# What do I need to know to understand the completion of the field of rational functions of a non-singular projective curve?

So the title gives the jist of my question. Specifically, let $X$ be a non-singular projective curve, $P$ a point on $X$, $v_P$ the discrete valuation associated to the ring $\mathcal{O}_P$. Then I have read that the completion of $k(X)$ with respect to the valuation $v_P$ is isomorphic to the field of formal Laurent series over $k$.

Stuff that might be relevant? I know some basic Galois theory, some very basic point set topology, and I'm just starting chapter 10 in An Introduction to Commutative Algebra by Atiyah and MacDonald.

I was hoping someone could tell me the material I will have to read to understand this along with good books that cover it. If there is an algebraic way of going about this I would prefer it as I'm really enjoying An Introduction to Commutative Algebra. Also if someone wanted to give me an overview of what is happening here that would be appreciated also.

Thanks for any help!

-

You need some geometric facts because the statement is false for singular curves. You also need to assume $P$ is a rational point of $X$ (automatically true if the base field $k$ is algebraically closed).

This being said, the fact that $X$ is a non-singular curve implies that the maximal ideal of the local ring $O_P$ is generated by one element $t$. The hypothesis $P$ is rational means that $O_P/tO_P=k$. So $$O_P=k+tO_P=k+t(k+tO_P)=...=k[t]+t^nO_P$$ for all $n\ge 1$. Passing to the limit (for the $t$-adic topology), we see that the completion of $O_P$ is $k[[t]]$. As $k(X)=\mathrm{Frac}(O_P)$, we get the desired result.

-

Ratinoal points: Let $k$ be a perfect field, fix an algebraic closure $\overline{k}$ of $k$, and let $X$ be a variety over $k$. Then is a standard way of obtaining "$P$ is rational $\Leftrightarrow\mathcal{O}_P/t\mathcal{O}_P=k$" to use the result that "the orbit of $P$ under the action of $G=Gal(\overline{k}/k)$ is equal to the residue extension of $\mathcal{O}_P/\mathfrak{m}_P$ over $k$"?

(Do you know of any books that cover this sort of stuff in an elementary manner? In particular, books that prove the above result I stated. I read the result in "Algebraic Geometry in Coding Theory and Cryptography" for finite fields and am guessing (hoping) it generalises to all perfect fields.)

$t$-adic topology: this is the bit that's really limiting my understanding at the moment. Would it suffice to read chapter 10 of An Introduction to Commutative Algebra to get an understanding of this (or is that more than necessary?!). For instance, it seems from your argument that completion commutes with localisation buts this is something I have no idea of.

Thanks again for any help.

-
Dear M Davolo, First, as a procedural matter, comments like the above should be left as comments to the answer you are addressing, not as separate answers. Second, completion does not commute with localization in general. However, it is true that completing at a maximal ideal gives a local ring, so that, when completing at a local ring, it is no loss of generality to localize at that maximal ideal first; QiL is using this in his answer. Regards, – Matt E Aug 26 '12 at 18:38
Hi, sorry for posting in the answers section. My post was too big. In future should I break it down into 2 (or more) comments? – M Davolo Aug 26 '12 at 19:17
@MDavolo: take an affine open neighborhood $V$ (embedded in some affine space ${\mathbb A}^n_k$) of $P$ and write the coordinates of $P$ in ${\mathbb A}^n_k$. Let $R$ be the ring of regular functions on $V$. Then the residue field $O_P/m_p$ is equal to $R/m$ where $m$ is the maximal ideal corresponding to the point $P$. A direct computation shows that the residue field is generated by the coordinates of $P$. So $P$ is rational (i.e. coordinates in $k$) iff $O_P/m_P=k$. In our case $m_P=tO_P$ by definition of $t$. – user18119 Aug 26 '12 at 21:07
Dear @MattE, thanks for clarifying the passage to localization. – user18119 Aug 26 '12 at 21:09
@MDavolo, yes A-M chapter 10 covers the necessary material about the completion. – user18119 Aug 26 '12 at 21:26