Difference between a stalk of a sheaf and a fiber of a vector bundle

Is there an analogy between fibers $\pi^{-1} ( x )$ of a vector bundle $\pi : E \to X$, and the stalk $\mathcal{F}_x$ of a sheaf $\mathcal{F}$ défined by : $\mathcal{F}_x = \displaystyle \lim \mathcal{F} ( U )$ : the direct limit over all open subsets of $X$ containing the given point $x$ ? Thanks a lot.

• Please consider accepting some answers to your previous questions. People will be less willing to respond if they think you won't appreciate their answers anyway. Dec 19, 2012 at 12:58
• I don't know how to do it, can you tell me how to do it please ?. Il don't speak and i don't undertand well english, i'm a moroccan men, sorry. Dec 19, 2012 at 13:04
• No need to be sorry. Next to each answer, under the arrows for up/downvoting, there is a little check mark. You can just click on the check mark belonging to the answer you like best to accept it. See here for some images explaining the process. Dec 19, 2012 at 13:08
• Ok, i did it. Thank you. :) Dec 19, 2012 at 13:23

The stalk of a sheaf $\mathscr{F}$ at a point $x$ is naturally isomorphic the fibre of the espace étalé over $x$. However, the espace étalé is in general a very strange space and is very rarely a vector bundle.
For example, let $\mathscr{T}_M$ be the sheaf of sections of the tangent bundle $T M \to M$ of a manifold (or smooth variety, if you prefer). The fibre of $T M \to M$ over a point $x$ of $M$ is automatically a vector space (by definition!) but the stalk of $\mathscr{T}_M$ at $x$ is in general only a module over the local ring $\mathscr{O}_x$, which need not be a field. More explicitly, the stalk $(\mathscr{T}_M)_x$ consists of germs of vector fields at $x$, while the fibre $T_x M$ consists of tangent vectors at $x$.
Addendum. In the algebraic context, we can get something akin to the fibre by taking the stalk $\mathscr{F}_x$ and tensoring it with the residue field $\kappa (x)$. As far as I know this is not something that can be defined as a direct limit. The fibre of a bundle is a limit in the general sense of category theory – it is just the pullback of the bundle along the inclusion of a point – but I don't think it's useful to think of it that way.
• Thank you very much @Zhen Lin. Can you tell me please, if it's true that the fiber $E_x = \pi^{-1} ( x )$ of a vector bundle $\pi : E \to X$ can be defined as the direct limit of maps $U \to E_{U} = \pi^{-1} ( U )$ like the stalk of a sheaf $\mathcal{F}_x$ ? In this case, what is the direct system which define this direct limit ? Thanks a lot. Dec 19, 2012 at 13:16