module.

- Is it true that $H^0(\mathbb{P}^n_A, \mathcal{E})$ is a projective module?

Let $B$ be a finitely generated $k$-algebra, $f: Spec B \to Spec A$ a morphism and $f^*\mathcal{E}$ the pullback of $\mathcal{E}$ to $Spec B$.

- Is it true that $H^0(\mathbb{P}^n_A, \mathcal{E})\otimes _A B= H^0(\mathbb{P}^n_B, f^*\mathcal{E})$ ?

1 and 2 above are true when $\mathcal {E}$ is a direct sum of line bundles of the form $\mathcal O(n)$. I was wondering if they are true for general $\mathcal{E}$.