# Ext sheaves as extension by zero of locally free sheaves

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 $$N$$?

• 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.... – Sándor Kovács Apr 22 '19 at 23:30

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".