Let $X$ be a smooth projective variety of dimension $n\geq 3$. Let $Z$ be a smooth subvariety of $X$ of codimension at least 3. Let $Y$ be the blow up of $X$ along $Z$ and let $f:Y\longrightarrow X$ be the blow up map. Suppose we have an exact sequence of sheaves:
$0\longrightarrow F\longrightarrow G\longrightarrow H\rightarrow 0$ on $Y$.
Consider the pushforward of this exact sequence:
$0\longrightarrow f_*F\longrightarrow f_*G\longrightarrow f_*H$ on $X$. By definition of pushforward, it is left exact. Is it exact as well?
We just need to check the surjectivity of the last map at stalk level. On $X\setminus Z$ it is surjective. If we take a point in $z\in Z$, how can we check this?