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It is a well-known result in algebraic geometry that if $X$ is an affine scheme and $\mathcal{F}$ is a quasi-coherent sheaf on $X$, then the higher cohomologies of $\mathcal{F}$ vanish, i.e. $$ H^i(X,\mathcal{F})=0, ~\forall~ i\geq 1. $$

In complex geometry we have a similar result of of Henri Cartan which claims that if $X$ is a Stein manifold and $\mathcal{F}$ is a coherent sheaf on $X$, then we have $$ H^i(X,\mathcal{F})=0, ~\forall~ i\geq 1. $$ See this nlab item: http://ncatlab.org/nlab/show/Stein+manifold#Forstneric11

$\textbf{My question}$ is if we consider the more general case that $\mathcal{F}$ is a quasi-coherent sheaf on a Stein manifold $X$, do we still have $$ H^i(X,\mathcal{F})=0, ~\forall~ i\geq 1? $$

Here by quasi-coherent I mean $\mathcal{F}$ locally can be written as the cokernel of $$ \bigoplus_I\mathcal{O}_X|_U\rightarrow \bigoplus_J\mathcal{O}_X|_U. $$ (I'm not sure whether there is other version of quasi-coherent sheaf in complex analytic context.)

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    $\begingroup$ a common trick to deal with these issues is to use the fact that any big object (ie quasi-coherent sheaf) is a filtered colimit of small objects (ie coherent sheaves). Since it vanishes for coherent sheaves it must vanish for all quasi-coherent sheaves as cohomology commutes with filtered colimits. (Problem: I know that in the algebraic world any qcoh is colimit of coh; I have no idea whether it holds in the analytic world...) $\endgroup$
    – bananastack
    Commented Mar 28, 2015 at 4:28
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    $\begingroup$ @Zhaoting Wei: In section 2.1 of eudml.org/doc/10167 (especially Lemmas 2.1.8 and 2.1.9) one finds the equivalence among several possible definitions of quasi-coherence in the analytic setting (written there for the rigid-analytic case but carrying over to the complex-analytic case without difficulty), such as locally being a direct limit of coherent sheaves, or of coherent subsheaves, or your proposed definition. They all fail your question, due to Gabber's example on the open unit disc in Example 2.1.6 of that link. $\endgroup$
    – user74230
    Commented Mar 29, 2015 at 8:52
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    $\begingroup$ @ZhaotingWei: Since cohomology commutes with direct limits on compact Hausdorff spaces (proved in Godement's book, for example), for your question in the compact case it would suffice to know that a quasi-coherent sheaf in some reasonable sense is globally a direct limit of coherent sheaves when the analytic space is compact. But I am doubtful that this could be true. $\endgroup$
    – user74230
    Commented Mar 29, 2015 at 19:26
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    $\begingroup$ @bananastack: Note also Example 2.1.10 in that link, which builds on Gabber's example to show that these various equivalent notions of quasi-coherence in the analytic setting aren't preserved under direct limits! (That might sound paradoxical, but it actually isn't.) So it is not really a "good" notion (though useful for some purposes nonetheless). $\endgroup$
    – user74230
    Commented Mar 29, 2015 at 19:27
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    $\begingroup$ @Zhaoting Wei ,There is a notion of quasi-coherent analytic Frechet sheaf due to Ramis and Ruget for which vanishing holds. $\endgroup$
    – Mohan Ramachandran
    Commented Mar 30, 2015 at 17:19

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