-1

When people write $\mathbb C((t))=\mathbb C[[t]][t^{-1}]$, what exactly do they mean?

Do they mean that the field of formal Laurent series $\mathbb C((t))$ is a polynomial ring in the variable $t^{-1}$ with values in the ring $\mathbb C[[t]]$ of formal power series or do they simply denote the fact that $\mathbb C((t))$ is localized at the multiplicative set generated by $t$? Every element in $\mathbb C((t))$ can be represented as a polynomial in $t^{-1}$ with coefficients in $\mathbb C[[t]]$, so it seems to make sense that we view $\mathbb C((t))$ as a polynomial ring with values in $\mathbb C[[t]]$. Does anyone disagree with such a statement?

Flavius Aetius
  • 914
  • 5
  • 13
  • 1
    Maybe $A_f \cong A[X]/(fX - 1) \cong A[f^{-1}]$? – xbh Oct 28 '22 at 13:16
  • Isn't it just the fraction field? For example, see the comment with $\Bbb Q[[x]][1/x]=\Bbb Q((x))$, – Dietrich Burde Oct 28 '22 at 13:16
  • As for the fraction field, we shall invert anything nonzero. For those with nonzero constant terms, they are already invertible, so we shall invert all those formal power series that is a multiple of $t$. But all such ones decompose as $t^nu$ where $u$ is a unit in $\Bbb C[[t]]$, so inverting all ${t^j}_{j=1}^\infty$ is equivalent to taking the fraction field for $\Bbb C[[t]]$. – xbh Oct 28 '22 at 13:25
  • @Dietrich Burde, Of course, it is the fraction field. The question is: Can we view the fraction field as a polynomial ring in $t^{-1}$ with coefficients in $\mathbb C[[t]]$? – Flavius Aetius Oct 28 '22 at 13:31
  • @xbh, What is $X$ and what does the notation $A[f^{-1}]$ mean for you? Polynomial ring in $f^{-1}$ with values in $A$? – Flavius Aetius Oct 28 '22 at 13:33
  • This is no longer called "polynomial ring", but rather the ring of Laurent polynomials. – Dietrich Burde Oct 28 '22 at 13:38
  • @FlaviusAetius $X$ an indeterminate, and $A[f^{-1}]$ the ring generated by $A$ and $f^{-1}$. – xbh Oct 28 '22 at 14:02
  • However, surely, you agree that the notation $A[f^{-1}]$ in the first comment looks like a polynomial ring in $t^{-1}$! – Flavius Aetius Oct 28 '22 at 14:29
  • @xbh, But every element in $A[f^{-1}]$ can be written as a polynomial in $t^{-1}$ with values in $A$, correct? Even if $A_f$ is not a polynomial ring, its elements can be expressed as polynomials. – Flavius Aetius Oct 28 '22 at 14:31

1 Answers1

1

It means the localization. The polynomial ring would be strictly larger, and in particular would not impose the relation $t \cdot t^{-1} = 1$.

Qiaochu Yuan
  • 419,620