Let $f: X\rightarrow Y$ be a morphism of varieties. Let $0\rightarrow F\rightarrow E\rightarrow G\rightarrow 0$ be a short exact sequence of locally free sheave of finite
rank. If direct images of above sheaves are locally free, then is it true that it
induces a short exact sequence $0\rightarrow f\_\*F\rightarrow f\_\*E\rightarrow f\_\*G\rightarrow 0$?



direct image functor
Let $X,Y$ be defined over the field $k$ and take $f$ to be the structure map $f:X\to {\rm Spec}\, k$. Then let $E\to G$ be a surjective morphism of sheaves that is not surjective on global sections, e.g., $$E=\mathcal O\_{\mathbb P^1}(-1)\oplus \mathcal O\_{\mathbb P^1}(-1)\to G=\mathcal O\_{\mathbb P^1}.$$
Then $f\_\*$ is just $H^0$ and the desired statement is false.


**EDIT:** *(to have an example mapping to an arbitrary scheme)*
Consider the base change of $f$ via $Y\to {\rm Spec}\, k$:
$$g: X\times\_{{\rm Spec}\, k} Y \to Y.$$ and let $\mathcal E:=p^\*E$ and $\mathcal G:=p^\*G$ where $p:X\times\_{{\rm Spec}\, k} Y \to X$ is the projection to $X$. Then $g\_\*\mathcal E\simeq H^0(X, E)\otimes\_k \mathcal O\_Y$ and $g\_\*\mathcal G\simeq H^0(X, G)\otimes\_k \mathcal O\_Y$, so again the desired statement is false.


