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Let $X$ be non-singular projective variety. Consider a smooth closed subvariety say $Y$ of $X$.

Let $F$ be a torsion-free coherent sheaf on $X$. Then is there any description of global sections of $F|_Y$ in terms of global sections of $F$ on $X$?

There is a short exact sequence $$0\rightarrow I_Y\rightarrow O_X\rightarrow O_Y.$$

However if we tensor with $F$ we get only right exactness. Another issue is that I don't know if $O_Y\otimes F$ can be written as $F|_Y$ since I know the projection formula only for $F$ locally free.

Amy insight would be helpful.

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1 Answer 1

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$O_Y\otimes F$ is precisely $F|_Y$ for any coherent sheaf. Even for locally free sheaves, in general, you only have a map $H^0(X,F)\to H^0(Y, F|_Y)$, which may be neither injective nor surjective.

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  • $\begingroup$ But the projection formula as given in Hartshorne is $f_*(F\otimes f^*E)\simeq f_*F\otimes E$ where $E$ is locally free of finite rank. Is it in general valid for any coherent sheaf? $\endgroup$
    – user349424
    Commented Apr 5, 2017 at 6:26
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    $\begingroup$ Is $f$ here your closed embedding map? In this particular case, what I wrote is correct. For general projection formula, one needs $E$ to be locally free. Try doing without the projection formula, just following definitions. $\endgroup$
    – Mohan
    Commented Apr 5, 2017 at 11:51

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