TITLE: Modules over Laurent series rings
QUESTION [9 upvotes]: Let $k[x]$ be the ring of polynomials over a field k in one variable x.  A $k[x]$-module is a k-vector space together with a linear endomorphism (the action of x).
The field $k(x)$ of rational functions is the maximal localization of $k[x]$, i.e. field of fractions.  (Edit: yes, I was an idiot in what I wrote here first; thanks James.)
The ring $k[[x]]$ of formal power series is the completion of $k[x]$ at the ideal $(x)$.  It is natural to consider only $k[[x]]$-modules which are likewise complete, meaning roughly that we can sum "infinite linear combinations" whose coefficients are increasing powers of x.  Completeness of a $k[x]$-module automatically makes it a $k[[x]]$-module.
Finally, both $k(x)$ and $k[[x]]$ embed into the ring $k((x))$ of formal Laurent series.  I have two questions, which I ask together because they seem related:

Is there a general ring-theoretic construction, akin to localization and completion, which produces $k((x))$ from $k[x]$?
Is there a natural condition to impose on $k((x))$-modules, akin to completeness for $k[[x]]$-modules, which would enable us to sum infinite linear combinations with coefficients increasing in powers of x?

REPLY [2 votes]: Sorry, the first version of this answer was broken in a few ways.
For your first question, it seems that there is more than one construction that specializes to what you want.  For example, you can take the completion $\hat{X}$ of a variety $X$ along a closed subvariety $Z$, and then take the tensor product $\mathscr{O}_{\hat{X}} \otimes_{\mathscr{O}_X} K_X$, where $K_X$ is the function field.  Alternatively, if you have an effective divisor $D$ in a variety $X$, you can take the scheme whose underlying topological space is $D$, and whose sheaf of rings is given by $U \cap D \mapsto \varinjlim_m \varprojlim_n \Gamma(U, \mathscr{O}_X(mD)/\mathscr{O}_X(-nD))$.  These two constructions are identical when we are presented with a codimension one subvariety, such as a point in a line.  I do not know a succinct name for either construction.
For your second question, you can ask for the modules to have a $k$-linear topology, i.e., there is a basis of neighborhoods of zero formed by $k$-submodules.  We can then demand that the action of $k((x))$ is continuous, where $k((x))$ is given the "usual" topology, with $k$ discrete and $ \{ x^n k[[t]] \}_{n \in \mathbb{Z} } $ forms a basis of neighborhoods of zero.  A continuous $k((x))$-module is the same as a topological $k$-module equipped with a continuous invertible topologically nilpotent endomorphism.