All Questions
Tagged with local-rings formal-completions
9 questions
2
votes
1
answer
97
views
Is $\mathbb C[[x, y]]/(x^3+y^4)$ an integral domain?
The polynomial $x^3+y^4$ is irreducible in $\mathbb C[x, y]$, thus $\mathbb C[x, y]/(x^3+y^4)$ is an integral domain. My question is: Is $\mathbb C[[x, y]]/(x^3+y^4)$ also an integral domain?
Further ...
- 1,682
3
votes
0
answers
162
views
Completion of primary ideal is primary
I have the following question.
Suppose $(R, \mathfrak{m})$ is a local Noetherian ring, $\mathfrak{a}$ is $\mathfrak{p}$-primary ideal. Is it true that $\hat{\mathfrak{a}}$ is a primary ideal in $\hat{...
- 105
3
votes
1
answer
109
views
Finitely generated torsion-free indecomposable modules over one-dimensional complete local domains are isomorphic to ideal?
Let $R$ be a complete local domain of dimension $1$. Let $M$ be a finitely generated torsion-free indecomposable $R$-module. Then, must $M$ be isomorphic to an ideal of $R$?
Also clearly, $R$ embeds ...
- 1,682
0
votes
1
answer
82
views
If a Noetherian local ring contains a field, then so does its $\mathfrak m$-adic completion? [closed]
Let $(R, \mathfrak m)$ be a Noetherian local ring containing a field. Then, does the $\mathfrak m$-adic completion $\widehat R$ of $R$ also contain a field?
- 435
2
votes
1
answer
122
views
Is $\widehat{IM}=\widehat I \widehat M$ for any finitely generated module $M$, where $\widehat {(-)}$ denotes completion w.r.t. maximal ideal?
Let $(R,\mathfrak m)$ be a Noetherian local ring, and let $(\widehat R, \widehat{\mathfrak m})$ be its $\mathfrak m$-adic completion. Let $M$ be a finitely generated $R$-module.
Then, is it true that $...
- 3,715
-1
votes
1
answer
216
views
Finitely generated module whose completion has finite length
Let $M$ be a finitely generated module over a Noetherian local ring $(R,\mathfrak m).$
If $n\ge 1$ is an integer such that $\widehat {\mathfrak m^n}\widehat M=0,$ then is it true that $\mathfrak m^n M=...
- 1,682
3
votes
1
answer
371
views
Completion of a polynomial ring over a complete ring
I'm learning about ring completions, and this question came to mind: If $R$ is a complete local ring with maximal ideal $\mathfrak{m}$ (e.g. $R = \mathbb{Z}_p$ or $R = k[[x]]$), is the completion of $...
- 33
1
vote
0
answers
126
views
Completion of finite local homomorphism.
Suppose that $(A,\mathfrak{m}_A)$ and $(B,\mathfrak{m}_B)$ are local rings and that $\varphi:A\rightarrow B$ is a local homomorphism (I am happy to assume that this morphism is quasifinite).
Let $M$ ...
- 671
1
vote
1
answer
411
views
Commutative Algebra "mess": recover the complete local rings of the normalization
Premise and main idea: I'm not an expert in the field of commutative algebra and when I encounter problems regarding local rings I try to solve them by following a sort of geometric intuition.
It was ...
- 13.7k