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4
votes
0answers
154 views

Completion of abelian topological groups

During some exercises I came upon the following questions that I cannot answer myself. Given a topological group $G$ we can build the completion using cauchy sequences and denote this completion by ...
1
vote
1answer
160 views

elementary question on a completion of a ring

Let $k$ a field, and $k[\epsilon]=k[X]/(X^{2})$ , what is the completion of the ring $k[\epsilon][t]$ with respect to the ideal $(t^{2}+\epsilon)$?
1
vote
1answer
209 views

Dimension of formal fiber

The question comes from my attempt to understand the following question. height of contracted prime ideals in power series rings $\bullet$ My original question: Let $(R,m)$ be a Noetherian local ring ...
3
votes
1answer
224 views

Artin approximation theorems over non-regular rings/non-Noetherian rings

In Artin1968 he considers $\underline{analytic}$ equations, but over the ring $R=k\{x_1,..,x_n\}$. In Artin1969 he works with $R=k\{x_1,..,x_n\}/I$, not necessarily regular, but considers ...
3
votes
0answers
317 views

Finite extensions of $\mathbb Q_p$

Is there any finite extension of $\mathbb Q_p$ which is not the completion of a finite extension of $\mathbb Q$ at some place over $p$ ? Analogously in equicaracteristic, if $k=\overline {\mathbb ...
3
votes
3answers
346 views

Does every Lindelof uniform space have a Lindelof completion?

Recall Lindelof = every open cover has a countable subcover. Is the question of the title answered by known material from some standard text on uniform spaces? Note it is well known to be true for ...
5
votes
1answer
361 views

Completion of local rings in the exceptional divisor of a blow-up

Let $X=\mathrm{Spec}(A)$ be an affine variety, $Z\subseteq X$ a closed, reduced subscheme. Let $$\beta:Y=\mathrm{Bl}_Z(X)\to X$$ be the blow-up of $X$ in $Z$. In other words, ...
1
vote
3answers
295 views

Metrizability of $\mathfrak{a}$-adic topology

Let $A$ be a ring, $\mathfrak{a}\subset A$ an ideal. Then is the $\mathfrak{a}$-adic topology on $A$ necessarily a metric space? I can see that it is true when $A$ is a DVR, but is it true in general? ...
1
vote
1answer
734 views

Does completion commute with localization?

Suppose $A$ is a Noetherian (not necessarily local) ring and $\mathfrak{m}\subset A$ a maximal ideal. Then is it true that $$\hat{A}_{\hat{\mathfrak{m}}}=\widehat{A _{\mathfrak{m}}},$$ where hats ...
9
votes
4answers
851 views

What kind of completion is this?

Let $X$ be a compact Hausdorff space, and $C(X)$ the unital commutative C*-algebra of continuous functions on it. The double Banach dual $C(X)^{**}$ is a commutative von Neumann algebra and hence has ...
2
votes
2answers
379 views

Comparing lower central series and augmentation ideal completions

Let G be a group. Let $G^p$ be the completion of G with respect to the mod p lower central series of G.i.e. $G^p=\varprojlim_{q} G/\gamma_qG$, where $\gamma_qG$ is generated by all ...
12
votes
2answers
846 views

Is completeness of a field an algebraic property?

Pretty straitforward: If a field has a metric in which it is complete can it have a metric in which it is not complete? By metric I mean field norm of course