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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?

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  • $\begingroup$ In the second, what do you mean by "much more" general than smooth manifolds? This equivalence depends on the smoothness (more precisely, the regularity) of the scheme. The two are different if the scheme is only an lci. Furthermore, I did not read SGA, but this seems to be algebraic, not analytic. $\endgroup$
    – Z. M
    Nov 28, 2022 at 7:13
  • $\begingroup$ So the original result in SGA 6, §1 (Corollaire 5.10) is for ringed toposes with enough points and such that all stalks are of finite tor-dimension — the specific example that lets you recover smooth complex manifolds is that of a ringed space with regular local rings. $\endgroup$
    – Tim
    Nov 28, 2022 at 12:37
  • $\begingroup$ so I agree that maybe "much more general" is a small exaggeration, but "more general" is fair, I think, in that it's not just for complex manifolds :-) $\endgroup$
    – Tim
    Nov 28, 2022 at 12:43
  • $\begingroup$ I am not familiar with complex geometry, but there is a stronger form of Oka's coherence for arbitrary compact Stein subsets. See Clausen–Scholze, Complex Thm 10.5 and Thm 10.10. It looks like that then you can pick a compact Stein neighborhood and pick a locally free resolution, just as if it were completely algebraic. $\endgroup$
    – Z. M
    Nov 28, 2022 at 17:14

1 Answer 1

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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.

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  • $\begingroup$ Thank you! So it seems like the key ingredient which lets you apply "classical" constructions is exactly this fact that vector bundles are projective over Stein spaces. I think a reference for this fact is Forster's "Zur Theorie der Steinschen Algebren und Moduln", but my German is not good enough to check... but like you say, it should follow from the local to global Ext spectral sequence. $\endgroup$
    – Tim
    Dec 13, 2022 at 15:02
  • $\begingroup$ Indeed, the argument as written uses the useful fact that vector bundles over Stein spaces are projective, although I suppose since you anyway need to shrink the neighborhood when obtaining the resolutions, by further shrinking the neighborhood, you could probably make do with the more elementary fact that trivial vector bundles are projective. $\endgroup$
    – Richard Lärkäng
    Dec 13, 2022 at 20:18
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    $\begingroup$ I am not aware of the history of the statement that vector bundles over Stein spaces are projective. I first learnt about it in O'Brian-Toledo-Tong, "The trace map and characteristic classes for coherent sheaves", where it is mentioned in passing after Lemma 1.6 without any indication of proof. $\endgroup$
    – Richard Lärkäng
    Dec 13, 2022 at 20:20
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    $\begingroup$ It seems indeed also to be proven in one direction of Satz 6.2 in the article of Forster. From a quick look, the key step in that proof seems to be Satz 2.1, which I believe shows that the global sections functor establishes an equivalence of categories between coherent sheaves on a Stein space and so-called "Stein modules". Then, Satz 6.2 is proven simply by the fact that sheaf $\mathcal{H}om(P,\bullet)$ is exact if P is a vector bundle, and taking global sections $\Gamma(X,\bullet)$ is exact on the category of coherent sheaves on a Stein space. $\endgroup$
    – Richard Lärkäng
    Dec 13, 2022 at 20:21
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    $\begingroup$ O'Brian–Toledo–Tong's work is also where I learnt basically everything I know about Stein spaces, but it's true that they often just state things without mentioning any proofs... but the Forster article seems like a good reference (shockingly I couldn't find this in Grauert and Remmert's "Theory of Stein Spaces", though maybe I just wasn't looking in the right places) $\endgroup$
    – Tim
    Dec 13, 2022 at 23:23

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