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Consider $\pi : X\rightarrow S$ a separated scheme of finite type over some base scheme $S$. The absolute Picard functor $\operatorname{Pic}_X$ from the category of $S$-schemes to that of abelian groups, is defined by $$\operatorname{Pic}_X(T):=\operatorname{Pic}(X_T)=\{\text{isomorphism classes of invertibles sheaves on } X_T=X\times_ST\}$$ It is claimed in every source I met (eg. on ncatlab, last paragraph of $2$) that this functor is never a separated presheaf in the Zariski topology, since given a non trivial invertible sheaf on $X_T$, it can be made trivial on some Zariski covering $\{T_i\rightarrow T\}$ so that the natural map $\operatorname{Pic}(X_T)\rightarrow \prod_i\operatorname{Pic}(X_{T_i})$ is not injective.

But can we be sure that there always exists a non trivial invertible sheaf on $X_T$ for some suitable $S$-scheme $T$ ? That is, could the Picard functor of some scheme $X$ actually send every $T$ to the trivial group ?

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Consider $T=\Bbb P^1\times S$. Then $X_T=X\times_S S\times \Bbb P^1=\Bbb P^1\times X$, and denoting $p: \Bbb P^1\times X\to \Bbb P^1$ the projection, we have that $p^*\mathcal{O}(1)$ is a nontrivial invertible sheaf on $X_T$.

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  • $\begingroup$ I see ! Thank you very much, that is a clever example =) $\endgroup$
    – Suzet
    Commented Feb 18, 2020 at 6:50

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