# global sections of locally free sheaf on projective space

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.

1. 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 $$Spec B$$.

1. 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 $$\mathcal{E}$$.

1) Let $$A = k[x,y,z]$$, $$n = 1$$. Note that $$H^1(\mathbb{P}^n_A,O(-2)) \cong A.$$ Consider the extension $$0 \to O(-2) \to E \to O \oplus O \oplus O \to 0$$ whose extension class is $$(x,y,z)$$. Then the cohomolopgy exact sequence $$0 \to H^0(\mathbb{P}^n_A,E) \to A \oplus A \oplus A \stackrel{(x,y,z)}\to A$$ shows that $$H^0(\mathbb{P}^n_A,E)$$ is reflexive but not locally free (hence not projective). In fact, this is the simplest example of a reflexive non-locally free sheaf.
2) Take $$A = k[x,y,z,w]$$ and define $$E$$ as the extension $$0 \to O(-2) \to E \to O \oplus O \oplus O \oplus O \to 0$$ whose extension class is $$(x,y,z,w)$$. Let $$B = k$$ with the morphism $$A \to B$$ defined by $$x,y,z,w \mapsto 0$$. Then $$f^*E \cong O(-2) \oplus O \oplus O \oplus O \oplus O$$, hence $$H^0(\mathbb{P}^n_B,f^*E) = B \oplus B \oplus B \oplus B.$$ On the other hand, tensoring $$0 \to H^0(\mathbb{P}^n_A,E) \to A \oplus A \oplus A \oplus A \stackrel{(x,y,z,w)}\to A \to B \to 0$$ by $$B$$ (over $$A$$), we deduce $$H^0(\mathbb{P}^n_A,E) \otimes_A B \cong Tor_2^A(B,B) \cong B^{\oplus 6}.$$