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 $pA_p$-...
2
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2answers
106 views
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 ...
3
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0answers
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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}...
5
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2answers
337 views
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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0answers
132 views
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 $\...
6
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1answer
162 views
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 ...
8
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1answer
268 views
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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1answer
468 views
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 $n$...
1
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1answer
171 views
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 ...
3
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161 views
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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1answer
170 views
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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0answers
541 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 ...
5
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1answer
199 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] \...
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1answer
213 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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2answers
159 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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2answers
441 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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4answers
2k 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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2answers
127 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 following?...
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779 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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1answer
198 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 $(t^{2}+\epsilon)$?
2
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1answer
599 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 ...
4
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1answer
394 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 $\underline{...
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0answers
348 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 F_p}...
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3answers
427 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
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1answer
542 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, $Y=\mathrm{Proj}(A[IT])...
2
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3answers
409 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?
6
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1answer
2k 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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4answers
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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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2answers
465 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 $\{[x_1,\cdots,x_s]^...
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2answers
1k 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
