Projectivity of a morphism between schemes implies the existence of an exact sequence

Question

If $X\longrightarrow S$ is projective over a Noetherian scheme $S$ and $\mathsf{E}$ is a coherent sheaf on $X$, then for any affine $U\subseteq S$ there exists an exact sequence \begin{equation*} E_{0}\longrightarrow E_{1}\longrightarrow\mathsf{E}\longrightarrow 0\text{,} \end{equation*} where $E_{0}$ and $E_{1}$ are vector bundles over $X_{U}$. Why is that the case?

Answer 1

This is an application of a theorem of Serre, available in Hartshorne as theorem II.5.17:

Theorem: Let $X$ be a projective scheme over a noetherian ring $A$, let $\mathcal{O}(1)$ be a very ample invertible sheaf on $X$, and let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then there is an integer $n_0$ such that for all $n\geq n_0$, the sheaf $\mathcal{F}(n)$ can be generated by a finite number of global sections.

Using the theorem, there exists a surjection $\bigoplus \mathcal{O}_X^N \to \mathcal{F}(n)$ for some $n,N\geq 0$ and then twisting by $-n$ we find a surjection from a vector bundle on to $\mathcal{F}$. Then we repeat with the kernel of the surjection from a vector bundle.

The property that any coherent sheaf admits a surjection from a locally free sheaf of finite rank is often called the resolution property (or sometimes "having enough locally frees").