The completion tag has no usage guidance.

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### Completeness of Localizations of Completions of Commutative Rings

Let $R$ be an integral domain. Let $x,y\in R\setminus\{0\}$ be distinct. Let $\hat R$ be the $x$-adic completion of $R$ (the ring of all sequences $(r_n+Rx^n)_{n\ge0}$ where for $n\ge0$, $r_n\in R$ ...

**7**

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**0**answers

341 views

### Lemma in Scholze-Weinstein

In the paper "Moduli of $p$-divisible groups" by Scholze and Weinstein (see http://math.bu.edu/people/jsweinst/Moduli/Moduli.pdf), one finds the following claim in Lemma 5.2.7:
Lemma: Let $K$ be a ...

**4**

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**1**answer

123 views

### Enriched Cauchy completions and underlying categories

The ordinary Cauchy completion $\overline{C}$ of a small category $C$ satisfies a number of conditions: Every idempotent in $\overline{C}$ splits, there's an equivalence of categories $[C^{op}, Set] ...

**0**

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**1**answer

140 views

### Is there a complete local analogue of the Artin-Tate lemma?

The Artin-Tate lemma states that if $A \subseteq B \subseteq C$ are commutative rings where $A$ and $C$ are Noetherian, $C$ is finitely generated as an $A$-algebra, and $C$ is finitely generated as a ...

**5**

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134 views

### Lie Algebras over DVRs and basechange to the completion

Let $R$ be a discrete valuation ring containing an algebraically closed field $K$ of characteristic zero and let $L$ be a Lie algebra over $R$ whose underlying $R$-module is finitely generated and ...

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226 views

### How is a MacNeille completion “universal” like a beta-compactification is “universal”?

The beta-compactification of a topological space is characterized as the largest space such that every mapping from the original space to another (range) space can be extended through to a mapping ...

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**4**answers

1k views

### Completion of a local ring of a curve

Let $X$ be a smooth projective irreducible curve defined over an algebraically closed field $\mathbb{K}$ (of arbitrary characteristic), and let $p\in X$ be a closed point. Denote by $\mathcal{O}_p(X)$ ...

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114 views

### Completion of the set of subsets with half volume.

Let $X$ be a measure space with finite $|X|=\int_X1$ and $f:X\rightarrow \mathbb{R}$ be a function. Under what condition on $X$ and $f$ does there exist a subset $Y \subset X$ satisfying the ...

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389 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 ...

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**1**answer

181 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 $(t^{2}+\epsilon)$?

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**1**answer

322 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 ...

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**1**answer

329 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 ...

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**0**answers

335 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 ...

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**3**answers

381 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 ...

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**1**answer

412 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, ...

**2**

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**3**answers

330 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?
...

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1k 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 ...

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977 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 ...

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421 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 ...

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923 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