Let $K$ be a field of characteristic $0$, $X\_K$ a smooth projective curve over $K$. Denote by $\bar{K}$ the algebraic closure of $K$. The base change morphism $X\_{\bar{K}} \to X\_K$, induces via the pull-back map, a morphism of the Picard groups, $\mbox{Pic}(X\_K) \to \mbox{Pic}(X\_{\bar{K}})$. Is this map injective? If not true in general, is there a known condition on $K$ under which this holds true?



Injectivity under flat base change of the Picard group on smooth projective curves
This map is injective. There is a Hochschild-Serre spectral sequence with $E^{pq}\_2=H^p(\mathrm{Gal}(\bar{K}/K), H^q(X\_{\bar{K}},\mathbb{G}\_m))$ converging towards $H^{p+q}(X\_{K},\mathbb{G}\_m)$. This gives a first terms exact sequence
$$0\rightarrow H^1(\mathrm{Gal}(\bar{K}/K), \bar{K}^\*)\rightarrow \mathrm{Pic}(X\_K)\rightarrow \mathrm{Pic}(X\_{\bar{K}})\ ,$$but $\ H^1(\mathrm{Gal}(\bar{K}/K), \bar{K}^\*)$ is zero by Hilbert theorem 90. 


