Complete local ring which is not complete valuation ring

Question

Complete valuation ring is obviously complete local, because valuation ring is always local ring.

I would like to know the example of

Complete local ring which is not complete valuation ring

Any examples? I could find example of local ring which is not valuation ring, but the example ring is not complete. Thank you for your help.

Answer 1

• $k[x]/(x^2)$ is a complete local ring, it is not a valuation ring because it is not an integral domain.

• $R=k[[x^2,x^3]]$ is a local ring, it is complete because $R\to \underset{n\to \infty}\varprojlim R/\mathfrak{m}^n$ is an isomorphism, it is not a valuation ring because $x,x^{-1}\in Frac(R),\not \in R$.

  Equivalently its principal ideals are not totally ordered: $(x^3)\not\subset (x^2)$, $(x^2)\not\subset (x^3)$.

• The completion at the maximal ideal doesn't have to be the same as the completion for the valuation: $S= \{ \sum_{j\ge 1} c_j x^{b_j},c_j\in k, b_j\in \Bbb{Q}_{\ge 0}, \lim_{j\to \infty} b_j=\infty\}$, then $\mathfrak{m}^n = \mathfrak{m}$ so $\underset{n\to \infty}\varprojlim S/\mathfrak{m}^n=S/\mathfrak{m}\cong k$ while $S$ is complete for the valuation (every Cauchy sequence converges in $S$).