Non-quasi-coherent sheaves of $\mathcal O_X$ modules on a scheme seem like a wild concept to me; are they actually used for something?
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14$\begingroup$ Canonical flasque resolutions, infinite direct products, extension by zero from a locally closed set (see the discussion of excision early in SGA2), sheaf-Hom (and sheaf-Ext) between quasi-coherent sheaves, topological pullbacks of sheaves (even q-coh. ones) along scheme morphisms,... $\endgroup$– BCnrdNov 2, 2010 at 15:08
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$\begingroup$ If $X$ is a scheme defined over a base $S$, and $G$ is a group scheme over $S$, then we get a sheaf on $X$ induced by $G$ (namely the sheaf of $S$ morphisms from $X$ to $G$). This is not in general quasi-coherent. The sheaf induced by $G_m$ in particular occurs a lot in nature, for example $H^1(X, G_m) = Pic(X)$. $\endgroup$– Daniel LoughranNov 2, 2010 at 15:17
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1$\begingroup$ Dear Daniel: that's not a sheaf of $O_X$-modules in most cases (e.g., not for $\mathbf{G}_m$). $\endgroup$– BCnrdNov 2, 2010 at 15:34
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1$\begingroup$ General module sheaves appear as soon as you want to consider schemes as a full subcategory of ringed spaces. And this happens, of course, very often, for example when some constructions leave the category of schemes. $\endgroup$– Martin BrandenburgNov 2, 2010 at 15:49
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$\begingroup$ @BCnrd: Woops sorry I misread the question. Thanks for pointing that out! $\endgroup$– Daniel LoughranNov 2, 2010 at 16:14
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One can think the adeles on a curve (or higher adeles on other spaces) as a sheaf of $\mathcal O$-algebras. That is, consider the sheaf $B(U)=\prod_{x\in U}\mathcal O_x$, where $\mathcal O_x$ is the completion of $\mathcal O$. Then the sheaf $A=B\otimes K$, where $K$ is the sheaf of rational functions, has the adeles as global sections. There is a short exact sequence $\mathcal O\to K\times B\to A$. One can tensor a quasicoherent sheaf with this to obtain a resolution to compute cohomology. Indeed, Weil introduced the adeles (after the earlier ideles) specifically to prove Riemann-Roch. I'm not sure when this was reinterpreted in terms of sheaves, which were only introduced later.
