It is a classical fact that a coherent analytic sheaf admits a local resolution by locally free sheaves (also known as a local syzygy). Griffiths and Harris' Principles of Algebraic Geometry (p. 696) gives a nice proof of this: by definition, at any point $z_0$ we have $$\mathscr{O}^p\to\mathscr{O}^q\to\mathscr{F}$$ on some open neighbourhood $U$ of $z_0$, and applying Oka's lemma gives $$\mathscr{O}^r\to\mathscr{O}^p\to\mathscr{O}^q\to\mathscr{F}$$ on some possibly smaller neighbourhood $U'\subseteq U$ of $z_0$, and we can repeat this process finitely many times, eventually terminating with an exact sequence, since the syzygy theorem tells us that eventually the stalk of the kernel at $z_0$ will be free.

It is natural to ask if this generalises to complexes of coherent sheaves. One answer to this is given in [SGA 6, §I, Exemples 5.11], which states that $$D^\mathrm{b}(X)_\mathrm{coh}\simeq D^\mathrm{b}(X)_\mathrm{perf}$$ or, in (vague) words, that complexes of coherent sheaves are perfect (i.e. locally quasi-isomorphic to a bounded complex of locally free sheaves). This is proved by what might fairly be called "general abstract methods" (in particular, it is proved in much more generality than just for smooth complex manifolds).

It is a classical fact that a coherent analytic sheaf admits a local resolution by locally free sheaves (also known as a local syzygy). Griffiths and Harris' Principles of Algebraic Geometry (p. 696) gives a nice proof of this: by definition, at any point $z_0$ we have $$\mathscr{O}^p\to\mathscr{O}^q\to\mathscr{F}$$ on some open neighbourhood $U$ of $z_0$, and applying Oka's lemma gives $$\mathscr{O}^r\to\mathscr{O}^p\to\mathscr{O}^q\to\mathscr{F}$$ on some possibly smaller neighbourhood $U'\subseteq U$ of $z_0$, and we can repeat this process finitely many times, eventually terminating with an exact sequence, since the syzygy theorem tells us that eventually the stalk of the kernel at $z_0$ will be free.

It is natural to ask if this generalises to complexes of coherent sheaves. One answer to this is given in [SGA 6, §I, Corollarie 5.10 & Exemples 5.11], which states that $$D^\mathrm{b}(X)_\mathrm{coh}\simeq D^\mathrm{b}(X)_\mathrm{perf}$$ or, in (vague) words, that complexes of coherent sheaves are perfect (i.e. locally quasi-isomorphic to a bounded complex of locally free sheaves). This is proved by what might fairly be called "general abstract methods" (in particular, it is proved in much more generality than just for smooth complex manifolds).