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# DeterminatDeterminant and exact sequences of sheaves.

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 $O_S(D)$ \mathcal{O}_S(D)$for some effective divisor$D$. Is it true that I have the following exact sequence:$0\to detE\to det G\to i_*\mathcal{O}_D\to 00$,$

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

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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 $\mathcal{O}_S(D)$ 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$0,$ where $i$ is the inclusion map from $D$ to $S$? In general, which is the behaviour of the determinat with respect to a short exact sequence of sheaves?

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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 $\mathcal{O}S(D)$ \mathcal{O}_S(D)$for some effective divisor$D$. Is it true that I have the following exact sequence:$0\to detE\to det G\to i*\mathcal{O}_D\to 0i_*\mathcal{O}_D\to 0$,$

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

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