Take the tour ×  suppose $X$ is a smooth variety, with a finite
open covering $U_i, i=1,....,k$. $F$ and $G$ are coherent
sheaves on $X$, and $G$ is locally free.  Suppose on each $U_i$,
there is a sheaf morphism $h_i: F_{U_i}-->G_{U_i}$.  Can this be
extended on X ?  i started by taking push-forward, for each i:
$$(h_i)_*: (h_i)_*F_{U_i} ---> (h_i)_*G_{U_i} = (h_i)_*O_{U_i}
\otimes G. $$ Restrict the map $(h_i)_*$ on the subsheaf $F$ to
get a morphism into right hand side.  Using the covering $U_i$
of $X$, can we conclude that the image of restriction, is in
$\cap_i (h_i)_*\cal {O_{U_i}} \otimes G$, (which is equal to
$G$) ?  share|improve this question This is true if $F$ is
locally free (and $G$ can be arbitrary, smoothness of $X$ is not
important either) and if morphisms $h_i$ agree on intersections
$U_i \cap U_j$. To prove replace $Hom(F,G)$ by $H^0(X,F^*\otimes
G)$. –  Sasha Nov 21 '12 at 11:17 I think the question -as it
stands now- is not well phrased. For example you give no
compatibility conditions