TITLE: Line bundles trivial after extension of the base-field
QUESTION [10 upvotes]: Let k be a field and let X be scheme over k. Let K be a field extension of k and denote by $X_K$ the base-change of X to Spec K. Under what conditions is the canonical map of Picard groups $Pic(X)\to Pic(X_K)$, induced by the projection, injective? I know that this is true if X is geometrically integral and proper over k, but what if X is only of finite type, separated and geometrically integral over k?

REPLY [8 votes]: Perhaps it is worth recording a simple example: let $X_k= \text{Spec}(k[x,y]/(x^2+y^2-1))$. Then $\text{Pic}(X_{\mathbb R}) = \mathbb Z/(2)$ while $\text{Pic}(X_{\mathbb C}) = \mathbb 0$, see Fossum's book "Divisor class groups of Krull domains", Prop 11.8.