Let $f:X\to Y$ be a morphism of schemes. Then there exists a functor $f_*:{Sh}X\to {Sh}Y$ with $f_*\mathcal{F}(U)=\mathcal{f^{-1}(U)}$ whenever $\mathcal{F}$ is asheaf on $X$.

It is proved that the direct image functor is a left adjoint functor. Now the question is which conditions are needed to impose on $f$, $X$, $Y$ or sheaves on $X$ to deduce that the direct image functor translate an exact sequence of sheaves on $X$ to an exact sequence of sheaves on $Y$? i.e. When the direct image functor is an exact functor?