Let $\mathbb{C}[[x,y]]$ be the ring of the formal series with coefficients in $\mathbb{C}$. I have to find $\operatorname{Spec}(\mathbb{C}[[x,y]])$.

I think that it is a local ring because it should represents germs of holomorphic functions.

If then I consider a point $f \in \mathbb{C}[[x,y]]$ what can I say about the Spec of localization: $\operatorname{Spec}(\mathbb{C}[[x,y]]_f)$?

  • 2
    $\begingroup$ See mathoverflow.net/questions/24082 $\endgroup$ – Martin Brandenburg Sep 21 '13 at 10:05
  • $\begingroup$ @MartinBrandenburg Is it true that $Spec(\mathbb{C}[[x]])=\{(0),(x)\}$? How can I prove it? $\endgroup$ – ArthurStuart Sep 21 '13 at 10:40
  • 1
    $\begingroup$ You can check directly that the ideals of $k[[x]]$ are given by $(0)$ and $(x^n)$ for $n \in \mathbb{N}$. Hint: $f \in k[[x]]^*$ is invertible iff $f(0) \in k^*$. The prime ideals are of course just $(0)$ and $(x)$. $\endgroup$ – Martin Brandenburg Sep 21 '13 at 10:50
  • $\begingroup$ @MartinBrandenburg I read that is very difficult to descrive the specrum of formal series. But if I localize them in a point is it easier? $\endgroup$ – ArthurStuart Sep 21 '13 at 10:55

$\mathbb{C}[[x,y]]$ is a regular local ring of dimension $2$. As such, it is a UFD, so the prime ideals are $0$, $(f)$ for $f$ an irreducible element of $\mathbb{C}[[x,y]]$, and the maximal ideal $(x,y)$. Note that being irreducible as a power series is substantially stronger than being irreducible as a polynomial: e.g. $y^2 - x^2 - x^3$ is an irreducible polynomial in $\mathbb{C}[x,y]$ (by Eisenstein's criterion), but is not irreducible as a power series (cf. Hartshorne I.5.6.3).

If $0 \ne f \in \mathbb{C}[[x,y]]$, factor $f$ into irreducibles $f = f_1^{e_1}...f_k^{e_k}$. Then $\text{Spec}(\mathbb{C}[[x,y]]_f)$ can be identified with $\text{Spec}(\mathbb{C}[[x,y]]) \setminus \{(x,y), (f_1), \ldots, (f_k) \}$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.