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Let $X$ be a ringed space. A quasi-coherent module on $X$ is a module which has locally a presentation, i.e. locally on $X$, it is the cokernel of a map between free modules. If $X$ is a scheme, then the category of quasi-coherent modules on $X$ or its derived category are quite important to understand $X$. In the stacks project (modules) you can find some theorems about quasi-coherent modules on arbitrary ringed spaces. Now I wonder if this is just abstract nonsense:

Question: Are there nontrivial, interesting examples of quasi-coherent modules on ringed spaces, which are no schemes? Or do they play no role outside algebraic geometry?

Of course, locally free modules of finite rank correspond to vector bundles and are important in other geometries as well (for example, manifolds when regarded as locally ringed spaces). Are there other quasi-coherent modules of interest?

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Quasi-coherent modules and algebras correspond to the "affine" objects over the structure sheaf, in a way that ends up being totally formal. That is, the quasi-coherent modules carry the data of the "relative commutative algebra" of the space. If you take a look at Monique Hakim's thesis, she gives everything in full generality (wrt relative schemes in arbitrary ringed topoi). If you have trouble finding a copy, shoot me an e-mail. –  Harry Gindi Nov 9 '10 at 11:57
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Perhaps you know but this already, but coherent sheaves originated in complex analysis and are still very important there. I suppose that if you applied the "an" functor to a quasicoherent sheaf on a complex scheme, you would get a quasicoherent sheaf on the analytic space. I'm not aware that people have actually considered this, however. –  Donu Arapura Nov 9 '10 at 12:25
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This notion arises in C-analytic and rigid-analytic geometry; e.g., useful theory of Proj with graded sheaves of algebras (coherent in each degree). For basic generalities see section 2 (esp. the serious defect noted in Remark 2.1.5, due to Gabber) in my paper "Relative ampleness in rigid-analytic geometry", and see section 3 for nontrivial applications (esp. Ex. 3.2.6 and Thm. 3.2.7, resting on Def. 3.2.2). Since Hakim's thesis is apparently limited to the "totally formal" aspects of the story, I doubt it incorporates Gabber's example or the work needed for section 3. –  BCnrd Nov 9 '10 at 14:10
    
I guess the problem is that, on schemes, quasi-coherent sheaves are the completion for direct limits of the category of coherent sheaves. But this may not be the case in a different context. This fails, for instance, in the case of noetherian formal schemes, see the paper on duality by Jeremías, Lipman and myself. –  Leo Alonso Nov 9 '10 at 15:17
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In D-module theory, one considers most of the time modules which are coherent over the sheaf of differential operators. –  Jan Weidner Mar 24 '11 at 15:06

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