Let $X$ be a scheme of finite type over $\mathbb C$. One might be interested in morphisms in the derived category $D(X)$ of coherent sheaves on $X$, that are morphisms $f:E^\bullet\to F^\bullet$ of complexes of vector bundles. However, these are morphisms in $D(X)$, thus not so easy (for me) to handle. I know that when $X$ has enough locally frees, one can choose a representative of $f$ such that $f$ is an actual map of complexes. This was just my motivation for the following question:
When does $X$, i.e. Coh$(X)$, have enough locally frees?
Thanks!
When $X$ has enough locally free sheaves, one says that $X$ has the resolution property. With this keyword you will find lots of literature. It is an open problem if every (semi)separated scheme of finite type over a field has the resolution property; at least no counterexamples are known. A simple example which illustrates the separated assumption is the affine plane with doubled origin, since here every locally free sheaf is free.
Related: math.SE/849958