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I have a question more about terminology than anything else: If $f:X\rightarrow Y$ is an isomorphism of schemes, then what does it mean to say "the invertible sheaf $\mathscr{L}$ on $X$ corresponds to the invertible sheaf $\mathscr{M}$ on $Y$?" Naturally, it makes sense to assume that this statement means $f_{\ast}\mathscr{L}=\mathscr{M}$ or $\mathscr{L}=f^{\ast}\mathscr{M}$, but I wasn't sure...

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Are you sure that it makes a difference, given that $f$ is an isomorphism? –  Mariano Suárez-Alvarez Feb 14 '11 at 20:45
    
If $f$ is an isomorphism, then there shouldn't be any problem in interpreting this statement as long as you believe that the concept of a sheaf on a scheme is well-defined at all. –  Qiaochu Yuan Feb 14 '11 at 21:28
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