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Let $X$ be an irreducible, reduced, projective curve over an algebraically closed field, with at worst nodes as singularities. Let $\mathcal{F}$ be a trivial vector bundle on $X$ of rank $r$. Consider a short exact sequence of the form $$0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0,$$ where $\mathcal{F}'$ and $\mathcal{F}''$ are coherent, torsion-free sheaves. What is the formula for $\chi(\mathcal{H}om(\mathcal{F}',\mathcal{F}''))$? Is there any reference for this?

I know the answer to the question in the case $X$ is smooth.

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  • $\begingroup$ For irreducible (reduced) curves, Riemann-Roch works essentially as in the smooth case: see for instance Hartshorne, ch. IV, exercise 1.9. $\endgroup$
    – abx
    Jul 6, 2015 at 17:08
  • $\begingroup$ @abx: The exercise only deals with line bundles and does not say anything about coherent sheaves in general. $\endgroup$
    – Ron
    Jul 6, 2015 at 17:09
  • $\begingroup$ We are talking about vector bundles here, aren't we? Then the extension is straightforward, any vector bundle is a successive extension of line bundles. $\endgroup$
    – abx
    Jul 6, 2015 at 17:11
  • $\begingroup$ Only $\mathcal{F}$ is locally free, I do not know about the rest. I have edited the question. Sorry. $\endgroup$
    – Ron
    Jul 6, 2015 at 17:12

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