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Analogue of the theorem on formal functions for henselization

Let $\pi:X\to S$ be a proper morphism of Noetherian schemes. For any integer $p\geq 0$ and any point $s\in S$ the theorem on formal functions gives us a description of the completion of $(R^p \pi_*\...
3
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1answer
342 views

Varieties with everywhere good reduction that are isomorphic over every completion have isomorphic generic fibers

Let $R$ be the ring of integers in a number field. Let $X$ and $Y$ be smooth and proper schemes over $R$. For a maximal ideal $\mathfrak{m}\subset R$ denote the completion of the localization at $\...
4
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0answers
170 views

Henselization and completions of local rings & schemes

That's the second part of my coarse becoming acquainted with Henselizations of fields and local rings. (in this question we focus on local rings as it is more algebro geometric motivated). So let $(R,...
2
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1answer
107 views

Completion and extension by scalars

Let $R\subset S$ be commutative rings, $I\trianglelefteq R$ an ideal and $M$ be an $R$-module. Suppose that 1) $R$ is Noetherian and $I$-adically complete. 2) $M$ is a finite $R$-module (hence $M$ ...
0
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107 views

Is the completion of an infinitely generated module, again infinitely generated

Let $A$ be a local noetherian ring with maximal ideal $m$. Let $M$ be an infinitely generated $A$-module and $\hat{M}$ be the $m$-adic completion of $M$. Denote by $\hat{A}$ the $m$-adic completion of ...
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0answers
67 views

Completion of infinite projective space

I would like to ask for a reference regarding the completion of infinite-dimensional Projective Space (both Real and Complex). Since in the infinite-dimensional projective space you can take sequences ...
1
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0answers
109 views

Riemann–Hurwitz Formula for Normal Projective Curves

My question refers to the proof of Theorem 7.4.16 from Liu's "Algebraic Geometry" at page 290 : QUESTION: Why does $e'_x= length_{\hat{B}}(W_{\hat{B}/\hat{A}}/\hat{B})$ hold? During the proof we ...
3
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0answers
200 views

Induced morphism of completions of local rings

Let $g: A \to B$ be a local ring morphism between local Noetherian (commutative) rings $A,B$ (so $g(m_A) \subset m_B$ for the unique maximal ideals of the corresponding rings). Assume that the induced ...
4
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0answers
168 views

A question related to bousfield localization and nilpotent completion

I am reading Bousfield's paper entitled "The localization of spectra with respect to homology" (MSN). In that paper, Corollary 6.13 states that, if a ring spectrum $E$ has countable homotopy and ...
5
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1answer
258 views

Is completion of isolated singularity isolated?

Let $K$ be an algebraically closed field and let $A=K[x_1,\dots,x_n]/I$ be a $K$-algebra of finite type which has only an isolated singularity at the origin. Let $\mathfrak{m}=(x_1,\dots,x_n)$ and ...
4
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1answer
149 views

Mapping cone and derived tensor product

This question is in some sense a continuation to this question: Derived Nakayama for complete modules For the setting: Let $A$ be a ring and let $I$ be some finitely generated ideal in $A$. Let $f\...
5
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1answer
334 views

Derived Nakayama for complete modules

I have encountered the following "Nakayama Lemma" recently: Let $A$ be a ring and $I$ some finitely generated ideal. Let $\mathcal C_\bullet$ be a chain complex of $I$-(derived) complete $A$-...
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75 views

Reference request : $I$-adic smoothness

The following result has been know for a while now: Let $f:\mathfrak X \rightarrow \mathfrak Y$ be an adic morphism of (locally) Noetherian formal schemes and $K\subset \mathcal O_{\mathfrak Y}$ and ...
1
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0answers
95 views

Completion in the non-noetherian case

Let $A$ be a non-noetherian, commutative $\mathbb{C}$-algebra and $X, Y$ be noetherian affine $\mathbb{C}$-schemes. Denote by $X_A:=X \times_{\mathbb{C}} \mbox{Spec}(A)$ and $Y_A:=Y \times_{\mathbb{C}}...
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0answers
181 views

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
169 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 ...
4
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0answers
185 views

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}...
6
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2answers
492 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 ...
5
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0answers
169 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
178 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
278 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 ...
13
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1answer
619 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
230 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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0answers
183 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 $\hat{R}$...
3
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0answers
86 views

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 ...
4
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1answer
189 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 $n\ge0$, $r_n\in R$ ...
8
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0answers
618 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
217 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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1answer
240 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
182 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 ...
1
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2answers
570 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 ...
16
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4answers
3k 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 $\mathcal{O}_p(X)$ ...
2
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2answers
130 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?...
4
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0answers
886 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 $\...
2
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1answer
200 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
674 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
424 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{...
3
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0answers
353 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}...
4
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3answers
443 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
610 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
450 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
3k 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 ...
11
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4answers
1k 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
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2answers
507 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]^...
12
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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