### 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, 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).


### 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?



Resolving complexes of coherent analytic sheaves
If I interpret your question correctly, then I believe there is indeed such a construction.


The construction relies first of all on the existence of local resolutions as in your point 1. Secondly, it relies on the fact that vector bundles are projective objects over Stein domains, for example over any ball in some local coordinates. This fact follows from the local to global spectral sequence of Ext. It follows from the second point that one has a *"Horseshoe lemma"* over any Stein domain, cf., i.e., Weibel, Homological Algebra, Lemma 2.2.8.


Then, one may construct a local *Cartan-Eilenberg resolution* $P\_{\bullet,\bullet}$ of any bounded complex $\mathcal{F}\_\bullet$ of coherent sheaves. This construction is based on taking local resolutions of each $\mathcal{F}\_k$, $B\_k(\mathcal{F})$, $H\_k(\mathcal{F})$, and using the Horseshoe lemma repeatedly in an explicit way. The Cartan-Eilenberg resolution is a double complex satisfying various nice properties. In particular, there exists an explicit quasi-isomorphism from the total complex $\mathrm{Tot}\_\bullet(P\_{\bullet,\bullet})$ of $P$ to $\mathcal{F}\_\bullet$, see i.e., Weibel, Section 5.7.


