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Determinat Determinant 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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