### Background

Throughout, let $X$ be a smooth complex manifold.

1. 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.

2. 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).

### Question

Is there a generalisation of the proof method of 1 to the setting of 2? That is, is there a nice manual construction of a local syzygy for a **complex** of coherent analytic sheaves?