Global sections of a sheaf restricted to a closed subscheme

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.

$O_Y\otimes F$ 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.
• 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? – user349424 Apr 5 '17 at 6:26
• 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. – Mohan Apr 5 '17 at 11:51