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 $N$?**