Given $\phi:X\rightarrow Y$ a surjective morphism of $k$-algebraic varieties ($k$ separably closed), I wanted to find how the write an exact sequence involving Pic(X) and Pic(Y). We can use the long exact sequence of the Leray spectral sequence for $\mathbb{G}_m$ to get:
$0\rightarrow H^1(Y, \phi_{*}\mathbb{G}_{m,X})\rightarrow H^1(X, \mathbb{G}_{m,X})\rightarrow H^0(Y, R^1\phi_{*}\mathbb{G}_{m,X})\cdots$
But I do not know how to use the morphism $\mathbb{G}_{m,Y}\rightarrow \phi_{*}\mathbb{G}_{m,X}$ to have $Pic(Y)$ involved instead of $H^1(Y, \phi_{*}\mathbb{G}_{m,X})$.

  • $\begingroup$ If $\phi$ is cohomologically flat in dimension 0, then the morphism $\mathbb{G}_{m,Y}\rightarrow \phi_{*}\mathbb{G}_{m,X}$ is an isomorphism. See mathoverflow.net/questions/61289/… $\endgroup$ – anon May 29 '15 at 13:03
  • $\begingroup$ Thank you for your help but I think the condition is too restrictive (for me). Do you know where it is written how to derive picard-brauer exact sequence for rings homomorphism from the Leray spectral sequence? $\endgroup$ – user052715 May 29 '15 at 16:20

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