It seems to me that the power series ring $\mathbb{C}[[t]]$ is isomorphic to the localization of $\mathbb{C}[t]$ at the maximal ideal $(t)$, but I am not sure.
9
-
$\begingroup$ Does $C[[t]]$ denote the ring of Laurent series? $\endgroup$– stressed outFeb 17, 2019 at 15:09
-
$\begingroup$ @stressedout I mean $\mathbb{C}[[t]]$ is the formal power series ring in one variable. $\endgroup$– RonFeb 17, 2019 at 15:10
-
$\begingroup$ No. It is its completion for the $t$-adic topology. $\endgroup$– BernardFeb 17, 2019 at 15:13
-
$\begingroup$ @Bernard In that case, will $\mathbb{C}[[t]]$ be finitely generated over $\mathbb{C}[t]_{(t)}$, the localization of $\mathbb{C][t]$ at $(t)$? $\endgroup$– RonFeb 17, 2019 at 15:15
-
$\begingroup$ @stressedout: The ring of (formal) Laurent series is the field of fractions of $\mathbf C[[t]]$. $\endgroup$– BernardFeb 17, 2019 at 15:15
|
Show 4 more comments
1 Answer
$\begingroup$
$\endgroup$
4
This is not true. The ring $\mathbb{C}[t]_{(t)}$ is naturally a subring of $\mathbb{C}[[t]]$, yet this inclusion is proper. For example one can solve $\sqrt{1-t}$ over $\mathbb{C}[[t]]$ and there is no solution over $\mathbb{C}[t]_{(t)}$.
-
$\begingroup$ Is it then correct to say that $\mathbb{C}[t]_{(t)}$ is isomorphic to $\mathbb{C}[t]$ itself? or is it bigger? $\endgroup$ Feb 17, 2019 at 15:29
-
$\begingroup$ $\mathbb{C}[t]_{(t)}$ certainly is bigger, for example it contains $\frac{1}{t-1}\in \mathbb{C}[t]_{(t)}$ $\endgroup$– ЖекаFeb 17, 2019 at 15:42
-
$\begingroup$ Sorry for asking dummy questions, I'm just interested in math. What does determine if a formal power series is in $\mathbb{C}[t]_{(t)}$ or not? $\endgroup$ Feb 17, 2019 at 15:44
-
$\begingroup$ @user881391 I am sorry but, $1/(t-1)$ is also an element of $\mathbb{C}[[t]]$ (take the power series expansion). Moreover, by the universal property of inverse limit, there is a natural morphism from $\mathbb{C}[t]_{(t)}$ to $\mathbb{C}[[t]]$. $\endgroup$– RonFeb 17, 2019 at 16:29