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Let $S$ be a smooth projective surface and $C$ a smooth, irreducible curve contained in $S$. Let $E_1$ and $E_2$ be two vector bundles on $S$ having the same rank and assume they lie in a short exact sequence $$ 0\to E_1\stackrel{f}{\to} E_2\to A\to 0, $$ with $A\in\mathrm{Pic}(C)$. In this situation $E_1$ is called an elementary transformation of $E_2$. Now, the map $f$ induces an injection $\mathrm{det}(f):\mathrm{det}E_1\to \mathrm{det}E_2$, whose cokernel is again supported on $C$ and coincides with $\mathrm{det}E_2\otimes\mathcal{O}_C$. Is there any relation between $A$ and $\mathrm{det}E_2\otimes\mathcal{O}_C$ (e.g, a map from one to the other one)?

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  • $\begingroup$ In this context, you might be interested in the theory of Fitting ideals. See for instance the appendix of Mazur, B.; Wiles, A. Class fields of abelian extensions of Q. Invent. Math. 76. $\endgroup$ – Damian Rössler Nov 29 '12 at 18:23
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There is no relation. In fact, if $E_2$ is fixed then $\det E_2\otimes O_C$ is fixed, while $A$ can be taken to be any invertible quotient of $E_{2|C}$.

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