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Let $X$ be a complex projective manifold and let $\phi \colon E \hookrightarrow F$ an injection of locally free sheaves. Then we have a sequence of coherent sheaves $$ 0 \to E \to F \to F/E \to 0 $$ that we can dualize: $$ 0 \to (F/E)^\vee \to F^\vee \to E^\vee \to \mathcal{Ext}^1_{\mathcal{O}_X}(F/E, \mathcal{O}_X) \to 0 $$ It is clear that the support of this ext sheaf is the locus $Z$ where $F/E$ is not locally free.

My questions are:

1. What are some natural conditions to have a locally free sheaf $N$ on $Z$ such that $\mathcal{Ext}^1_{\mathcal{O}_X}(F/E, \mathcal{O}_X) = j_! N$, where $j\colon Z \to X$ is the inclusion?

2. Once we are in the situation above, how to compute $N$?

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  • $\begingroup$ Since $\mathcal{Ext}^1_{\mathcal{O}_X}(F/E, \mathcal{O}_X)$ is supported on $Z$, $N=\mathcal{Ext}^1_{\mathcal{O}_X}(F/E, \mathcal{O}_X)$ (or you can add "restricted to $Z$") should work, no? So, this does not simplify the computation.... $\endgroup$ – Sándor Kovács Apr 22 '19 at 23:30
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Assume $r(E) = e$, $r(F) = f$ (with $e \le f$). Then the natural scheme structure of $Z$ is given by the Fitting ideal, i.e., the image of the map $$ \Lambda^e(E) \otimes \Lambda^e(F^\vee) \to \mathcal{O}_X. $$ Such $Z$ is usually called the degeneracy locus for the morphism of sheaves.

If $Z$ is defined like that then $$ \mathcal{E}xt^1(F/E,\mathcal{O}_X) \cong j_*N $$ for a sheaf $N$. If, moreover, the next degeneracy locus defined by the ideal $$ \Lambda^{e-1}(E) \otimes \Lambda^{e-1}(F^\vee) \to \mathcal{O}_X $$ is empty, then the sheaf $N$ is locally free. You can find these results in the "Commutative Algebra with a View Towards Algebraic Geometry" by Eisenbud, search there for "Fitting Ideals".

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