Let $X$ be a normal, complex, projective variety, and let $X'\subset X$ be an open subset such that the complement $X\setminus X'$ has codimension 2. Let $j:X'\to X$ be the inclusion map. If $0\to\mathcal{F}\to\mathcal{G}\to\mathcal{H}\to0$ is a short exact sequence of locally free sheaves on $X'$, will $0\to j_*\mathcal{F}\to j_*\mathcal{G}\to j_*\mathcal{H}\to0$ be an exact sequence on $X$? In my case I take $\mathcal{F}=\mathcal{O}_{X'}$, but I suspect the statement should hold in general. I have been trying to show that $R^1j_*\mathcal{O}_{X'}=0$ but didn't get very far.
Any hints/answers/suggestions are appreciated, thanks!
In general this is false. As an example, take $X=\mathbb{P}^2$ and $X'$ the complement of $p=(0,0,1)$. You have a complex, $0\to O_X\to O_X(1)^2\to O_X(2)$, given by the first map $(y,-x)$ and the second by $(x,y)$, where $x,y,z$ are the coordinates of $X$. You can easily check that restricted to $X'$, this is exact and surjective on the right, but when you apply $j_*$, you just get back the above sequence which is not surjective on the right.