Completion of DVR

Question

Let $R$ be a DVR with a uniformizing parameter $t$, that is, $(R,(t))$ is a local ring. Let $\hat{R}$ to be the $(t)$-adic completion of $R$. Then it seems that every element $\alpha\in\hat{R}$ can be written in $$\alpha=\sum_{i=0}^\infty a_i t^i=a_0+a_1t+a_2t^2+\cdots\quad(1)$$ where $a_i\not\in(t)$ so that $a_i$'s are unit. $t$ just works as it is an indeterminate w.r.t $R^\times$. Indeed, if $t$ satisfies an algebraic equation $c_nt^n+\cdots+c_1t=c_0$ for some $c_i\in R^\times$, then by comparing the valuation of each sides, it can be shown that all $c_i=0$. So this is my question: is it really possible to write the elements of $R^\times$ as in (1)?

Answer 1

Indeed, elements of $\hat{R}$ may be written as power series in $t$ with coefficients in some set of representatives of $R/t.$ By definition, $$\hat{R} := \varprojlim(\dots\to R/(t^i)\to R/(t^{i-1})\to\dots\to R/(t^2)\to R/t),$$ and we may represent elements of this inverse limit concretely as compatible sequences $$(a_1, a_2, a_3,\dots)\in\prod_{i = 1}^\infty R/(t^i).$$ Concretely, this means that if $i > j,$ $a_i - a_j = a_{i,j}t^j$ (or $a_i = a_j + a_{i,j} t^j$) for some $a_{i,j}\in R.$ Inductively, our sequence looks like $$(a_1, a_1 + a_{1,2}t, a_1 + a_{1,2}t+a_{2,3}t^2, \dots).$$ So the data of our sequence $(a_1,a_2,a_3,\dots)$ is equivalent to the data of the sequence $(a_1, a_{1,2}, a_{2,3},\dots),$ which gives you the translation that lets you write elements of $\hat{R}$ as power series in $t.$ That you can take each $a_{i,i+1}$ to be a unit in $R$ or $0$ follows because each nonzero element $a\in R$ can be written uniquely as $u t^i$ for some $u\in R^\times,$ $i\in\Bbb Z_{\geq 0}.$ (See here or here for more details.)

Now, $\hat{R}$ is also a DVR with uniformizer $t,$ and so units are precisely elements of $\hat{R}$ which are not in the maximal ideal $t\hat{R}$ - when written as power series, these are precisely the power series with nonzero leading coefficient.

Some good references include Serre's "Local Fields," although I couldn't find this proposition. Neukirch's "Algebraic Number Theory" (in particular, chapter 2) is another good source, and has a detailed proof of the above in chapter II, section 4 (proposition 4.4).