Let us consider $0\to E\to G\to H\to 0$ an exact sequence of coherent sheaves on a surface $S$ such that $rk G=rk E +1$ and the double dual of $H$ is $\mathcal{O}\_S(D)$ for some effective divisor $D$. Is it true that I have the following exact sequence:


$0\to detE\to det G\to i\_\*\mathcal{O}\_D\to 0$,


where $i$ is the inclusion map from $D$ to $S$?
In general, which is the behaviour of the determinant with respect to a short exact sequence of sheaves? 



Determinant and exact sequences of sheaves.
Provided that $S$ is smooth and $E,G$ are locally free, I find an exact sequence $0\to\det E\to\det G\to i\_\*\mathcal O\_D(\det G\vert\_D)\to 0$, as follows. Tensor the exact sequence $0\to\mathcal O\_S\to\mathcal O\_S(D)\to\mathcal O\_D(D)\to 0$ coming from the existence of a section with $\det E$ and use $\det E\otimes \mathcal O\_S(D)=\det G$. 


