# notation for formal Laurent series

I've found a few articles that write the ring of formal Laurent series in $t$ as $R((1/t))$, but what's the underlying meaning of $\cdot ((\cdot))$?

A mathematician of my acquaintance swears that $R((t))$, not $R((1/t))$, should be used to denote the ring of formal Laurent series in $t$. We can't decide who's right without knowing what $\cdot((\cdot))$ means. (We both agree that $R[[t]]$ denotes the ring of formal power series in $t$ with coefficients in $R$.)

• $(())$ should be the fraction field of $[[]]$, just as $()$ is of $[]$. – Mariano Suárez-Álvarez Jul 24 '12 at 22:45
• ... so $R((1/t))$ denotes the field of formal power series of the form $\sum_{k=-\infty}^ r c_k t^k$ (possibly infinitely many terms of negative degree, but only finitely many terms of positive degree) – Pietro Majer Jul 24 '12 at 22:54
• I've taken a closer look at the articles I mentioned in the original post, and it appears that they all are looking at formal sums in which the exponent of t is bounded from above, rather than from below. So it seems that my acquaintance was quite right about $R((1/t))$. Is it fair to say, then, that the definition of $R((x_1,x_2,…))$ is the quotient ring of $R[[x_1,x_2,…]]$? – James Propp Jul 25 '12 at 0:25
• @James: What you write in your last comment is true if $R$ is a field. Note though that taking fraction fields of formal power series rings gets tricky: e.g. $K[[t_1,t_2]] = K[[t_1]][[t_2]]$, but $K((t_1,t_2)) \subsetneq K((t_1))((t_2))$. – Pete L. Clark Jul 25 '12 at 0:57
• Just a (related) question: for $k$ a field, $k[[t]]$ is $\hat{\mathcal{O}}_{\mathbb{A}^1_k,0}$. $k((t))$ is the quotient field of $\hat{\mathcal{O}}_{\mathbb{A}^1_k,0}$. Is $k((t))$ also the completion of the stalk at the generic point of $\mathbb{A}^1_k$? Which of the definitions of "formal Laurent series" $k((x_1,\ldots,x_n))$ in the answer by @eithil has an analogous geometric interpretation in terms of $\mathbb{A}^n_k$? – Qfwfq Sep 8 '18 at 17:32

## 3 Answers

I agree with the mathematician of your acquaintance -- well, okay, I am the mathematician of your acquaintance.

Here are some references for the notation $K((x))$ for the field of formal Laurent series $\sum_{n \geq n_0} a_n x^n$ over $K$:

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Jacobson's Basic Algebra II, $\S 9.12$.

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Lam's Introduction to Quadratic Forms Over Fields, $\S VI.1$

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Neukirch's Algebraic Number Theory, $\S II.4$.

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Serre's Corps Locaux, $\S 1.1$.

I've seen $k((x,y))$ used to mean two different things: the field of fractions of $k[[x,y]]$ and also

$k((x,y)) = \{\sum_{i \geq n} \sum_{j \geq m} f_{i,j}x^iy^j : f_{i,j} \in k; n,m \in \mathbb{Z}\}$

These rings are distinct, e.g. the second one does not contain $\sum_{i \geq 0}y^{-(i+1)}x^i = (y-x)^{-1}$.

In one variable the distinction disappears, and then the only difference between $R((t))$ and $R((1/t))$ is whether the powers of $t$ go to $\infty$ or $-\infty$.

The ring of formal Laurent series is a particular case, for the group order $$\ldots 1/x^{n+1} < 1/x^{n} < \ldots 1/x^2 <1/x < 1 < x < x^2 < \ldots x^n < x^{n+1} < \ldots$$ of Malcev-Neumann series (series with well-ordered support on an ordered group $\Gamma$). These series are denoted $k((\Gamma))$. So I agree with the mathematician of your acquaintance because this is in accordance with the double bracketing'' and also with the the order. Indeed $k((x))$ is well-suited for the classical Laurent series whereas $k((1/x))$ tends to indicate $1/x>1$ and then the series with finite number of positive powers and (possibly) infinite number of negative ones (see Pietro Majer's comment).