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.