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Let $f:X\rightarrow Y$ be a proper surjective morphism over some base scheme $S$ of finite type, suppose $f$ restricts to an isomorphism over some open $U$ of $X$, we also suppose both $X$ and $Y$ are integral for simplicity.

I guess now $f$ is a covering for the cdh-topology. Now is there anyway to get an injection $$0\rightarrow \mathrm{Pic}^{\tau}_{Y/S}\rightarrow \mathrm{Pic}^{\tau}_{X/S}$$.

By the way is $\mathbb{G}_m$ a sheaf for the cdh-topology?

Could anyone give me a good reference for cdh topology treated in a general way instead serving only for motivic cohomology?

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  • $\begingroup$ i made a very stupid mistake, now i have editted my question. $\endgroup$
    – Heer
    Commented Jan 24, 2015 at 19:22
  • $\begingroup$ Are you sure that all (non-generic) points of $Y$ lift to $X$? $\endgroup$
    – Mikhail Bondarko
    Commented Jan 24, 2015 at 21:44
  • $\begingroup$ no, thanks for that point. i guess i need f to be surjective. i have edited my question $\endgroup$
    – Heer
    Commented Jan 25, 2015 at 8:52
  • $\begingroup$ I believe that a proper generically surjective morphism is always surjective. Yet it not clear that above any point $y$ of $Y$ there is an isomorphic point $x$ of $X$ (so, the question is whether the field for $x$ is isomorphic to that for $y$ and not just a finite extension of it). $\endgroup$
    – Mikhail Bondarko
    Commented Jan 25, 2015 at 9:00
  • $\begingroup$ now i am confused. According to wiki page for h-topology, this is just the example of cdh-cover given there $\endgroup$
    – Heer
    Commented Jan 25, 2015 at 12:59

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