Let $X$ be a variety over a field $k$, $\mathcal F$ a locally free $\mathcal O_X$-module. Has the functor on locally free sheaves 'tensoring with $\mathcal F$' any exactness property (is it right/left exact or not at all) ?
If $\mathcal F$ is of rank $1$, it is known that this functor is exact. What happens for greater rank ?