Nonquasicoherent sheaves of $\mathcal O_X$ modules on a scheme seem like a wild concept to me; are they actually used for something?

13$\begingroup$ Canonical flasque resolutions, infinite direct products, extension by zero from a locally closed set (see the discussion of excision early in SGA2), sheafHom (and sheafExt) between quasicoherent sheaves, topological pullbacks of sheaves (even qcoh. ones) along scheme morphisms,... $\endgroup$– BCnrdNov 2 '10 at 15:08

$\begingroup$ 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 quasicoherent. 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 '10 at 15:17

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 '10 at 15:34

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 '10 at 15:49

$\begingroup$ @BCnrd: Woops sorry I misread the question. Thanks for pointing that out! $\endgroup$– Daniel LoughranNov 2 '10 at 16:14
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 RiemannRoch. I'm not sure when this was reinterpreted in terms of sheaves, which were only introduced later.