I'm trying remark 2.11 of this.
Let $f : X \to Y$ be a morphism of schemes, and assume that $\mathscr O_Y \to f_* \mathscr O_X$ is an isomorphism.
By Leray spectral sequence we have the canonical map $H^1(Y, f_*\mathscr O^*) \to H^1(X, \mathscr O^*)$.
Since $H^1$ is equal to its Cech version, and since $\check{H}^1$ is equal to $\operatorname{Pic}$, compositing $\mathscr O_Y \to f_* \mathscr O_X$, we have $\operatorname{Pic}Y \to \operatorname{Pic}X$.
Does this coincide with the pullback $f^* : \operatorname{Pic}Y \to \operatorname{Pic}X$?
I know that the Cech-derived isomorphism is induced by a spectral sequence $\check{H}^p(X, \mathscr{H}^q (-)) \Rightarrow H^n(X, -)$. (where $\mathscr{H}^q$ is the right derived functor of the forgetfull functor from the category of sheaves to one of presheaves on $X$.)
I think if we can take "nice" injective resolution of $\mathscr{O}^*$, we can compute the desired map between Pic.
Thank you very much!