It is well known that for an affine scheme $X$, every finitely generated locally free sheaf $\mathcal{E}$ is projective in the category Qcoh$(X)$. i.e. the functor $\text{Hom}_{\text{Qcoh}(X)}(\mathcal{E},\cdot)$ is exact. Of course this is no longer true if $X$ is a general, non-affine scheme.

Now we go beyond schemes: Let $(X,\mathcal{O}_X)$ be a ringed space. We still have a definition of quasi-coherent sheaf: A sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ is called quasi-coherent if for every point $x\in X$ there exists an open neighbourhood $x\in U\subset X$ such that $\mathcal{F}|_U$ is isomorphic to the cokernel of a map $$ \bigoplus_{j\in J} \mathcal{O}_U\rightarrow \bigoplus_{i\in I} \mathcal{O}_U. $$ See http://stacks.math.columbia.edu/download/modules.pdf Section 10.

My question is: do we have a condition on the ringed space $(X,\mathcal{O}_X)$ to guarantee it has the similar property as an affine scheme, i.e. every finitely generated locally free sheaf $\mathcal{E}$ is projective in the category Qcoh$(X)$?

  • $\begingroup$ Clearly $\mathcal F \to \Gamma(X, \mathcal F)$ is an exact functor from quasi coherent sheaves to $\Gamma(X, \mathcal O_X)$-modules. There is an adjoint functor, also exact. I believe they are inverses if and only if there are no nonzero quasicoherent sheaves with no global sections. One might guess that, in this case, your ringed space is related in some way to the affine scheme $\operatorname{Spec} \Gamma(X, \mathcal O_X)$. On the other hand you might not be able to say enough about ringed spaces in this generality. $\endgroup$ – Will Sawin Mar 17 '15 at 21:09
  • $\begingroup$ Smooth manifolds. $\endgroup$ – Fernando Muro Mar 17 '15 at 23:44
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    $\begingroup$ @FernandoMuro Yes smooth manifolds with the sheaf of smooth functions gives an affirmative example. However I'm looking for a more general criterion. For example, complex manifold in general does not satisfy my requirement. Only Stein manifolds work here. So I wonder what is the general condition here. I believe it's some cohomological requirement but I'm not sure. $\endgroup$ – Zhaoting Wei Mar 18 '15 at 0:12
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    $\begingroup$ @ZhaotingWei If we restrict to ringed spaces in which the structure sheaf is compact and every quasi-coherent is filtered colimit of coherent (might be true in this generality anyway) then I think what you're looking for are precisely those whose structure sheaf is projective in $QCoh(X)$. Compact ensures that the global sections commute with filtered colimits and then projective is equivalent the property: - For all coherent $\mathcal{O}_X$-modules $\mathcal{E}$ exactness of $\mathcal{Hom}_{\mathcal{O}_X}(\mathcal{E},-)$ implies exactness of $Hom_{\mathcal{O}_X}(E,-)$. $\endgroup$ – Saal Hardali May 14 '17 at 21:51

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