Is there any example of usage for a vector space over the field of formal Laurent series?

Question

The formal Laurent series over a field is a field.

Is there any example where vector spaces over that field occur naturally?

Answer 1

If $R$ is a ring, $M$ an $R$-module, and $f : R \to S$ a ring homomorphism, then we naturally get an induced $S$-module $S \otimes_R M$. This construction is important, among other places, in algebraic geometry. So whenever we have a ring homomorphism $f : R \to k((t))$ from a ring into a formal Laurent series ring, we have a way to turn $R$-modules into $k((t))$-modules. A natural example is when $R$ is the ring of functions on some variety over $k$ and $f$ comes from taking germs of functions at some point on the variety, then adjoining an inverse to $t$. The resulting construction on modules will give you slightly more information than taking the stalk of $M$, regarded as a sheaf, at the point; roughly speaking you are "restricting $M$ to a formal neighborhood" of the point.