# Questions tagged [completion]

The completion tag has no usage guidance.

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### Completion of localization of completion

Let $(A,m)$ be a noetherian local ring,
and let $p \subseteq A$ be a prime ideal.
From this data, we can construct two rings:
1. We may localize $A$ at $p$, and then complete,
obtaining the $pA_p$-...

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### Is a filtered colimit of complete module complete?

This is probably a textbook question but i haven't been able to find a reference. Let $R$ be a complete commutative Noetherian local ring and $I$ its unique maximal ideal (I'm mostly interested in the ...

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### completion of finitely generated module over non-Noetherian ring

Let $A$ be a commutative ring with unity and fix $f \in A$. Any $A$-module $M$ has its $f$-adic completion, the $\hat{A}$-module $\hat{M} = \underset{n}{\lim} M/f^nM$. There is a canonical map $\hat{A}...

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### Why is $K_{\upsilon}|K$ separable for a global field $K$?

I asked this question on math.stackexchange but maybe it fits here better. If not, I apologize in advance and will remove the question.
Let $K$ be a global field and $\upsilon$ a prime of $K$. Then ...

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### Can we recover the completion of a local ring $R$ if its associated graded is the coordinate ring of a Veronese variety?

Suppose $R$ is a localization of a normal closed point of a variety of dimension $n$ over an algebraically closed field $k$ with maximal ideal $\mathfrak{m}$. Suppose also that the associated graded $\...

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### Is the completion of a CAT(0) open ball a closed ball?

It is well-known that the completion of a metric space which is homeomorphic to a ball can be very wild; in fact, I think, every compact manifold is the closure of an open ball!
But CAT(0) spaces are ...

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### Doesn't completion of a representation ring preserve its indecomposables?

For $G = PSU(3)$, it is known that $\dim I(G;\mathbb Q) / I(G;\mathbb Q)^2 = 3$, while $H^{**}(BG;\mathbb Q)$ is obviously a power series ring in two indeterminates since $G$ has rank 2. This would ...

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### Can an intersection of ideals in a Noetherian ring be replaced by a countable intersection?

Let $(R,\frak m)$ be a Noetherian local ring, and let $X$ be a set of ideals in $R$. Assume $\bigcap_{I \in X} I = 0$. Is there some sequence $\{I_n\}_{n \in \mathbb N}$, with $I_n \in X$ for all $n$...

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### link between completion of the universal enveloping algebra and an endomorphism of functor

My question could be resume in the following way :
Let $\mathfrak{t} \to \mathrm{End}(V)$ a representation of an abelian Lie algebra into an infinite dimensional vector space.
What can we say ...

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### Is the completion of an injective local homomorphism still injective

Let $R$ be a regular local ring (say of dimension 2) with quotient field $F$. Let $A$ be a discrete valuation ring with the same quotient field such that $R\subset A\subset F$, $A\neq F$. Let $\hat{R}$...

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### Example of R-bad space

I have been looking around for examples of $R$-bad spaced in the sense of Bousfield and Kan. In their book "Homotopy limits, completions and localizations] they give several examples of such spaces ...

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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 $n\ge0$, $r_n\in R$ ...

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### 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 ...

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### 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] \...

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### 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 $...

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### 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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### 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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### 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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### 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 following?...

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### 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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### 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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### 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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### 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 $\underline{...

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### 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 F_p}...

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### 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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### 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, $Y=\mathrm{Proj}(A[IT])...

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### 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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### 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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### 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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### 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 $\{[x_1,\cdots,x_s]^...

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### 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