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 $N$?**



Ext sheaves as extension by zero of locally free sheaves
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".


