Let $X \to S$ be a morphism of schemes. The relative Picard functor from schemes over $S$ to abelian groups is usually defined by the formula $T \mapsto \text{Pic}(X \times_S T)/\text{Pic}(T)$ (e.g. in Kleiman's article and the book *Neron Models* by Bosch et al.). In favorable situations, this functor is representable, and in some less favorable situations its fppf-sheafification is.

There seems to be something wrong with this formula. Namely, it makes it look like $\text{Pic}(T)$ is a subgroup of $\text{Pic}(X \times_S T)$, which it need not be. For example, let $S = \text{Spec } \mathbb{R}$, $X = \text{Spec } \mathbb{C}$, and for $T$ take $\mathbb{P}^1_{\mathbb{R}}$ minus a degree two point. Then $\text{Pic}(T) \cong \mathbb{Z}/2\mathbb{Z}$, but $X \times_S T$ is $\mathbb{P}^1_{\mathbb{C}}$ minus two points and therefore has trivial Picard group.

So I have two questions. First, is it appropriate to simply take the cokernel of the map $\text{Pic}(T) \to \text{Pic}(X \times_S T)$ as the definition of the relative Picard functor? Second, under what hypotheses on $f : X \to S$ is $\text{Pic}(T) \to \text{Pic}(X \times_S T)$ injective for all $T$? Proposition 4 in Chapter 8 of *Neron Models* says that this is true e.g. if $f$ is quasi-compact and quasi-separated, and $f_*\mathcal{O}_X = \mathcal{O}_S$ universally. But what if $X$ is just, say, a geometrically connected variety over a field?