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.

Solution 1:

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.