Exercise II 7.9 (a) in Hartshorne's Algebraic geometry is
Let $X$ be a regular noetherian scheme, and $E$ a locally free coherent sheaf of rank $\geq 2$ on $X$.
- Show that $\DeclareMathOperator{\Pic}{Pic} \Pic \mathbb P(E) \cong \Pic X \times \mathbb Z$
- If $E'$ is another locally free coherent sheaf on $X$, show that $\mathbb P(E') \cong \mathbb P(E)$ (over $X$) if and only if there is an invertible sheaf $L$ on $X$ such that $E' \cong E \otimes L$.
First note that for 1. we clearly need that $X$ is connected, otherwise we would get more factors of $\mathbb Z$. So let's assume $X$ is connected.
I wonder if I need the regularity of $X$ to show part 1. Here is how my proof goes:
Denote by $\pi: \mathbb P(E) \to X$ the canonical map and consider the sequence $$\tag{$*$} 0 \to \Pic X \xrightarrow{\pi^*} \Pic \mathbb P(E) \xrightarrow{|_F} \Pic F \to 0,$$ where $F = \pi^{-1}(x_0) \cong \mathbb P^n$ is some fiber. If we can show that this sequence is exact we are done, because $\Pic F \cong \mathbb Z$, so the sequence splits. Clearly the restriction to $F$ is surjective, because $\mathcal O_\pi(1)|_F$ generates $\Pic F$. So we even see that the splitting $1 \mapsto \mathcal O_\pi(1)$ is canonical.
To show that the pull-back $\pi^*$ is injective, I don't have a really good argument which fits the scope of the exercise, but this is certainly true. For example this is part of the splitting principle (cf. Fulton, Intersection Theory, Corollary 3.1). Does anyone know of a simpler argument?
Clearly ($*$) is a complex, the pull-back of a line bundle is trivial on the fiber $F$. To see the exactness in the middle, let $L$ be any line bundle on $\mathbb P(E)$ that restricts to the free sheaf $\mathcal O_F$. Then $L$ is also free in a neighbourhood $V = \pi^{-1}(U)$ of $F$. In particular, the set $$Y = \{\,x \in X : L|_{\pi^{-1}(x)} \cong \mathcal O_{\pi^{-1}(x)}\,\}$$ is open. Twisting $L$ with $\mathcal O_\pi(1)$ we see that the sets $$\{\,x \in X : L|_{\pi^{-1}(x)} \cong \mathcal O_{\pi^{-1}(x)}(d)\,\} = \{\,x \in X : (L \otimes \mathcal O_\pi(-d))|_{\pi^{-1}(x)} \cong \mathcal O_{\pi^{-1}(x)}\,\}$$ are open as well. Clearly those sets form a disjoint open cover of $X$, so the connectedness yields $Y = X$. So we can obtain an open affine cover $\{U_i\}$ of $X$ such that $V_i = \pi^{-1}(U_i) \cong U_i \times \mathbb P^n$, and there are sections $s_i: \mathcal O_{V_i} \xrightarrow{\cong} L|_{V_i}$. The transition functions $$g_{ij} = s_i^{-1} \circ s_j \in \Gamma(V_i \cap V_j, \mathcal O_{\mathbb P(E)})$$ are pull-backs of functions $f_{ij} \in \Gamma(U_i \cap U_j, \mathcal O_X)$ (if $X$ is separated this follows from Prop 5.13, I hope this is still true in general?), which define a line bundle $M$ on $X$ such that $\pi^* M \cong L$.
So I don't think I need that $X$ is regular for this (at least if $X$ is separated). Did I make a mistake?
It could be that we only need the regularity in part 2., I still have to think a bit more about that.