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It is a "well known" fact that

locally free sheaves over a Stein space $X$ are projective as $\mathcal{O}_X$-modules

(see e.g. just after Lemma 1.6 in O'Brian-Toledo-Tong's "The trace map and characteristic classes for coherent sheaves").

As mentioned in this comment, it seems like this is proven in Satz 6.2 in Forster's "Zur Theorie der Steinschen Algebren und Moduln" (and the linked comment gives a sketch proof), but my German is essentially non-existent, and I'm hesitant to cite a paper that I can't fully understand (though I have no doubt of its veracity). Not only that, but the proof is really spread out over previous lemmas as well, including the definition of "Stein modules".

Is there a proof of this fact elsewhere in the literature? I had expected to find it in Grauert and Remmert's "Theory of Stein Spaces" but couldn't see it.

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    $\begingroup$ By Grauert--Remmert, Theory of Stein Spaces, V 2. Theorem 3, p. 152, Stein is equivalent to having exact global sections function on coherent sheaves. This should give you what you want for finite rank. Do you need the result for all projective modules? $\endgroup$
    – Ben
    Commented Dec 6 at 19:10
  • $\begingroup$ what do you mean "the result for all projective modules"? are you saying that the converse is true as well?? $\endgroup$
    – Tim
    Commented Dec 8 at 17:50
  • $\begingroup$ Sorry, I meant infinite rank. $\endgroup$
    – Ben
    Commented Dec 9 at 11:04
  • $\begingroup$ oh no, I'm happy with finite rank! $\endgroup$
    – Tim
    Commented Dec 9 at 14:54

2 Answers 2

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I suppose you were given essentially a reference in the comment. Alternatively, to expand a little on the proof of this fact that I mention in the answer above the linked comment, to me this argument seems simple enough that you could sketch it without a reference:

Indeed, if $F$ is a vector bundle and $\mathcal{G}$ is any coherent sheaf on the Stein space $X$, then $\mathcal{E}xt^k(F,\mathcal{G})=0$ for $k>0$ (since $F$ is a locally free resolution of itself). Furthermore, $H^k(\mathcal{H}om(F,\mathcal{G}))=0$ for $k>0$ by Cartan's theorem B (as $\mathcal{H}om(F,\mathcal{G})$ is a coherent sheaf). By the local to global spectral sequence of Ext, since $H^p(\mathcal{E}xt^q(F,\mathcal{G})) \implies \textrm{Ext}^{p+q}(F,\mathcal{G})$, it follows that $\textrm{Ext}^k(F,\mathcal{G}) = 0$ for $k>0$, i.e., $\textrm{Hom}(F,\bullet)$ is exact, so $F$ is a projective object.

(I implicitly assume in your question that the category consists of coherent $\mathcal{O}_X$-modules, as that seems to be assumed in all the mentioned sources.)

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  • $\begingroup$ yes, this would all be inside the category of coherent modules — thanks for the details! $\endgroup$
    – Tim
    Commented Dec 11 at 14:02
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Corollary 2.6.5 in Forstnerič "Stein Manifolds and Holomorphic Mappings". (2nd Edition)

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