Let $X=\mathbb P(\mathcal E)$, where $\mathcal E$ is a locally free sheaf of rank $n+1$ on $Y$, an integral scheme of finite type over an algebraically closed field $k$. I'm trying to show that $\text{Pic }X\cong \text{Pic }Y\times \mathbb Z$. The only small point I'm stuck on is showing that every invertible sheaf on $X$ is of this form. I consider an invertible sheaf on $X$, $\mathcal M$, and it's restriction $\mathcal M_y$ to the fiber $X_y$ over a point $y$. Since this is an invertible sheaf on $\mathbb P^n$ we get that it must be $\mathcal O_{\mathbb P^n}(m)$ for some $m$. So I consider $\mathcal M\otimes \mathcal O_X (m)$, where the second term in the tensor product comes from the natural invertible sheaf $\mathcal O_X(1)$ on the projective bundle. My only question is how do I know that on another fiber, say $X_{y'}$, $\mathcal M\otimes \mathcal O_X(m)$ will be isomorphic to $\mathcal O_{X_{y'}}$ like it is on the fiber over $y$. Once I have this I can use something else I've shown to finish up.

Dear HNuer, a fundamental theorem on Chow groups describes the relation between the Chow ring $CH^\ast (\mathbb P(\mathcal E))$ of $X=\mathbb P(\mathcal E)$ and that $CH^\ast (Y)$ of $Y$ when $Y$ is a regular variety over a not necessarily algebraically closed field. If we call $p:\mathbb P(\mathcal E) \to Y $ the projection and $\xi$ the relative hyperplane bundle $\mathcal O_{\mathbb P(\mathcal E)}(1)$, we have $$ CH^\ast(\mathbb P(\mathcal E) )= CH^\ast (Y)[\xi]/ < \xi^n +c_1 (p^\ast \mathcal E)\xi^{n1} +\cdots+c_n (p^\ast \mathcal E)> $$ In particular $CH^1(\mathbb P(\mathcal E) )=p^\ast CH^1(Y)\oplus \mathbb Z \xi. $ (This is true even if $Y$ is not regular) If you remember that locally factorial varieties (for example regular or smooth varieties) satisfy $Pic(P)=CH^1 (P)$ , your formula is proved under this hypothesis of local factoriality. Edit: As the OP remarks in his comments below, the formula $Pic(\mathbb P(\mathcal E) )=p^\ast Pic(Y)\oplus \mathbb Z \xi $ is also true for any integral variety $Y$ over an algebraically closed field, locally factorial or not. The tool is then Grauert's semicontinuity theorem (cf. Hartshorne Chapter III, §12) rather than Chow groups. 


Thanks to Piotr Achinger for the idea to consider the euler characteristic. I was looking for an answer that doesn't use fancy machinery beyond what's presented in the main text in Hartshorne (so no generalized RiemannRoch). Here is one based on his suggestion: Denote by $\mathcal F$ the line bundle $\mathcal M\otimes \mathcal O_X(m)$ with notation as above. Then we have that on the fiber above our point $y$, $\mathcal F_y=\mathcal O_{X_y}$. Now since $Y$ is an integral scheme, it's connected, and since the Euler characteristic is constant in this case, we see that $\chi(\mathcal F)(y')$ is the constant function with value 1 since it takes that value at the point $y$. But since on $\mathbb P^n$ lines bundles have no cohomology between $H^0$ and $H^n$, we get that $1=\chi(\mathcal F)(y')=h^0(y',\mathcal F)+(1)^n h^n(y',\mathcal F)$. But this implies that on each fiber $\mathcal F_y'$ is the trivial line bundle or the canonical bundle (if $n$ is even, otherwise we get the result immediately since then the Euler characteristic would be 1) since in every othercase either both $h^0$ and $h^n$ vanish, or just $h^n$ vanishes but then $h^0$ is too large. Now to prove that we in fact always get the trivial line bundle on fibers, we consider $h^0(y',\mathcal F)$. By semicontinuity we get that since the only values possibly taken are 0 and 1, the set $S$ upon which 0 is achieved by $h^0(y',\mathcal F)$ is open (being the complement of the closed set when this function is $\geq 1$). Now considering everything above with $\mathcal F^{1}$ instead, we get that the set upon which 0 is acheieved for $h^0(y',\mathcal F^{1})$ is also open. But this must be the complement of $S$. So $S$ is both open and closed in a connected space, and thus it's either empty or the entire space. It can't be the entire space since our point $y$ is not in it. Hence it's empty and $h^0(y',\mathcal F)=1$ everywhere. This gives us that $\mathcal F_y=\mathcal O_{X_y}$ on every fiber. 

