Assume that $f:X\to Y$ is a morphism of complex varieties, and the homomorphisms $H^i_\text{sing}(f)$ are bijective for $0\le i\le 3$ (though possibly $3$ is too much here:)). Under which assumptions on $X$ and $Y$ one can deduce that $\operatorname{Pic}(f)$ and $\operatorname{Br}(f)$ are bijective?
Probably, a well-known idea here is to apply the exponential sequence. Consequently, I would like to know whether the corresponding $H^i(-,\mathcal{O})(f)$ are bijective. Respectively, which assumptions are needed to deduce that $f$ induces isomorphisms on cohomology with coefficients in the structure sheaves? This is clearly true when $X$ and $Y$ are smooth projective; yet can one weaken these assumptions?
Lastly, what can one do about the characteristic $p>0$ case? I actually consider étale cohomology (with any prime to $p$ torsion coefficients) instead of singular cohomology. Can one say anything better than that the corresponding kernels and cokernels are uniquely prime-to-$p$-divisible?
